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This invaluable book is renowned for its fully worked examples and numerous applications. Concepts are presented either graphically, analytically, or numerically (sometimes using more than one approach) depending on which seems the most appropriate to make the material clear and easy to understand. Technology is also fully integrated into problem sets and examples, providing a real-world perspective. And the authors encourage readers to begin the problem solving process by estimating the solution and checking their work by examining their answers for "reasonableness".
About the Author
Paul Calter is Professor Emeritus of Mathematics at Vermont Technical College and Visiting Scholar at Dartmouth College. A graduate of The Cooper Union, New York, he received his M.S. from Columbia University and a MFA from Norwich University. Professor Calter has taught technical mathematics for over twenty-five years. He is a member of the American Mathematical Association of Two Year Colleges, the Mathematical Association of America, the National Council of Teachers of Mathematics, the College Art Association, and the Author's Guild.
Calter is involved in the Mathematics Across the Curriculum movement, and has developed and taught a course called Geometry in Art and Architecture at Dartmouth College under an NSF grant.
Professor Calter is the author of several other mathematics textbooks, among which are the Schaum's Outline of Technical Mathematics, Problem Solving with Computers, Practical Math Handbook for the Building Trades, Practical Math for Electricity and Electronics, Mathematics for Computer Technology, Introductory Algebra and Trigonometry, Technical Calculus, and Squaring the Circle: Geometry in Art and Architecture.
Michael Calter is Associate Professor at Wesleyan University. He received his B.S. from the University of Vermont. After receiving his Ph.D. from Harvard University, he completed a post-doctoral fellowship at the University of California at Irvine. Michael has been working on his father's mathematics texts since 1983, when he completed a set of programs to accompany Technical Mathematics with Calculus. Since that time, he has become progressively more involved with his father's writing endeavors, culminating with becoming co-author on the second edition of Technical Calculus and the fourth edition of Technical Mathematics with Calculus. Michael also enjoys the applications of mathematical techniques to chemical and physical problems as part of his academic research. Michael is a member of the American Mathematical Association of Two Year Colleges, the American Association for the Advancement of Science, and the American Chemical Society.
Michael and Paul enjoy hiking and camping trips together. These have included an expedition up Mt. Washington in January, a hike across Vermont, a walk across England on Hadrian's Wall, and many sketching trips into the mountains. | 677.169 | 1 |
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CHAPTER ZERO PrefaceThis is a free and open source differential calculus book. The "free and opensource" part means you, as a student, can give digital versions of this book to any-one you want (for free). It means that if you are a teacher, you can (a) give or printor xerox copies for your students, (b) use potions for your own class notes (if theyare published then you might need to add some acknowledgement, depending onwhich parts you copied), and you can xerox even very large portions of it to yourhearts content. The "differential calculus" part means it covers derivatives andapplications but not integrals. It is heavily based on the first half of a classic text,Granville's "Elements of the Differential and Integral Calculus," quite possibly abook your great grandfather might have used when he was college age. Some ma-terial from Sean Mauch's excellent public domain text on Applied Mathematics, also included. Calculus has been around for several hundred years and the teaching of it has notchanged radically. Of course, like any topic which is taught in school, there aresome modifications, but not major ones in this case. If x(t) denotes the distance atrain has traveled in a straight line at time t then the derivative is the velocity. Ifq(t) denotes the charge on a capacitor at time t in a simple electrical circuit thenthe derivative is the current. If C(t) denotes the concentration of a solvent in achemical mixture at time t then the derivative is the reaction rate. If P (t) denotesthe population size of a country at time t then the derivative is the growth rate.If C(x) denotes the cost to manufacture x units of a production item (such as abroom, say) then the derivative is the marginal cost. ix
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Some of these topics, electrical circuits for example, were not studied in calcu-lus when Granville's book was first written. However, aside from some changesin grammar and terminology (which have been updated in this version), the math-ematical content of the calculus course taught today is basically the same as thattaught a hundred years ago. Terminology has changed, and no one talks about"versines" any more (they were used in navigation tables before the advent ofcomputers), but the basic techniques have not. Therefore, to make the book moreuseful to current students, some modification and rearrangement of the materialin Granville's old text is appropriate. Overall, though the rigor and detailed expla-nations are still at their same high level of quality. Here is a quote form Granville's original preface: The author has tried to write a textbook that is thoroughly modern and teachable, and the capacity and needs of the student pursuing a first course in the Calculus have been kept constantly in mind. The book contains more material than is necessary for the usual course of one hundred lessons given in our colleges and engineering schools; but this gives teachers an opportunity to choose such subjects as best suit the needs of their classes. It is believed that the volume contains all topics from which a selection naturally would be made in prepar- ing students either for elementary work in applied science or for more advanced work in pure mathematics. WILLIAM A. GRANVILLE GETTYSBURG COLLEGE Gettysburg, Pa. For further information on William Granville, please see the Wikipedia article at has a short biography and links for further information. Granville's book "Elements of the Differential and Integral Calculus" fell intothe public domain (in the United States - other countries may be different) andthen much of it (but not all, at the time of this writing) was scanned into by P. J. Hall. This wikisource document uses MathML and LTEX and Asome Greek letter fonts. x
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In keeping with the "free and open source" aspect of this textbook, and thetheme of updating to today's much more technologically-aware students, a freeand open sourse mathematical software package Sage was used to illustrate ex-amples throughout. You don't need to know Sage to read the book (just ignorethe Sage examples if you want) but it certainly won't hurt to learn a little aboutit. Besides, you might find that with some practice Sage is fun to "play with"and helps you with homework or other mathematical problems in some of yourother classes. It is a general purpose mathematical software program and it mayvery likely be the only mathematical software you will ever need. This LTEX'd version is due to the second-named author, who is responsible for Aformatting, the correction of any typos in the scanned version, significant revisionfor readability, and some extra material (for example, the Sage examples andgraphics). In particular, the existence of this document owes itself primarily tothree great open source projects: TEX/LTEX, Wikipedia, and Sage . All the fig- Aures were created using Sage and then edited and converted using the excellentopen source image manipulation program GIMP ( Sage code for each image can be found in the LTEX source code, available Aat information on Sage can be found at the Sage website (located at or in the Appendix (Chapter 13) below. Though the original text of Granville is public domain, the extra material addedin this version is licensed under the GNU Free Documentation License (repro-duced in an Appendix below), as is Wikipedia. Acknowledgements: I thank the following readers for careful proofreading andreporting typos: Mario Pernici, Jacob Hicks, Georg Muntingh, and Minh VanNguyen. I also thank Trevor Lipscombe for excellent stylistic advice on the pre-sentation of the book. However, any remaining errors are solely my responsibility.Please send comments, suggestions, proposed changes, or corrections by email towdjoyner@gmail.com. xi
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CHAPTER ONE Variables and functions1.1 Variables and constantsA variable is a quantity to which an unlimited number of values can be assigned.Variables are denoted by the later letters of the alphabet. Thus, in the equation ofa straight line, x y + =1 a bx and y may be considered as the variable coordinates of a point moving along theline. A quantity whose value remains unchanged is called a constant.√Numerical or absolute constants retain the same values in all problems, as 2, 5, 7, π, etc. Arbitrary constants, or parameters, are constants to which any one of an unlim-ited set of numerical values may be assigned, and they are supposed to have theseassigned values throughout the investigation. They are usually denoted by the ear-lier letters of the alphabet. Thus, for every pair of values arbitrarily assigned to aand b, the equation x y + =1 a brepresents some particular straight line. 1
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1.2. INTERVAL OF A VARIABLE1.2 Interval of a variableVery often we confine ourselves to a portion only of the number system. Forexample, we may restrict our variable so that it shall take on only such values aslie between a and b, where a and b may be included, or either or both excluded.We shall employ the symbol [a, b], a being less than b, to represent the numbersa, b, and all the numbers between them, unless otherwise stated. This symbol[a, b] is read the interval from a to b.1.3 Continuous variationA variable x is said to vary continuously through an interval [a, b], when x startswith the value a and increases until it takes on the value b in such a manner as toassume the value of every number between a and b in the order of their magni-tudes. This may be illustrated geometrically as follows: Figure 1.1: Interval from A to B.The origin being at O, layoff on the straight line the points A and B correspondingto the numbers a and b. Also let the point P correspond to a particular value of thevariable x. Evidently the interval [a, b] is represented by the segment AB. Nowas x varies continuously from a to b inclusive, i.e. through the interval [a, b], thepoint P generates the segment AB.1.4 FunctionsA function f of the real numbers R is a well-defined rule which associated to eachx ∈ R a unique value f (x). Usually functions are described algebraically usingsome formula (such as f (x) = x2 , for all real numbers x) but it doesn't have to beso simple. For example, 2
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1.5. NOTATION OF FUNCTIONS x2 , if x is an integer, f (x) = 0, otherwise,is a function on R but it is given by a relatively complicated rule. Namely, the rulef tells you to associate to a number x the value 0 unless x is an integer, in whichcase you are to associate the value x2 . (In particular, f (x) is always an integer, nomatter what x is.) This type of rule defining a function of x is sometimes calleda piecewise-defined function. In this book, we shall usually focus on functionsgiven by simpler symbolic expressions. However, be aware that piecewise-definedfunctions do arise naturally in applications. For example, in electronics, when a6 volt battery-powered flashlight is powered on or off using a switch, the voltageto the lightbulb is modeled by a piecewise-defined function which has the value 0when the device is off and 6 when it is switched on. When two variables are so related that the value of the first variable depends onthe value of the second variable, then the first variable is said to be a function ofthe second variable. Nearly all scientific problems deal with quantities and relations of this sort, andin the experiences of everyday life we are continually meeting conditions illus-trating the dependence of one quantity on another. For instance, the weight a manis able to lift depends on his strength, other things being equal. Similarly, the dis-tance a boy can run may be considered as depending on the time. Or, we may saythat the area of a square is a function of the length of a side, and the volume of asphere is a function of its diameter.1.5 Notation of functionsThe symbol f (x) is used to denote a function of x, and is read "f of x". In orderto distinguish between different functions, the prefixed letter is changed, as F (x),φ(x), f ′ (x), etc. During any investigation the same functional symbol always indicates the samelaw of dependence of the function upon the variable. In the simpler cases thislaw takes the form of a series of analytical operations upon that variable. Hence,in such a case, the same functional symbol will indicate the same operations orseries of operations, even though applied to different quantities. Thus, if f (x) = x2 − 9x + 14, 3
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1.6. INDEPENDENT AND DEPENDENT VARIABLES1sage: f = lambda x,y: xˆ2+yˆ2sage: f(3,4)25sage: R.<x> = PolynomialRing(CC,"x")sage: f = xˆ2+2sage: f.roots()[(1.41421356237309*I, 1), (2.77555756156289e-17 - 1.41421356237309*I, 1)]1.6 Independent and dependent variablesThe second variable, to which values may be assigned at pleasure within limitsdepending on the particular problem, is called the independent variable, or argu-ment; and the first variable, whose value is determined as soon as the value of theindependent variable is fixed, is called the dependent variable, or function. Though we shall wait to introduce differentiation later, please keep in mind thatyou differentiate the dependent variable with respect to the independent variable.Example 1.6.1. In the equation of an upper half-circle of radius 1, √ y= 1 − x2 ,we typically call x the independent variable and y the dependent variable. Frequently, when we are considering two related variables, it is in our powerto fix upon whichever we please as the independent variable; but having oncemade the choice, no change of independent variable is allowed without certainprecautions and transformations. One quantity (the dependent variable) may be a function of two or more otherquantities (the independent variables, or arguments). For example, the cost ofcloth is a function of both the quality and quantity; the area of a triangle is afunction of the base and altitude; the volume of a rectangular parallelepiped is afunction of its three dimensions. In the Sage example below, t is the independent variable and f is the dependentvariable. Sagesage: t = var('t')sage: f = function('f', t)sage: f = cos 5
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1.7. THE DOMAIN OF A FUNCTIONsage: f(pi/2)0sage: (f(-3*pi)-2*f(1))ˆ2(-2*cos(1) - 1)ˆ21.7 The domain of a functionThe values of the independent variable for which a function f (x) is defined isoften referred to as the domain of the function, denoted domain(f ). Consider the functions x2 − 2x + 5, sin x, arctan xof the independent variable x. Denoting the dependent variable in each case by y,we may write y = x2 − 2x + 5, y = sin x, y = arctan x.In each case y (the value of the function) is known, or, as we say, defined, for allvalues of x. We write in this case, domain(f ) = R. This is not by any meanstrue of all functions, as the following examples illustrating the more commonexceptions will show. a y= (1.1) x−bHere the value of y (i.e. the function) is defined for all values of x except x = b.When x = b the divisor becomes zero and the value of y cannot be computed from(1.1). We write in this case, domain(y) = R − {b}. √ y= x. (1.2)In this case the function is defined only for positive values of x. Negative valuesof x give imaginary values for y, and these must be excluded here, where we areconfining ourselves to real numbers only. We write in this case, domain(y) ={x ∈ R | x ≥ 0}. y = loga x. a>0 (1.3) 6
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CHAPTER TWO Theory of limitsIn this book, a variable denotes a quantity which takes values in the real numbers.2.1 Limit of a variableIf a variable v takes on successively a series of values that approach nearer andnearer to a constant value L in such a manner that |v − L| becomes and remainsless than any assigned arbitrarily small positive quantity, then v is said to approachthe limit L, or to converge to the limit L. Symbolically this is written limv=L , ormore commonly lim . v→L The following familiar examples illustrate what is meant: 1. As the number of sides of a regular inscribed polygon is indefinitely in- creased, the limit of the area of the polygon is the area of the circle. In this case the variable is always less than its limit. 2. Similarly, the limit of the area of the circumscribed polygon is also the area of the circle, but now the variable is always greater than its limit. 3. Hold a penny exactly 1 meter above the ground and observe its motion as you release it. First it travels 1/2 the distance from the ground (at this stage its distance fallen is 1/2 meter), then it travels 1/2 that distance from the 9
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2.1. LIMIT OF A VARIABLE Hence, by definition of the limit of a variable, it is seen that both S2n and S2n+1 are variables approaching 2 as a limit as the number of terms in- 3 creases without limit. Summing up the first two, three, four, etc., terms of (2.1), the sums are found by ((2.2) and ((2.3) to be alternately less and greater than 2 , illustrating 3 the case when the variable, in this case the sum of the terms of ((2.1), is alternately less and greater than its limit. Sage sage: S = lambda n: add([(-1)ˆi*2ˆ(-i) for i in range(n)]) sage: RR(S(1)); RR(S(2)); RR(S(5)); RR(S(10)); RR(S(20)) 1.00000000000000 0.500000000000000 0.687500000000000 0.666015625000000 0.666666030883789 You can see from the Sage example that the limit does indeed seem to approach 2/3. In the examples shown the variable never reaches its limit. This is not by anymeans always the case, for from the definition of the limit of a variable it is clearthat the essence of the definition is simply that the absolute value of the differencebetween the variable and its limit shall ultimately become and remain less thanany positive number we may choose, however small.Example 2.1.1. As an example illustrating the fact that the variable may reachits limit, consider the following. Let a series of regular polygons be inscribed ina circle, the number of sides increasing indefinitely. Choosing anyone of these,construct. the circumscribed polygon whose sides touch the circle at the verticesof the inscribed polygon. Let pn and Pn be the perimeters of the inscribed andcircumscribed polygons of n sides, and C the circumference of the circle, andsuppose the values of a variable x to be as follows: Pn , pn+1 , C, Pn+1 , pn+2 , C, Pn+2 , etc.Then, evidently, lim x = C x→∞and the limit is reached by the variable, every third value of the variable being C. 11
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2.2. DIVISION BY ZERO EXCLUDED2.2 Division by zero excluded00 is indeterminate. For the quotient of two numbers is that number which multi-plied by the divisor will give the dividend. But any number whatever multipliedby zero gives zero, and the quotient is indeterminate; that is, any number whatevermay be considered as the quotient, a result which is of no value. a 0 has no meaning, a being different from zero, for there exists no number suchthat if it be multiplied by zero, the product will equal a. Therefore division by zero is not an admissible operation. Care should be taken not to divide by zero inadvertently. The following fallacyis an illustration. Assume that a = b.Then evidently ab = a2 .Subtracting b2 , ab − b2 = a2 − b2 .Factoring, b(a − b) = (a + b)(a − b).Dividing by a − b, b = a + b.But a = b, therefore b = 2b, or, 1 = 2. The result is absurd, and is caused by thefact that we divided by a − b = 0, which is illegal.2.3 InfinitesimalsDefinition 2.3.1. A variable v whose limit is zero is called an infinitesimal1 . This is written lim, or, lim , v=0 v→0and means that the successive absolute values of v ultimately become and remainless than any positive number however small. Such a variable is said to become"arbitrarily small." 1 Hence a constant, no matter how small it may be, is not an infinitesimal. 12
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2.4. THE CONCEPT OF INFINITY (∞) If lim v = l, then lim(v − l) = 0; that is, the difference between a variable andits limit is an infinitesimal. Conversely, if the difference between a variable and a constant is an infinitesi-mal, then the variable approaches the constant as a limit.2.4 The concept of infinity (∞)If a variable v ultimately becomes and remains greater than any assigned positivenumber, however large, we say v is "unbounded and positive " (or "increaseswithout limit"), and write lim , or, lim , or, v → +∞. v=+∞ v→+∞If a variable v ultimately becomes and remains smaller than any assigned negativenumber, we say "unbounded and negative " (or "v decreases without limit"), andwrite lim , or, lim , or, v → −∞. v=−∞ v→−∞If a variable v ultimately becomes and remains in absolute value greater than anyassigned positive number, however large, we say v, in absolute value, "increaseswithout limit", or v becomes arbitrarily large2 , and write lim , or, lim , or, v → ∞. v=∞ v→∞Infinity (∞) is not a number; it simply serves to characterize a particular mode ofvariation of a variable by virtue of which it becomes arbitrarily large. Here is a Sage example illustrating limt=∞ 1/t = limt=−∞ 1/t = 0. Sagesage: t = var('t')sage: limit(1/t, t = Infinity) 2 On account of the notation used and for the sake of uniformity, the expression v → +∞ issometimes read "v approaches the limit plus infinity". Similarly, v → −∞ is read "v approachesthe limit minus infinity", and v → ∞ is read "v, in absolute value, approaches the limit infinity".While the above notation is convenient to use in this connection, the student must not forget thatinfinity is not a limit in the sense in which we defined it in §2.2, for infinity is not a number at all. 13
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2.5. LIMITING VALUE OF A FUNCTION0sage: limit(1/t, t = -Infinity)02.5 Limiting value of a functionGiven a function f (x). If the independent variable x takes on any series of valuessuch that lim x = a,and at the same time the dependent variable f (x) takes on a series of correspond-ing values such that lim f (x) = A,then as a single statement this is written lim f (x) = A. x→a Here is an example of a limit using Sage : Sagesage: limit((xˆ2+1)/(2+x+3*xˆ2),x=infinity)1/3 x2 +1This tells us that limx→∞ 2+x+3∗x2 = 1. 32.6 Continuous and discontinuous functionsA function f (x) is said to be continuous for x = a if the limiting value of thefunction when x approaches the limit a in any manner is the value assigned to thefunction for x = a. In symbols, if lim f (x) = f (a), x→a 14
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2.6. CONTINUOUS AND DISCONTINUOUS FUNCTIONSthen f (x) is continuous for x = a. Roughly speaking, a function y = f (x) iscontinuous if you can draw its graph by hand without lifting your pencil off thepaper. In other words, the graph of a continuous function can have no "breaks."Example 2.6.1. The piecewise constant function 1, x ≥ 0, u(x) = 0, x < 0,is not continuous since its graph has a "break" at x = 0 where it "steps up" from0 to 1. This function models "on-off" switches in electrical engineering and iscalled the unit step function or the Heaviside function (after the brilliant engineerOliver Heaviside, 1850 1925). The function is said to be discontinuous for x = a if this condition is not satis-fied. For example, if lim f (x) = ∞, x→athe function is discontinuous for x = a. Sagesage: t = var('t')sage: P1 = plot(1/t, (t, -5, -0.1))sage: P2 = plot(1/t, (t, 0.1, 5))sage: show(P1+P2, aspect_ratio=1)sage: limit(1/t,t=0,dir="plus")+Infinitysage: limit(1/t,t=0,dir="minus")-InfinityThe graph in Figure 2.1 suggests that limx→0+ 1/x = +∞ and limx→0− 1/x =−∞, as the above Sage computation confirms. The attention of the student is now called to the following cases which occurfrequently.CASE I. As an example illustrating a simple case of a function continuous for aparticular value of the variable, consider the function x2 − 4 f (x) = . x−2 15
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2.6. CONTINUOUS AND DISCONTINUOUS FUNCTIONSCASE II. The definition of a continuous function assumes that the function isalready defined for x = a. If this is not the case, however, it is sometimes possibleto assign such a value to the function for x = a that the condition of continuityshall be satisfied. The following theorem covers these cases.Theorem 2.6.1. If f (x) is not defined for x = a, and if lim f (x) = B, x→athen f (x) will be continuous for x = a, if B is assumed as the value of f (x) forx = a. Thus the function x2 − 4 x−2is not defined for x = 2 (since then there would be division by zero). But forevery other value of x, x2 − 4 = x + 2; x+2and lim (x + 2) = 4 x→2 2therefore limx→2 x −4 = 4. Although the function is not defined for x = 2, if we x−2assign it the value 4 for x = 2, it then becomes continuous for this value. Sagesage: x = var('x')sage: limit((xˆ2-4)/(x-2), x = 2)4A function f (x) is said to be continuous in an interval when it is continuous forall values of x in this interval3 . 3 In this book we shall deal only with functions which are in general continuous, that is, con-tinuous for all values of x, with the possible exception of certain isolated values, our results in 17
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2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONSILLUSTRATED BY THEIR GRAPHS2.7 Continuity and discontinuity of functions illus- trated by their graphs 1. Consider the function x2 , and let y = x2 (2.4) If we assume values for x and calculate the corresponding values of y, we can plot a series of points. Drawing a smooth line free-hand through these points: a good representation of the general behavior of the function may be obtained. This picture or image of the function is called its graph. It is evidently the locus of all points satisfying equation (2.4). It is very easy to create the above plot in Sage , as the example below shows: Sage sage: P = plot(xˆ2,-2,2) sage: show(P) Such a series or assemblage of points is also called a curve. Evidently we may assume values of x so near together as to bring the values of y (and therefore the points of the curve) as near together as we please. In other words, there are no breaks in the curve, and the function x2 is continuous for all values of x. 2. The graph of the continuous function sin x, plotted by drawing the locus of y = sin x, It is seen that no break in the curve occurs anywhere. 3. The continuous function exp(x) = ex is of very frequent occurrence in the Calculus. If we plot its graph fromgeneral being understood as valid only for such values of x for which the function in question isactually continuous. Unless special attention is called thereto, we shall as a rule pay no attentionto the possibilities of such exceptional values of x for which the function is discontinuous. Thedefinition of a continuous function f(x) is sometimes roughly (but imperfectly) summed up in thestatement that a small change in x shall produce a small change in f (x). We shall not considerfunctions having an infinite number of oscillations in a limited region. 18
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2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONS ILLUSTRATED BY THEIR GRAPHS Figure 2.2: The parabola y = x2 . y = ex , (e = 2.718 · · · ),we get a smooth curve as shown.From this it is clearly seen that, (a) when x = 0, limx→0 y(= limx→0 ex ) = 1; (b) when x > 0, y(= ex ) is positive and increases as we pass towards the right from the origin; (c) when x < 0, y(= ex ) is still positive and decreases as we pass towards the left from the origin. 19
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2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONSILLUSTRATED BY THEIR GRAPHS Figure 2.3: The sine function. 4. The function ln x = loge x is closely related to the last one discussed. In fact, if we plot its graph from y = loge x, it will be seen that its graph is the reflection of the graph of y = ex about the diagonal (the x = y line). (This is because they are "inverses" of each other: loge (ex ) = x and eloge x = x.) Here we see the following facts pictured: (a) For x = 1, loge x = loge 1 = 0. (b) For x > 1, loge x is positive and increases as x increases. (c) For 1 > x > 0, loge x is negative and increases in absolute value as x, that is, limx→0 log x = −∞. (d) For x ≤ 0, loge x is not defined; hence the entire graph lies to the right of OY . 1 5. Consider the function x , and set 20
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2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONS ILLUSTRATED BY THEIR GRAPHS Figure 2.4: The exponential function. 1 y= x If the graph of this function be plotted, it will be seen that as x approaches the value zero from the left (negatively), the points of the curve ultimately drop down an infinitely great distance, and as x approaches the value zero from the right, the curve extends upward infinitely far. The curve then does not form a continuous branch from one side to the other of the axis of y, showing graphically that the function is discontinuous for x = 0, but continuous for all other values of x.6. From the graph (see Figure 2.7) of 2x y= 1 − x2 21
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2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONSILLUSTRATED BY THEIR GRAPHS Figure 2.5: The natural logarithm. 2x it is seen that the function 1−x2 is discontinuous for the two values x = ±1, but continuous for all other values of x. 22
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2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONS ILLUSTRATED BY THEIR GRAPHS Figure 2.6: The function y = 1/x.7. The graph of y = tan x shows that the function tan x is discontinuous for infinitely many values of the independent variable x, namely, x = nπ , where n denotes any odd 2 positive or negative integer.8. The function arctan x has infinitely many values for a given value of x, the graph of equation y = arctan x consisting of infinitely many branches. If, however, we confine ourselves to any single branch, the function is con- tinuous. For instance, if we say that y shall be the smallest angle (in radians) whose tangent is x, that is, y shall take on only values between − π and π , 2 2 then we are limited to the branch passing through the origin, and the condi- tion for continuity is satisfied. 23
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2.7. CONTINUITY AND DISCONTINUITY OF FUNCTIONSILLUSTRATED BY THEIR GRAPHS Figure 2.7: The function y = 2x/(1 − x2 ). 1 9. Similarly, arctan x , is found to be a many-valued function. Confining our- selves to one branch of the graph of 1 y = arctan , x we see that as x approaches zero from the left, y approaches the limit − π , 2 and as x approaches zero from the right, y approaches the limit + π . Hence 2 the function is discontinuous when x = 0. Its value for x = 0 can be assigned at pleasure. 10. As was previously mentioned, a piecewise-defined function is one which is defined by different rules on different non-overlapping intervals. For exam- ple, 24
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2.8. FUNDAMENTAL THEOREMS ON LIMITS Figure 2.9: The arctangent (or inverse tangent) function. sage: f Piecewise defined function with 3 parts, [[(-5, -2), -1], [(-2, 3), 3], [(3, 5), 2]] Functions exist which are discontinuous for every value of the independent vari-able within a certain range. In the ordinary applications of the Calculus, however,we deal with functions which are discontinuous (if at all) only for certain iso-lated values of the independent variable; such functions are therefore in generalcontinuous, and are the only ones considered in this book.2.8 Fundamental theorems on limitsIn problems involving limits the use of one or more of the following theorems isusually implied. It is assumed that the limit of each variable exists and is finite. 26
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2.8. FUNDAMENTAL THEOREMS ON LIMITS Figure 2.10: The function y = arctan(1/x).Theorem 2.8.1. The limit of the algebraic sum of a finite number of variables isequal to the algebraic sum of the limits of the several variables. In particular, lim [f (x) + g(x)] = lim f (x) + lim g(x). x→a x→a x→aTheorem 2.8.2. The limit of the product of a finite number of variables is equalto the product of the limits of the several variables. In particular, lim [f (x) · g(x)] = lim f (x) · lim g(x). x→a x→a x→a Here is a Sage example illustrating these facts in a special case. Sagesage: t = var('t')sage: f = expsage: g = sinsage: a = var('a')sage: L1 = limit(f(t)+g(t), t = a) 27
40.
2.8. FUNDAMENTAL THEOREMS ON LIMITS Figure 2.12: Another piecewise defined function. In particular, limx→a f (x) lim [f (x)/g(x)] = , x→a limx→a g(x)provided limx→a g(x) = 0. Before proving these theorems it is necessary to establish the following proper-ties of infinitesimals (Definition 2.3.1). 1. The sum of a finite number of infinitesimals is an infinitesimal. To prove this we must show that the absolute value of this sum can be made less than any small positive quantity (as ǫ) that may be assigned (§2.3). That this is possible is evident, for, the limit of each infinitesimal being zero, each ǫ one can be made less than, in absolute value, n (n being the number of infinitesimals), and therefore the absolute value of their sum can be made less than ǫ. 2. The product of a constant c = 0 and an infinitesimal is an infinitesimal. For the absolute value of the product can always be made less than any small 29
41.
2.8. FUNDAMENTAL THEOREMS ON LIMITS positive quantity (as ǫ) by making the absolute value of the infinitesimal less ǫ than |c| . 3. If v is a variable which approaches a limit L different from zero, then the quotient of an infinitesimal by v is also an infinitesimal. For if v → L, and k is any number in absolute value less than L, then, by definition of a limit, v will ultimately become and remain in absolute value greater than ǫ k. Hence the quotient v , where ǫ is an infinitesimal, will ultimately become ǫ and remain in absolute value less than k , and is therefore, by the previous item, an infinitesimal. 4. The product of any finite number of infinitesimals is an infinitesimal. For the absolute value of the product may be made less than any small positive quantity that can be assigned. If the given product contains n factors, then since each infinitesimal may be assumed less than the n − th root of ǫ, the product can be made less than ǫ itself. Proof of Theorem 2.8.1. Let v1 , v2 , v3 , . . . be the variables, and L1 , L2 , L3 , . . .their respective limits. We may then write v 1 − L1 = ǫ1 , v 2 − L2 = ǫ2 , v 3 − L3 = ǫ3 ,where ǫ1 , ǫ2 , ǫ3 , . . . are infinitesimals (i.e. variables having zero for a limit).Adding (v1 + v2 + v3 + . . . ) − (L1 + L2 + L3 + ...) = (ǫ1 + ǫ2 + ǫ3 + . . . ).Since the right-hand member is an infinitesimal by item (1) above (§2.8), we have,from the converse theorem (§2.3), lim(v1 + v2 + v3 + . . . ) = L1 + L2 + L3 + . . . ,or, lim(v1 + v2 + v3 + . . . ) = lim v1 + lim v2 + lim v3 + . . . ,which was to be proved. Proof of Theorem 2.8.2. Let v1 and v2 be the variables, L1 and L2 their respectivelimits, and ǫ1 and ǫ2 infinitesimals; then v 1 = L1 + ǫ1 30
45.
2.11. THE NUMBER E Figure 2.13: Comparing x and sin(x) on the unit circle. Although the function is not defined for x = 0, yet it is not discontinuous whenx = 0 if we define sin 0 = 1 (see Case II in §2.6). 0 Finally, we show how to use the Sage command limit to compute the limitabove. Sagesage: limit(sin(x)/x,x=0)12.11 The number eOne of the most important limits in the Calculus is 1 lim (1 + x) x = 2.71828 · · · = e x→0 34
46.
2.11. THE NUMBER E sin(x) Figure 2.14: The function x .To prove rigorously that such a limit e exists, is beyond the scope of this book.For the present we shall content ourselves by plotting the locus of the equation 1 y = (1 + x) x 1and show graphically that, as x=0, the function (1 + x) x (= y) takes on values in ˙the near neighborhood of 2.718 . . . , and therefore e = 2.718 . . . , approximately. x -.1 -.001 .001 .01 .1 1 5 10 y = (1 + x)1/x 2.8680 2.7195 2.7169 2.7048 2.5937 2.0000 1.4310 1.0096 As x → 0− from the left, y decreases and approaches e as a limit. As x → 0+from the right, y increases and also approaches e as a limit. As x → ∞, y approaches the limit 1; and as x → −1+ from the right, yincreases without limit. Natural logarithms are those which have the number e for base. These loga-rithms play a very important rle in mathematics. When the base is not indicatedexplicitly, the base e is always understood in what follows in this book. Thusloge v is written simply log v or ln v. Natural logarithms possess the following characteristic property: If x → 0 inany way whatever, 35
52.
CHAPTER THREE Differentiation3.1 IntroductionIn this chapter, we investigate the ways in which a function can change in valueas the independent variable changes. For instance, if f (t) is a function of t (time),we want to quantify what it means to talk about the "rate of change" of f (t).A fundamental problem of differential calculus is to establish a mathematicallyprecise measure of this change in the function. It was while investigating problems of this sort that Newton1 was led to thediscovery of the fundamental principles of calculus. Today, Gottfried Leibniz(1646-1716) is generally credited with independently discovering calculus aroundthe same time2 . 1 Sir Isaac Newton (1642-1727), an Englishman, was a man of the most extraordinary genius.He developed the science of calculus under the name of "Fluxions." Although Newton had discov-ered and made use of the new theory as early as 1670, his first published work in which it occurs isdated 1687, having the title Philosophiae Naturalis Principia Mathematica. This was Newton'sprincipal work. Laplace said of it, "It will always remain preeminent above all other productionsof the human mind." See frontispiece. 2 However, see v. Leibniz calculus controversyand the footnote in §3.9 below. 41
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3.4. DERIVATIVE OF A FUNCTION OF ONE VARIABLEIt is apparent that as ∆ x decreases, ∆ y also diminishes, but their ratio takes on ∆ythe successive values 9, 8.8, 8.6, 8.4, 8.2, 8.1, 8.01; illustrating the fact that ∆xcan be brought as near to 8 in value as we please by making ∆ x small enough.Therefore3 , ∆y lim = 8. ∆x→0 ∆x3.4 Derivative of a function of one variableThe fundamental definition of the Differential Calculus is:Definition 3.4.1. The derivative4 of a function is the limit of the ratio of the incre-ment of the function to the increment of the independent variable, when the latterincrement varies and approaches the limit zero. When the limit of this ratio exists, the function is said to be differentiable, or topossess a derivative. The above definition may be given in a more compact form symbolically asfollows: Given the function y = f (x), (3.2)and consider x to have a fixed value. Let x take on an increment ∆ x; then thefunction y takes on an increment ∆ y, the new value of the function being y + ∆ y = f (x + ∆ x). (3.3)To find the increment of the function, subtract (3.2) from (3.3), giving ∆ y = f (x + ∆ x) − f (x).Dividing by the increment of the variable, ∆ x, we get ∆y f (x + ∆x) − f (x) = . (3.4) ∆x ∆x 3 The student should guard against the common error of concluding that because the numeratorand denominator of a fraction are each approaching zero as a limit, the limit of the value of thefraction (or ratio) is zero. The limit of the ratio may take on any numerical value. In the aboveexample the limit is 8. 4 Also called the differential coefficient or the derived function. 44 | 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). | 677.169 | 1 |
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The Algebra Survival Guide comes with an Algebra Wilderness "Bored Game" that gives children a fun way to practice their lessons and also with a tear-out Emergency Fact Sheet poster with all of algebra's secret rules and formulas at your fingertips. The book has been written by Josh Rappaport who has been a winner of […]
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We know very well what a variable is. We use letters x, y, l, m, … etc. to denote variables. A variable can take various values. Its value is not fixed. On the other hand a constant has a fixed value. Examples of constant are: 4, 100, -17, etc. We combine variables and constants to […] | 677.169 | 1 |
Description: The purpose of the course is to give students a broad exposure to combinatorial mathematics, using applications to emphasize fundamental concepts and techniques. Our approach to the course is to show students the beauty of combinatorics and how combinatorial problems naturally arise in many settings, particularly in computer science.
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Product Description: Algebra is much easier to teach when your students can work through the self-directed lessons in this book. Students can explore mathematical concepts on the TI-84 Plus while you walk around the room and facilitate their learning. Keying sequences guide them through the labyrinth of menus on the TI-84 Plus without the need to page through the calculator manual. Detailed solutions at the back of the book help students over obstacles to their learning and the calculator functions are absorbed with minimal effort as the mathematics is learned. Some programs are included to showcase the power of iteration in solving some otherwise intractable problems. Historical vignettes are included to add a human dimension to the mathematical excursions. The topics include sequences & series, finding roots of polynomial equations, the fundamental theorem of algebra, complex numbers, fractals and mathematics of chaos, combinatorics, probability, linear systems, matrices, linear programming and various applications such as the mathematics of investment. A new section on Phi and the Golden Ratio has been added for current interest stimulated by The DaVinci Code. For those with the TI-84 Plus Silver Edition, this book includes references to the probability simulations involving coin and dice tossing experiments found on the Applications menu. | 677.169 | 1 |
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Numeracy and mathematics
Experiences and outcomes
My learning in mathematics enables me to:
develop a secure understanding of the concepts, principles and processes of mathematics and apply these in different contexts, including the world of work
engage w
CHAPTER 8 | LINEAR MOMENTUM AND COLLISIONS
8
LINEAR MOMENTUM AND COLLISIONS
Figure 8.1 Each rugby player has great momentum, which will affect the outcome of their collisions with each other and the ground. (credit: ozzzie, Flickr)
Learning Objectives
8.1
International Baccalaureate
Mathematics
Higher Level
2012 Year 5
August 2012
50 minutes
Trigonometry
Marks
Name
Class
Subject
Group
HLMA 1 / 2 / 3
39
Full marks are not necessarily awarded for a correct answer with no working. Answers must
be supported by
International Baccalaureate
Mathematics
Higher Level
2011 Year 5
August 2011
45 minutes
Trigonometry
Marks
Name
Class
Subject
Group
HLMA
36
Full marks are not necessarily awarded for a correct answer with no working. Answers must
be supported by working a
International Baccalaureate
Mathematics
Higher Level
2012 Year 5
21 February 2012
30 minutes
Quadratics, Exponents and Logarithms
Marks
Name
Class
Subject Group
5MaHL 1 / 2 / 3
25
Full marks are not necessarily awarded for a correct answer with no working
IB
Mathematics HL
Year 5
Common Test 1
24th February 2014
60 minutes
Logarithm and Polynomials
Name
Marks
Class
55
Write your answers on foolscap paper. Full marks are not necessarily awarded for a correct
answer with no working. Answers must be supported | 677.169 | 1 |
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Discrete Math
This compulsory course teaches the use of Discrete Math in Computer Science.
The main focus of the course is on training problem solving skills: each student gets a problem which he has never seen before, for which, on first sight, he has no idea how to solve it, in fact, he may have trouble understanding what is asked from him in the exercise.
Another focus is to learn problem solving in small groups.
The course is based on the textbook "Invitation to Discrete Mathematics" by Matousek and Nesetril. We will cover Chapters 2--8 and 10 in class.
WORK THROUGH CHAPTER 1 BEFORE THE FIRST CLASS MEETING to revive your UG math. SOLVE ALL THE EXERCISES!!
You need a copy of the book for the whole course! Moreover, you will need a hard copy of the book for the exams! (The exams are open book.) There should be sufficiently many copies (~35) in the libraries.
Course prerequisites
This is a graduate level course. We take undergraduate level math for computer science for granted. Computer Science majors at the University of Tartu have 24 ECTS worth of undergraduate math education, covering, among others, logic & proofs, calculus, (linear) algebra, and basic probability.
Make sure your UG math for CS is up to speed!
If it isn't you'll have to work very hard for this MTAT.05.008. Here's how you can make sure your UG math is sufficient.
(1) You can test whether your UG math background is sufficient by reading Chapter 1 of the textbook, and solving all the exercises in that chapter on your own. If you can do that (possibly with considerable effort), you're fine. If you find that you couldn't solve the exercises if your life depended on it, then the course will be very challenging, even overwhelming most of the time, and passing the exams will require a lot of very frustrating work. But don't worry about it: you're smart, so, if you put in the work, and find yourself a study group, we'll get you there. In fact, last year's experience shows that you can even get an A. (No promises here.)
(2) Check out the ATI Math Wiki for a subset of the facts you're supposed to know. What the Math Wiki doesn't give you is experience in solving problems.
(3) A more comprehensive source is the MIT undergraduate course "Mathematics for Computer Science". It has video lectures. Check out the lectures 1-3, 6-23. Keep in mind that that is an undergraduate course. Solve the problems in the assignments.
Course organization
The course has one compulsory weekly meeting (you have to attend), two voluntary weekly sessions -- and a lot of individual study time (same as "Advanced Algorithms", but more frustrating). The individual study consists of (a) reading the assigned sections of the textbook and (b) solving as many of a list of the recommended exercises as you can. For (b), find yourself a study group: a few people consuming large amounts of pizza while discussing the recommended exercises, trying to come up with solutions, making sure everybody understands the solutions, etc.
In the compulsory class meetings, you'll solve problems in small groups, under time pressure, and hand them in at the end of the class. The solutions will be marked and returned to you. The results will not influence the final grade.
In the practice session classes, you will solve problems in small groups, but we'll be there to give you hints etc. These sessions are for guided learning of how to solve problems.
Finally, in the textbook session classes, we'll discuss the assigned reading, trying to make sure everybody understands all the concepts and arguments.
The final grade
The final grade will be compiled from your results of one midterm exam and one final exam. In the exam, you'll have to solve problems on your own (no group), but there will be less time pressure. | 677.169 | 1 |
ISBN 13: 9789380570310
Concept Attainment Testing in Mathematics
Mathematics has become an integral part of general education and has been taught as compulsory subject up to secondary level. As a result of the recommendations of various Education Commission especially the Kothari Commission (1964-66) Mathematics is one of the compulsory components of general education up to secondary level in Indian Schools. Mathematics is a complex system of concepts and not a collection of facts. The pupil should be helped to have a carefully chosen set of mathematical experiences through teaching in order to help him/her to form a new concept. | 677.169 | 1 |
Graphs and Matrices provides a welcome addition to the rapidly expanding selection of literature in this field. As the title suggests, the book's primary focus is graph theory, with an emphasis on topics relating to linear algebra and matrix theory. Information is presented at a relatively elementary level with the view of leading the student into further research. In the first part of the book matrix preliminaries are discussed and the basic properties of graph-associated matrices highlighted. Further topics include those of graph theory such as regular graphs and algebraic connectivity, Laplacian eigenvalues of threshold graphs, positive definite completion problem and graph-based matrix games. Whilst this book will be invaluable to researchers in graph theory, it may also be of benefit to a wider, cross-disciplinary readership.
From the Back Cover
Whilst it is a moot point amongst researchers, linear algebra is an important component in the study of graphs. This book illustrates the elegance and power of matrix techniques in the study of graphs by means of several results, both classical and recent. The emphasis on matrix techniques is greater than other standard references on algebraic graph theory, and the important matrices associated with graphs such as incidence, adjacency and Laplacian matrices are treated in detail.
Presenting a useful overview of selected topics in algebraic graph theory, early chapters of the text focus on regular graphs, algebraic connectivity, the distance matrix of a tree, and its generalized version for arbitrary graphs, known as the resistance matrix. Coverage of later topics include Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph.
Such an extensive coverage of the subject area provides a welcome prompt for further exploration, and the inclusion of exercises enables practical learning throughout the book. It may also be applied to a selection of sub-disciplines within science and engineering.
Whilst this book will be invaluable to students and researchers in graph theory and combinatorial matrix theory who want to be acquainted with matrix theoretic ideas used in graph theory, it will also benefit a wider, cross-disciplinary readership. | 677.169 | 1 |
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Fourier research is a crucial region of arithmetic, specifically in gentle of its value in physics, chemistry, and engineering. but it sounds as if this topic isn't provided to undergraduates. This e-book introduces Fourier research in its 3 so much classical settings: The Discrete Fourier remodel for periodic sequences, Fourier sequence for periodic capabilities, and the Fourier rework for features at the actual line. The presentation is out there for college kids with simply 3 or 4 phrases of calculus, however the e-book can also be meant to be appropriate for a junior-senior path, for a capstone undergraduate direction, or for starting graduate scholars. fabric wanted from genuine research is quoted with out facts, and problems with Lebesgue degree thought are handled fairly informally. integrated are a few functions of Fourier sequence, and Fourier research in greater dimensions is in brief sketched. A pupil could ultimately are looking to circulation directly to Fourier research mentioned in a extra complicated method, both when it comes to extra common orthogonal structures, or within the language of Banach areas, or of in the neighborhood compact commutative teams, however the event of the classical surroundings offers a psychological photograph of what's occurring in an summary environment.
This 5th variation of Lang's booklet covers all of the subject matters regularly taught within the first-year calculus series. Divided into 5 elements, every one component to a primary path IN CALCULUS comprises examples and purposes when it comes to the subject coated. additionally, the rear of the ebook includes unique ideas to various the routines, letting them be used as worked-out examples -- one of many major advancements over prior variations.
As an introductory clarification of the idea of Fourier's sequence, this transparent, certain textual content is phenomenal. The 3rd revised version, that is right here reprinted unabridged, includes exams for uniform convergence of sequence, an intensive remedy of term-by-term integration and the second one theorem of suggest price, enlarged units of examples on countless sequence and integrals, and a bit facing the Riemann Lebeague theorem and its results.
"A convenient ebook like this," famous The Mathematical Gazette, "will fill a superb wish. " dedicated to totally labored out examples, this certain textual content constitutes a self-contained introductory direction in vector research for undergraduate and graduate scholars of utilized arithmetic. commencing chapters outline vector addition and subtraction, exhibit the best way to get to the bottom of and make certain the path of 2 or extra vectors, and clarify platforms of coordinates, vector equations of a aircraft and immediately line, relative pace and acceleration, and infinitely small vectors.
Completely revised, up to date, extended, and reorganized to function a first-rate textual content for arithmetic classes, creation to Set idea, 3rd version covers the fundamentals: relatives, features, orderings, finite, countable, and uncountable units, and cardinal and ordinal numbers. It additionally presents 5 extra self-contained chapters, consolidates the cloth on actual numbers right into a unmarried up-to-date bankruptcy affording flexibility in direction layout, provides end-of-section difficulties, with tricks, of various levels of trouble, contains new fabric on general types and Goodstein sequences, and provides very important fresh rules together with filters, ultrafilters, closed unbounded and desk bound units, and walls. | 677.169 | 1 |
Tag: mathematicsA list of courses and exercises for learning and practicing the following topics in multivariable calculus: Thinking about multivariable functions Derivatives of multivariable functions Applications of multivariable derivatives Integrating multivariable functions Green's, Stoke's, and the divergence theorems
From Youtube: "Videos about math, at the graduate level or upper-level undergraduate", these videos cover topics in mathematics and statistics that are less than 15 minutes long, with narration over written text and equations. Topics include: Machine Learning Probability Primer Information Theory
From Amazon: "Often calculus and mechanics are taught as separate subjects. It shouldn't be like that. Learning calculus without mechanics is incredibly boring. Learning mechanics without calculus is missing the point. This textbook integrates both subjects and highlights the profound connections between them. This is the deal. Give me 350 pages of your attention, and I'll teach you everything you need to know about functions, limits, derivatives, integrals, vectors, forces, … Continue Reading
From Amazon: "The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are … Continue Reading | 677.169 | 1 |
It was three years ago that my colleague, Tim Sipka, and I decided to begin incorporating computer algebra system (CAS) technology into our calculus program. We opted to use the CAS, not to supplant hand computation altogether as some good programs have done, but instead to supplement our existing traditional program. Our simple guiding goal, we decided, would be to try to find ways to use the new technology to help students understand the concepts of calculus better. After experimenting with a few systems, we chose Maple and began developing laboratory assignments which students would carry out in groups of two or three. Of the sixteen topics we've used Maple to teach, I've found that the ones which are most enhanced by the technology are the multivariable topics. In this paper, I share two of the laboratory activities we have used to teach them. Also, I share the results of a student opinion survey about our project.
The first of these is the topic of space curves. The purpose of the lab we wrote on this was to help students visualize three-dimensional curves by
using Maple to plot the curve,
rotating the plot to see the curve from several points of view,
projecting the curve onto the coordinate planes.
From previous work, the students know parametric form for curves in the plane. We start by showing that, if z is forced to be 0, as in
then the curve will be confined to the xy -plane. See Figure 1. If, however, we allow z to vary as t does, as in
then the spiral will stretch up out of the plane. See Figure 2a. We want the students to see the projection of the curve in the xy -plane, so we rotate the z -axis forward (see Figures 2b and 2c) until our viewpoint is directly over the xy -plane (Figure 2d) and we see the same curve as when z was forced to be 0.
Maple has a feature called "tubeplot" which plots the curve as a hollow tube. See Figure 3. Notice how this feature especially helps in interpreting the apparent intersections of the curve. It is clear in this plot which portions of the curve are in front of others. We'll use this feature now to plot another space curve.
An advantage of using the CAS is that students can start analyzing some interesting curves right away--much earlier than we would have expected them to do just plotting by hand. One of the curves we have them analyze has parametric equations
A tube plot of this curve is shown in
Figure 4a. We tell the students to rotate the curve so that they are looking straight down each axis. Let's start by rotating the x -axis upward and to the right (see Figures 4b and 4c) until we see the projection of the curve onto the yz -plane (Figure 4d). As it should be, this is a circle.
Now, back the original view (Figure 5a), and we'll rotate the y -axis upward and to the left (Figures 5b and 5c) until we see the projection of the curve in the xz -plane (Figure 5d). We can see in this projection that z is between -1 and 1 as it should be, and that x is between 0 and 2. (Actually, x is between about 0.0001 and 2.)
Finally, we'll rotate the z -axis forward (Figures 6a, 6b, 6c) until we can see the projection of the curve in the xy -plane (Figure 6d). As it should, y falls between -1 and 1, and x is between 0 (actually about 0.0001) and 2.
We feel that this kind of exercise really gives the students a better feel for the way these curves sit in space and that, after doing this lab, they are able to visualize a space curve reasonably well by inspection of the parametric equations.
It's interesting to see what happens if we increase the domain of t from [-pi, pi] to [-2 pi, 2 pi]. This graceful curve is shown in Figure 7. We can see that, as t takes on larger absolute values, the x -coordinates become very small, so the curve hugs the yz -plane. The curve is traversed from the right-hand side near the y -axis (when t = -2 pi), upward and to the left.
Probably the best spot of all that we've found to use a CAS is to teach the concept of limits of real-valued functions of two real variables. Before we turn them loose on the lab, we tell the students that the concept of limit is complicated by the requirement of path-independence. That is, the single-variable notion of approach from either side becomes approach from every direction and along every possible curve. See Figure 8. Typically, calculus text books include limit problems in which a function nears the same value along all lines of approach, but it tends to a different value along some curve of approach. The student is supposed to find such a curve by inspection. While this is a valid activity, another way is to use the graphics capability of a CAS--and especially the ability to rotate the surface--to help find such a curve or, at least, determine that there is probably one to be found. Here is a problem from our lab:
A plot of this appealing surface is shown in
Figure 9a. Again, the activity is to look at the surface from many points of view. See Figures 9b and 9c. We can see that something fairly precipitous is happening near the origin, and that the function nears a different value along the ridge than it does along lines of approach to the origin. (It also tends to a different value along the trough, but we'll concentrate on the ridge.) To get a better look at the ridge, we'll rotate the z -axis forward. See Figures
9d, 9e, and 9f. We hope this leads students to recognize the ridge curve
(or, possibly, the trough curve) in the surface and then to
We find that the capability the CAS provides to graph and manipulate the surface gives students something concrete to associate with this analytic process.
To learn whether the students thought things were going as well as we did, I surveyed a Calculus I class and a Calculus III (Multivariable Calculus) class. Among other things, I asked,
Do you believe that using Maple is assisting you in learning calculus?
with possible responses
Yes, considerably.
Yes, it helps some.
No, not much.
No, not at all.
The results are shown in Figure 10. In Calculus I, 71% of the students thought that Maple was helping them to learn calculus. In the multivariable course, 86% thought that using Maple was helping them. So students are generally favorable and, in the multivariable course, highly so.
The survey confirmed my own suspicion that a CAS is particularly helpful in the multivariable course. Certainly, I'm convinced that this is the way to teach the course. I can't imagine teaching it now without a computer algebra system.
Acknowledgment:
I thank Myles McNally who helped me to format this paper in HTML. | 677.169 | 1 |
Solving Equations Using Distributive Property Lesson 2 of 2
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This is a Pre-Algebra Common Core Lesson on Solving Linear Equations by Using the Distributive Property. After a few teacher led examples, students will work independently or with a partner to practice solving equations.
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Like this:
PhotoMath, by MicroBlink Ltd., is a neat little utility available from the iOS App Store, and Android Play Store, that can make solving elementary Math homework less of a chore. Besides giving the answer outright, it provides a detailed step-by-step procedure for arriving at the answer.
Try it while the following equation is displayed on a second screen
It does have its limitations though. An equation such as the following is currently unsolvable
Wolfram Alpha makes short work of that equation when you query it using its representation. | 677.169 | 1 |
is Joshua B.
Pre Algebra is based on a simple set of rules professional | 677.169 | 1 |
The student is able to construct a small MATLAB program to perform simple tasks. The student can explain and contrast different numerical methods. The student is able to solve physical problems as, a fall with air resistance, by applying those and to assess the correctness and accuracy of the outcome.
Omschrijving
In this course the programming language of MATLAB is taught. The following subjects will be treated: variables and functions, the concept of datatype, operations with variables, loops and conditions, formatted input and output, programming techniques to master the complexity and ensure the correctness of a program. Furthermore, a number of numerical methods for solving problems as, numerical integration, linear algebra, and differential equations, will be treated in this course. Finally, the students will be introduced to the use of the Symbolic Toolbox of MATLAB to perform symbolic calculations. | 677.169 | 1 |
Algebra 1 Semester 2 Final Exam
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This is a comprehensive final exam that starts with foundational Algebra 1 skills and builds up to finding roots of quadratic equations by factoring and the quadratic formula. This final took my students approximately 50 minutes for fast finishers and 75 minutes for my slower workers.
This exam is 58 questions that are a mix of multiple choice and free response. | 677.169 | 1 |
Mathematics and Statistics
Research
Teaching
About my teaching
One of my main aims in teaching is to get students to see the connections between various parts of mathematics, thereby glimpsing the Big Picture which represents a mathematicianís view of the world. I also encourage students to think of mathematics both symbolically and visually, as these two modes of thought often support one another. Examples of this can be found in some of the resources below.
Teaching resources
MATLAB experiments in linear algebra (2010) to help students explore the meaning of basic terms from linear algebra, and to develop a feeling for typical behaviour of vectors and matrices.
MAPLE experiments in advanced calculus (2001-2007). The following MAPLE worksheets are modelled on similar resources from the Connected Curriculum Project.
As part of my sabbatical year in 2012, I spent a month at
Quest University in Canada teaching a mathematics course
for liberal arts students. Quest students do just one course at a time, with each course squeezed into a three and half week block,
so I had the whole course to myself. This style and the actual course content was all new to me, so it was an exhausting experience.
Quest recently topped all five categories in
Canada's national student engagement survey, so the generally positive
student comments about my teaching were quite pleasing. A fascinating learning experience
for all of us. | 677.169 | 1 |
CHAPTER
4
Numerical Differentiation
4.1
INTRODUCTION
Numerical differentiation deals with the following problem: given the function y = f (x) find one of its derivatives
at the point x = xk. Here, the term given implies that we either have an algorithm fo
Numerical Solution of System of Linear Equations
A system of linear equation is a set of linear equations to be solved simultaneously.
For example:
To simplify notation, rewrite the above equations in matrix notation: A x = b
The elements of the mn of mat
COURSE PRO FORMA
IMPORTANT:
Contents of this Pro Forma shall not be changed without the Senates approval for items indicated with*.
Changes to the other items can be approved at the Academy/Faculty/Institute/Centre level.
Academy/Faculty/Centre
Faculty of | 677.169 | 1 |
I know its subjective, but what is the definition of "mathematical maturity"?
Comfort with abstract mathematical concepts. In general, if you haven't taken calculus yet, then you don't have any (because you've never even encountered abstraction in mathematics). Rudin is something you read after you've already been introduced to real analysis; you are not ready. There are a few good introductory analysis texts (e.g. Elementary Analysis by Ross, which is quite good), but some of the motivation may be lacking if you haven't studied calculus (which isn't to say that you couldn't do it, just that you may not understand the importance of the some of the topics quite yet). | 677.169 | 1 |
Math 210-01: Linear Algebra: Reading Homework 5.4
Mathematical Models and Least Squares Analysis : what is this section concerned with?
Least Squares Regression Line : what is a least squares regression line? what does this have to do with the phrase ``best possible''?
least squares problem : what do we mean by the ``least squares problem''?
Orthogonal Subspaces : when are two subspaces orthogonal?
Orthogonal Complement : what is the orthogonal complement of a subspace? how do we find the orthogonal complement of a subspace?
Direct Sum : what is the direct sum of two subspaces?
Orthogonal Subspaces : what properties do orthogonal subspaces have?
Projection onto a Subspace : what do we mean when we project a vector onto a subspace? How is this related to the orthogonal complement of the subspace and to direct sums? Graphically, what do we mean by the projection of a vector (e.g., in R2, onto another vector)?
Orthogonal Projection and Distance : why is theorem 5.15 titled this?
Fundamental Subspaces : what are the fundamental subspaces of a matrix? how are they related?
Least Squares : what equation does the solution to the least squares problem solve? what properties of projections and fundamental subspaces are used in the derivation of this equation? | 677.169 | 1 |
Larson's PRECALCULUS is known for delivering sound, consistently structured explanations and exercises of mathematical concepts to expertly prepare students for the study of calculus. With the Tenth Edition, the author continues to revolutionize the way students learn the material by incorporating more real-world applications, ongoing review, and innovative technology. How Do You See It? exercises give students practice applying the concepts, and new Summarize features and Checkpoint problems reinforce understanding of the skill sets to help students better prepare for tests. The companion website at LarsonPrecalculus.com offers free access to multiple tools and resources to supplement students' learning. Stepped-out solution videos with instruction are available at CalcView.com for selected exercises throughout the text. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. | 677.169 | 1 |
Course content
Algebraic methods are very important in many areas of
combinatorial mathematics. On the other hand, there are several
situations in algebra and related fields where combinatorial
constructions hold the key to effective understanding. In this
course we will explore some classical material of these two kinds.
Also, glimpses of current research in the area will be given.
An important part of the course is formed by the theory of
symmetric functions, that is, polynomials that are invariant under
permutation of its indeterminates. This is a classic topic in
algebra, however the theory turns out to be of a mainly
combinatorial character. The ring L of symmetric
functions has a distinguished basis consisting of the Schur
functions. These are generating functions for certain
combinatorial gadgets called Young tableaux.
One reason for the broad mathematical significance of symmetric
functions is their role in representation theory and algebraic
geometry. Briefly, the representation ring of the symmetric groups
is isomorphic to L under an isomorphism that
carries the irreducible representations to the Schur functions. A
similar statement is true for the general linear groups.
Furthermore, the cohomology ring of a Grassman variety is
isomorphic to a quotient of L in a way that
matches Schubert cycles with Schur functions. Hence, in all these
situation the multiplication can be described in terms of Schur
functions, which means that ultimately it is governed entirely by
the combinatorics of Young tableaux.
On the combinatorial side the course will cover several topics
from classical enumerative combinatorics. This concerns in the
first instance partitions, permutations, plane partitions and
tableaux, where some of the highlights are the hook-length formula
for enumerating standard Young tableaux, MacMahon's enumeration
formula for plane partitions, the Robinson-Schensted-Knuth
correspondence between permutations (and more generally,
nonnegative integer matrices) and pairs of tableaux, jeu de
taquin, the theory of monotone subsequences, enumeration using
non-crossing lattice paths, etc
References
The indicated chapters of the
following books are recommended for side reading (not required). | 677.169 | 1 |
Providing Strategies for Success: This text provides numerous strategies for success for both students and instructors. Instructors will find the book easier to use with such additions as an Annotated Instructor's Edition, instructor notes within the exercise sets, and an Insider's Guide. Students will find success through features including highlights, exercise hints, art annotations, critical thinking exercises, and pop quizzes, as well as procedures, strategies, and summaries.
"synopsis" may belong to another edition of this title.
About the Author:
Mark Dugopolski was born in Menominee, Michigan. After receiving a B.S. from Michigan State University, he taught high school in Illinois for four years. He received an M.S. in mathematics from Northern Illinois University at DeKalb. He then received a Ph.D. in the area of topology and an M.S. in statistics from the University of Illinois at Champaign-Urbana. Mark taught mathematics at Southeastern Louisiana University in Hammond for 25 years and now holds the rank of Professor Emeritus of Mathematics. He has been writing textbooks since 1988. He is married and has two daughters. In his spare time he enjoys tennis, jogging, bicycling, fishing, kayaking, gardening, bridge, and motorcycling. | 677.169 | 1 |
Section 2.1
Quadratic Functions
and Polynomials
EXAMPLE
Suppose a baseball is dropped
from the top of the Empire
State Building. The table to the
right shows (based on
measurements) the distance d
(in feet) the baseball has fallen
after t seconds.
t (sec)
Math 1160: Linear Functions
Name_Key_
This worksheet will not be due, it is meant to be extra practice. There will be a key posted on my website.
1. The management of a company that manufactures surfboards has fixed costs (at 0 output) of $200
per day and
Math 1160
Exam 1 Info
Spring 2014
Exam 1 will include material from sections 2.1-2.3, 3.1-3.5, 4.1-4.4, and 5.1-5.5.
I have posted a review in MyMathLab. I would recommend working through these
problems on paper and only using MyMathLab to check your answ
Finite Mathematics and Introduction to C... Advice
Showing 1 to 2 of 2
Aaron did an outstanding job of teaching us how to do all the calculations and making sure we understood the material. He had a knack for always saying "unlucky" when something bad happened. It was awesome, and he was a great professor!
Course highlights:
We had a math project that we worked on throughout the semester, and it involved everything from calculating sales to using calculus to figure out exponential bacteria growth. It was really interesting to apply it the real world.
Hours per week:
6-8 hours
Advice for students:
If you study the material for even a half hour a night then you will do fine.
Course Term:Winter 2016
Professor:Aaron Shepanik
Course Required?Yes
Jan 04, 2017
| Would highly recommend.
Not too easy. Not too difficult.
Course Overview:
A teacher can really change a students perspective on a class and thats exactly what tom sojberg did for me. He's funny and likes to have fun but is also serious about his teaching
Course highlights:
i learned a lot of algebra and trig functions that will help me later as i take calculus I and II
Hours per week:
6-8 hours
Advice for students:
If you do the homework assigned and go to discussion one will fly through this class very easy | 677.169 | 1 |
1.4. DIVISION ALGORITHM AND THE FUNDAMENTAL THEOREM OF ARITHMETIC13
1.4
Division Algorithm and the Fundamental Theorem of Arithmetic
In the next few pages, we will try to study properties of integers that will be required later. We
start with a lemma, com
Chapter 1
Properties of Integers and Basic
Counting
We will use the following notation throughout these notes.
1. The empty set, denoted , is the set that has no element.
2. N := cfw_0, 1, 2, . . ., the set of Natural numbers;
3. Z := cfw_. . . , 2, 1, 0,
1.3. STRONG FORM OF THE PRINCIPLE OF MATHEMATICAL INDUCTION
1.3
11
Strong Form of the Principle of Mathematical Induction
We are now ready to prove the strong form of the principle of mathematical induction.
Theorem 1.3.1 (Principle of Mathematical Induct
UNIT I: QUEUEING THEORY
1. Define a queue and a customer.
Queue: The flow of customers waiting for service in a system rendering some service is
called a queue.
Customer: The arriving unit requires some service to be performed is called a customer.
2. Wha
UNIT-II: NON MARKOVIAN QUEUES &QUEUE NETWORKS
1. Derive Pollaczek-Khintchine formula
2. A patient goes to a single doctor clinic for a general check up with 4 phases. The doctor
takes on average 4 minutes for each phase of the check up and time taken for
UNIT-III: SIMULATION
1. Define simulation
Simulation is the imitation of the operation of a real world process or system
over time. It is a numerical technique for conducting experiments that involve certain
types of mathematical and system over extended
UNIT IV INTRODUCTION TO LINEAR PROGRAMMING
PART - A
1. What is OR? Give some applications.
Operations Research has been variously described as the Science of Use,
Quantitative Common sense Scientific approach to decision making
problemetc. But only a few | 677.169 | 1 |
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This is an Algebra 1 Common Core Lesson on factoring a trinomial with a leading coefficient greater than 1 using the grouping strategy. Students will be guided through specific steps how to factor by grouping. The class will complete four whole class examples. After a few teacher led examples, students will complete examples using an interactive whiteboard activity. Students can work individually, groups, or in pairs. This is a great form of formative assessment! | 677.169 | 1 |
Assignments on presentations
If one of your objectives is for students to learn to communicate as mathematicians, then consider creating a presentation context that is comparable to those encountered by mathematicians (e.g., conference talk, colloquium talk, poster presentation, lecture to students, informal presentation to collaborators).
Short Talks
Assigning short talks at the start of the term
acts as an ice breaker and helps the class get to know each other
gives you a quick way to assess which students will need the most help with presenting
helps the students to start becoming more comfortable speaking in front of the class
If students will do a large project as part of the course, a short talk could be an introduction to what they plan to work on.
Chalk Talks
Students may assume that chalk talks require little preparation; however, many students find that juggling speaking and writing is much more challenging than they anticipate. Consider recommending or requiring the following forms of preparation:
Schedule a rehearsal enough in advance for the student to recover once it becomes clear just how challenging presenting will be.
Advise students to plan exactly what they will write on each board. They should review these "board sketches" to ensure that the boards will make sense to anyone who stops listening for a moment to think.
The few students who have particular trouble focusing their speaking may benefit from the exercise of writing out and revising what they plan to say, but they should not memorize this script or use it during the presentation.
Slide Talks
In many branches of mathematics (and science in general), it is now expected that talks be given using computer slides projected on a screen. Although preparing the slides requires students to think through the talk ahead of time, students often make the mistakes of including dense mathematics on the slides and rushing through the slides when they present. Consider reviewing a draft of the slides. Again, a rehearsal is very helpful.
Choosing suitable content
There are various issues to consider depending on whether you assign the material to present or students choose their own presentation topics.
Assigned Topic
If you assign students the material, consider the following questions:
Is the source material at a suitable level? How much mathematical maturity is needed to absorb (and then teach) the material presented? Is the exposition in the source material clear, correct, and complete?
Is the amount of material assigned reasonable given the time constraints on the presentation? An important presentation skill is deciding what to present: how much flexibility are you willing to give the students about omitting or restructuring the material? If you are open to them deciding to omit certain portions, make sure they know.
How is the material related to what has been discussed in class already? It may be helpful to speak with the student about what they might need to review and what they can assume the audience knows. If the material foreshadows future material, let the student know so s/he may give the audience this helpful information.
Student Choice of Topic
If students choose their own material, many of the issues for assigned topics are still relevant. However, the added complication is that the students may not be able to answer these questions themselves when choosing their material. It may be helpful to require consultation on the subject matter with you well ahead of the presentation.
Talks on Final Project
A capstone project (final paper) in the course is a good opportunity for a final student presentation. Students often enjoy the opportunity to speak on material they have worked on for a while. If most presentations up to this point in your course have been chalk talks, the final presentation is a good time to introduce the students to slides and to talk about how changing the "props" of a talk impacts the style of presentation. The transition from giving chalk talks to giving a slide talk is challenging, so a practice presentation is essential.
Should students present alone or with a collaborator?
It's common for each student to present alone, but an advantage to having students present in pairs or teams is that the students can help each other to present well. Below are some observations from educators who have assigned pair and team presentations.
Pair presentations in M.I.T.'s Seminar in Number Theory
Scott Carnahan experimented with having students present math in pairs. He assigned a general topic for each presentation, and the students covered some amount of material that seemed reasonable for a 30 minute talk, coming up with their own strategies for presentation.
Scott writes, "In almost all of the pair presentations, the students decided to split the talk roughly in two, and each student talked for about 15 minutes. In some of the cases, one of the students in the pair had a weaker background or less confidence, and just gave some background information while the more confident student presented the "meat" of the talk. I don't know how much time they spent preparing, so I can't tell if this is a just division of labor.
"There were a few attempts at more innovative presentation styles. For one pair, they allocated half of the board space to each student, and took turns talking, where one student introduced the general theory, and the other student wrote examples. It wasn't an exceptionally polished performance (they had some synchronization problems), but it showed some potential.
"I should mention that one key advantage of pair presentations is that it forced the students to work with each other for preparation, instead of only spending time with me. For the solo presentations, I had been telling them what I personally liked to see in a presentation, but I think hearing another student's opinion is valuable."
Teams of 3 in M.I.T.'s Project Lab in Mathematics
Susan Ruff writes, "In Project Lab students present in teams of 3. Here are some of the challenges they face:
If the entire presentation is 45 minutes, each student presents for only 15 minutes. It's very difficult for students to judge an appropriate amount of material to try to include in 15 minutes. If the timing isn't flexible, doing a timed practice presentation is essential–usually more than one practice presentation is needed. We schedule a practice presentation with the team mentor and me for a few days before the in-class presentation; then students are encouraged to do at least one more practice presentation on their own.
"Here are a few superficial issues the students often face:
If the later presenter needs to refer to content presented by an earlier presenter, the presenters need to ensure that the necessary earlier material isn't erased.
When one presenter makes a mistake, the others typically either sit silently and cringe or try to whisper the correction. You can point out to the presenters that they should help each other and that they can interrupt professionally: "Excuse me, …" "Thank you."
Often the presenters aren't prepared for how to transition between presenters.
One of the greatest advantages of presenting in teams is that the teammates can help each other to plan a good presentation–they can point out for each other ways to present information more clearly, and they can together come up with good ideas for how to structure the presentation and how to handle the most challenging parts of the content.
"A few years ago one 821 team did a shorter presentation in which the teammates assumed different roles: one was the narrator (the person who provided verbal guiding text or meta explanation) while the others did the mathematical heavy lifting as needed. This dynamic worked remarkably well for the short presentation. For a longer presentation I could imagine that one challenge of this approach might be that the narrator is on stage all of the time, so must remain still to avoid being distracting."
Writing a clear assignment
Specify the details of the assignments: content, purpose, timing, format (chalk? slides?), grading, educational objectives. If students don't yet know each other or if you would like students to target an audience other than each other, also indicate how much background knowledge they may assume.
Examples
See the examples above under "Should students present alone or with a collaborator?"
Please contribute additional examples of presentation assignments, along with commentary on the effectiveness of the assignment | 677.169 | 1 |
Math 1051-11
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Fall 2011
CRN: 90297
HOMEWORK I DUE Tuesday, September 13, at the beginning of the lecture
1. Find equations for the following graphs:
(a) The straight line through the point (3, 1) an
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 1051
Fall 2016
HOMEWORK IV DUE Wednesday, November 2, at the beginning of the lecture
1. A city council is planning to build several hotels along its beach-front. They will be of two
types: co
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 1051
Fall 2016
HOMEWORK III DUE Wednesday, October 12, at the beginning of the lecture
1. Use Gauss-Jordan to find the inverse of the following matrix:
Clearly depict your elementary operation
We 5
h I FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 1051-10 Fall 2013 CRN: 50279
QUIZ 11 NAME:
, TOEQ, 0:33
In each of the following applications of the Simplex Method,
0 perform the next pivot, and
0 read off the x, y, and 2 values of
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 105111 Fall 2012 CRN: 30283
QUIZ 2 G NAME:
1. If possible, determine just by looking how many points of intersection the following three systems
have. In each case briefly describe the geometr
we"
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 105110 Fall 2013 CRN: 50279
QUIZ 3 NAME:
3am Was:
1. If possible, determine just by looking the number of points of intersection of the following
three 3-D systems. If impossible to tell,
we 5
FTNITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 1051-10 Fall 2013 CRN: 50279
omzs NAME: jam wens
1. After the rst Gauss-Jordan pivot on a 3-D dependent system I obtained the matn'x:
6
(a) Perform the next pivot;
(b) Depict the genera
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 105111 Fall 2012 I CRN: 30283
QUIZ 1 G NAME:
There is no need to offer any explanation for what you do but clearly show all the steps of your
working (if any).
1. Where does the graph of the e
:0
wo
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 1051-10 Fall 2013 CRN: 50279
U12 2 NAME:
Q (Ma, L061 35
1. Determine just by looking how many points of intersection the following three systems have.
In each case state the number of po
we
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 1051-10 Fall 2013 CRN: 50279
QUIZ 1 NAME:
ma, \DUSS
There is no need to offer any explanation for what you do but clearly show all the steps of your
working (if any).
1. Where does the grap
FINITE MATHEMATICS
FOR THE SOCIAL AND MANAGEMENT SCIENCES
Math 1051
Fall 2016
PRACTICE PROBLEMS V
1. Determine the Linear Programming Problem whose solution would answer the question asked in
each of the following real-world problems. Clearly tell me what | 677.169 | 1 |
Polynomial Functions
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Polynomial Functions guided notes with answer sheet. These notes included vocabulary with examples of graphing polynomial functions, applying the leading terms test, finding the zeros of a polynomial function, zeros of a polynomial function in quadratic form, polynomial functions with repeated zeros, and modeling data using polynomial functions. | 677.169 | 1 |
Complex Analysis: The Complex Logarithm
In this complex logarithm worksheet, students compute the log of given functions. They sketch the sketch the set of points and investigate the analyticity of the logs. This two-page worksheet contains approximately five problems, with explanations and examples. | 677.169 | 1 |
Larson Algebra 1 Practice has been writing in one form or another for most of life. You can find so many inspiration from Larson Algebra 1 Practice also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Larson Algebra 1 Practice book for free.
Algebra 1 Grades 9 12 Practice Workbook has been writing in one form or another for most of life. You can find so many inspiration from Algebra 1 Grades 9 12 Practice Workbook also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Algebra 1 Grades 9 12 Practice Workbook book for free.
Holt McDougal Larson Algebra 1 Virginia has been writing in one form or another for most of life. You can find so many inspiration from Holt McDougal Larson Algebra 1 Virginia also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Holt McDougal Larson Algebra 1 VirginiaAlgebra 1 Common Core Assessment Book has been writing in one form or another for most of life. You can find so many inspiration from Algebra 1 Common Core Assessment Book also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Algebra 1 Common Core Assessment Book book for free.
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Algebra 1 Objectives Review and Practice Grades 9 12 has been writing in one form or another for most of life. You can find so many inspiration from Algebra 1 Objectives Review and Practice Grades 9 12 also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Algebra 1 Objectives Review and Practice Grades 9 12 Practice Workbook Student Bundle of 5 has been writing in one form or another for most of life. You can find so many inspiration from Algebra 1 Grades 9 12 Practice Workbook Student Bundle of 5 also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Algebra 1 Grades 9 12 Practice Workbook Student Bundle of 5 Test Preparation and Practice Workbook has been writing in one form or another for most of life. You can find so many inspiration from Algebra 1 Grades 9 12 Test Preparation and Practice Workbook also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Algebra 1 Grades 9 12 Test Preparation and Practice Workbook book for free.
Holt McDougal Larson Pre-Algebra provides clear, comprehensive coverage of the new Common Core State Standards, including the Standards of Mathematical Practice. The program gives students a strong preparation for Algebra 1, and may be used with advanced students enrolled in the Accelerated Pathway.--Publisher.
This guide offers step-by-step solutions for all odd-numbered text exercises, Chapter and Cumulative Tests, and Practice Tests with solutions. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
Larson | 677.169 | 1 |
Revision Guide Search Result
CGP KS2 Maths Important Bits
Qualification: Key stage 2 Subject: Maths Exam Board(s): All Level: All Description: This book is ideal for last minute key stage 2 SATs revision. All the main topics are covered in quick, clear and easy to understand summaries all written and presented in that now famous simple CGP style. | 677.169 | 1 |
Welcome to MAT 515
MAT 515 is devoted to elementary geometry. The study will focus on the features of
the subject which develop logical skills and geometric imagination of students.
For more information, please select General information link in
the menu to the left.
Announcements
Announcements are listed in reverse choronological order:
most recent announcement at the top.
12/11/10
The final exam will be on Thursday, December 16, at 2:15 pm - 4:45 pm
in the room 4-130.
The exam will cover the whole content of the course, except the part
covered by the second midterm (vectors and analytic geometry).
It will include theoretical questions about theorems and thier proofs.
One of the questions will suggest to formulate and prove
a few theorems from the course closely related to each other.
Here is the list of topics
that are recommended for revisiting to this end.
One question will require just formulation of definitions
and statements of theorems without proof.
Here is the list
of topics recommended for revisiting in this connection.
This list is longer, it provides also a reliable skeleton for
preparation to the whole exam.
Other questions in the exam will be in the traditional shape of a problem.
On Tuesday, December 14, at 5:20 in room 4-130, we will have a review
session.
11/28/10
We start studying of the similarity geometry. This is an integral part of
Euclidean Geometry.
The main notions of the similarity geometry emerge in traditional presentations of Euclidean
Geometry (in particular in the textbook by Hadamard that we use) in a
very indirect way. A short modern introduction can be found here.
11/14/10
The second midterm exam will be on Thursday November 18.
It will cover vector algebra and a piece of so-called Analytic Geometry related to
lines and planes. It will consist of 4 problems.
This material is included into the course, because a teacher should be
aware about it and should be able to solve elementary geometric problems
in coordinates or using vector algebra.
All these sources miss vector equations, and I could not find a web site
with appropriate text. That is why I wrote a
concise text about various
equations presenting lines and planes.
10/23/10
The first midterm exam will be on Thursday October 28. It will consist
of 5 problems. In the homework, which is due by Tuesday, you have to
compose your own version of this exam. On the page of homework you may find
a detailed description of the kinds of problems. Below the material covered
by the exam is described.
The exam covers the book 1 of the textbook, the first two chapters
from the book 2 of the textbook and the material about isometries presented
on this web site. I recommend revisiting all these texts. Pay a special
attention to definitions and formulations of theorems. | 677.169 | 1 |
Intro to graphs notebook file
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This notebook is what I use as an introduction to graphing in science. It can also be addressed in math. Key ideas: all graphs need labels and data, types of graphs, misleading graphs, and interpreting graphs. | 677.169 | 1 |
Welcome to Mathematics Department
Mathematics is one of the primary tools used to describe the world and make sense
out of its various parts. Most of the incredible achievements of our society in the
20th century were facilitated by our understanding of mathematics and its ability
to be applied to many different situations from computers and rockets to skyscrapers
and credit cards. Mathematics is also a necessary tool for scientific research, actuarial
science , medical testing such as MRI and CAT scans, business planning, construction,
finances, aviation, electronics, engineering and many other areas of practical importance
too numerous to mention. At ELAC you will have the opportunity to acquire a solid
base in mathematics to fit your needs and enable you to successfully pursue the career
of your dreams | 677.169 | 1 |
22A - Section 001
Midterm 2 May 20 9
22A - Section 4
Midterm 2 March 6 Tot
MATH 22A
ADDITIONAL HOMEWORK PROBLEMS
FALL 2012
This is list of problems in the textbook that will be of importance later in the course. In particular,
you may nd similar problems on one of the Midterms or the Final Exam. Most of these problems have
solut
LIST OF NONSINGULAR EQUIVALENCES
Definition 1. Let n 2 be any positive integer greater than or equal to 2. A square n n matrix is called
nonsingular if and only if the matrix A has an inverse, A1 .
The following statements are all equivalent for an n n ma
MAT 22A
MT 01 PRACTICE PROBLEMS this exam
MAT 22A
MT 01 PRACTICE PROBLEM SOLUTIONS
Homework 10
due Friday Dec 2, 10pm (or Friday in class)
Review of determinants and their properties
A
If A is a square n n matrix, it represents a linear function Rn Rn (with the same
Rn as input and as output). The determinant of A is defined as the fact
Math 22A (De Loera)
Mid-term exam 2
November 19, 2012
Name:
Student ID#
INSTRUCTIONS
(1) READ INSTRUCTIONS CAREFULLY!
(2) DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO DO SO
(3) FILL IN THE INFORMATION ON THIS PAGE (your name, etc) NOW!
(4) SHOW YOUR WORK on
Math 22A
Practice Final Exam
March 22, 2012
Name:
Student ID Number:
Read each problem carefully.
Write each step of your reasoning clearly.
The best strategy is to solve the easiest problem rst, the second easiest problem
next, etc., working your way
Math 22A:
Exam 1 Study guide
5 sets of webwork problems
4 practice problem sets
14 problems from the textbook
1. Write parametric equations for a line in R3 given two points on the line.
2. Find the equation of a plane in R3 given three points on the p
Practice problems set 4:
Matrix multiplication, transposition, and inversion
1. Suppose A is an nn matrix so that each entry of A is given by the formula aij = i2 +j 2 .
So, for example, the entry in row 3, column 2 is given by a32 = 32 + 22 = 13. Prove
t
Practice problems set 5:
Elementary matrices
If the RREF of A is I, then you can nd elementary matrices so that
En E2 E1 A = I.
1 1
1
Solving the equation for A we obtain A = E1 E2 En . So, if the RREF of
A is I, we can write A as a product of elementary
Practice problems set 2:
RREF and systems of linear equations
1. Write all possible forms for the RREF of a 2 4 matrix.
2. Let k be an arbitrary real number. Suppose we have a system of equations whose
augmented matrix is given by
2
1 2
1
0 1 1/6 1/2
0
Practice problems set 7:
Subspaces and bases
1. Determine which of the following sets are a subspace of R3 . For those that are, nd a
basis. For those that are not, explain which subspace property fails.
a) cfw_[a, b, c] where a + b + c = 0
b) cfw_[x, y,
Linear Algebra Advice
Showing 1 to 1 of 1
it has valuable information about matrices and matrix manipulation. Also teaches you how to use matlab.
Course highlights:
Highlights were matlab and learning about matrices. Watching MIT lectures help a lot.
Hours per week:
9-11 hours
Advice for students:
The teacher is not easy to follow. I recommend watching MIT lectures instead of trying to gain anything out of this professors lectures, as they are very unorganized. Homework was on web works so very easy to get 100 on all of them. best way to study for exams is to thoroughly understand the practice exams. Do it repetitively until you can do it be heart. You might not understand the logic behind most things in this class. Focus on memorization if you don't have time. | 677.169 | 1 |
Operations with Polynomials Lesson
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In this lesson, students add, subtract, multiply, and divide polynomials. This includes vertical and horizontal addition and subtraction.
Included • Video Warm-Up: Students preview the lesson by watching a short video on YouTube and then come to class with some prior knowledge. Warm-up handout included to review what students learned from the video. • Notes & Practice: Three pages of guided notes and practice. • Homework: Two pages of 16 homework problems. • Answer Keys: Completely worked out answer keys are included.
Please look at the preview to find out more information about this resource. | 677.169 | 1 |
Ordinary Differential Equations
Instructor: Elizabeth Bradley
This tutorial contains excerpts from Elizabeth Bradley's Nonlinear Dynamics course. Note that Complexity Explorer tutorials are meant to introduce students to various important techniques and to provide illustrations of their application in complex systems. A given tutorial is not meant to offer complete coverage of its topic or substitute for an entire course on that topic.
This tutorial is designed for more advanced math students who have taken Calculus.
Syllabus
- Ordinary Differential Equations
- Nonlinearity and nonintegrability
- ODEs, vector fields, and dynamical landscapes
- Introduction to ODE Solvers
- Forward and Backward Euler
- Solving Simple Harmonic Oscillator ODEs
- ODE Solvers: Error and Adaptation
- Production ODE Solvers
- Numerical Dynamics
- Unit 3 Homework
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Program Details
Understand what ordinary differential equations look like and how they relate to dynamical systems. | 677.169 | 1 |
501 Math Word Problems was designed to provide students with review and practice for the math word problems that are found on many standardized tests, including high school exit exams, the PSAT, SAT, and GRE, and professional and civil service qualifying exams. The book's 501 problems are organized into six sections: Miscellaneous Math, Fractions, Decimals, Percents, Algebra, and Geometry. The book teaches both basic and advanced math concepts and every problem comes with a complete answer explanation. For problems that require numerous steps, a thorough step-by-step explanation is provided | 677.169 | 1 |
In some courses, all it will require to pass an examination is be aware having, memorization, and recall. Nevertheless, exceeding in a very math class requires a different sort of exertion. You can not simply just display up for the lecture and observe your teacher "talk" about math and . You find out it by undertaking: being attentive in school, actively finding out, and fixing math challenges – even when your instructor hasn't assigned you any. In case you find yourself battling to carry out effectively in your math course, then stop by finest website for resolving math challenges to learn how you could become an improved math student.
Low-priced math experts on the internet
Math courses follow a pure progression – every one builds on the information you have acquired and mastered from the former system. When you are acquiring it difficult to observe new principles in class, pull out your aged math notes and evaluate earlier substance to refresh oneself. Be sure that you meet the conditions just before signing up for any class.
Evaluate Notes The Night Just before Course
Detest any time a instructor phone calls on you and you've forgotten how to fix a particular challenge? Avoid this instant by reviewing your math notes. This will assist you to figure out which ideas or issues you'd wish to go more than in school the next working day.
The considered carrying out homework each and every night time could appear aggravating, but when you'd like to achieve , it truly is essential that you repeatedly follow and master the problem-solving methods. Use your textbook or on-line guides to operate as a result of leading math difficulties on the weekly basis – even when you may have no research assigned.
Use the Nutritional supplements That include Your Textbook
Textbook publishers have enriched contemporary publications with extra substance (including CD-ROMs or on the net modules) that can be utilized to aid pupils attain more follow in A few of these materials may include an answer or explanation guideline, which often can enable you to with doing the job through math difficulties all by yourself.
Study Forward To stay In advance
If you need to minimize your in-class workload or the time you expend on homework, use your free time soon after college or on the weekends to browse forward into the chapters and concepts that may be coated the subsequent time that you are in school.
Evaluation Outdated Checks and Classroom Examples
The function you do in school, for research, and on quizzes can present clues to what your midterm or remaining test will look like. Make use of your outdated exams and classwork to make a particular research tutorial on your future exam. Glimpse at the way your teacher frames inquiries – this is probably how they can surface on your test.
Learn how to Operate Via the Clock
This can be a well known analyze idea for folks getting timed exams; particularly standardized exams. When you only have forty minutes for the 100-point exam, then you can certainly optimally devote four minutes on each 10-point issue. Get details regarding how extended the test will be and which kinds of thoughts is going to be on it. Then strategy to attack the better issues 1st, leaving yourself sufficient the perfect time to devote to the far more demanding types.
Maximize your Methods for getting math research assist
If you are acquiring a hard time being familiar with principles at school, then you'll want to get assist beyond course. Inquire your friends to produce a review group and visit your instructor's place of work several hours to go over hard challenges one-on-one. Attend examine and critique periods when your teacher announces them, or use a personal tutor if you need a person.
Discuss To Oneself
Whenever you are reviewing troubles for an test, try out to explain out loud what technique and techniques you utilized to obtain your answers. These verbal declarations will arrive in handy through a test when you must remember the steps you ought to just take to locate a option. Get extra follow by attempting this tactic with a mate.
Use Study Guides For Excess Apply
Are your textbook or course notes not serving to you understand anything you really should be finding out in school? Use study guides for standardized exams, including the ACT, SAT, or DSST, to brush up on previous content, or . Review guides usually occur outfitted with comprehensive explanations of the way to address a sample problem, , so you can typically uncover the place will be the far better acquire mathproblems. | 677.169 | 1 |
Sunday, April 27, 2014
Algebra Tech Tool: GeoGebraTube-Polynomial Exploration
Polynomials are a concept that I always have a difficult time working on with my students, largely because it is difficult to find a real-world context relevant to them that is polynomial based. Students have an easy enough time memorizing polynomial rules, but they often have a difficult time conceptualizing these rules or understanding where they came from. While my Algebra 1 students aren't going to be manipulating the function form of the polynomial expression anytime soon, I really like this tool because it helps them to visualize the changes that occur within a polynomial expression and how that change affects the graphic version of the polynomial.
Overview
Brief Description of Tech Tool: With this tool, students use sliders to manipulate coefficient values for constant through quintic polynomial functions to analyze the graphic changes. Students are prompted with a series of questions to guide their explorations.
This activity would be best used for an individual or pair to explore various changes that can occur within a polynomial function graph for polynomials of various degrees (Constant through quintic). Students would take notes on the graphic changes for the various shifts in coefficient values to come up with generalized rules for graphing polynomial functions.
1. Learning Activity Type:
LA-Explore - This tool is particularly useful for helping students to explore graphic changes in polynomial functions of various degrees (constant through quintic) as different coefficient values are manipulated.
2. What mathematics is being learned?
NCTM Standards
Proficiency Strands
PS-conceptual understanding Students develop a conceptual understanding of changes in a graphical context to develop generalized rules for these changes.
Additional comments on what is being learned
This tool allows students to view multiple function graphs at the same time allowing them to compare/contrast between even and odd degree functions or between various functions of the same type of degree. Students can make generalizations based on the various coefficient changes for all functions using this feature as well.
3. How is the mathematics represented?
This tool is a virtual manipulative that relies on slide values to alter the coefficient for each of the terms within a polynomial function. Students can slide these values to view the changes in the graphic form of the function.
4. What role does technology play?
Technology enables the users to view graphic changes automatically, an activity that would eat up a lot of class time if done by hand, or even if done by graphing calculator. The slide values are essential to allowing the student to work on developing their conceptual understanding without being bogged down by the procedures of choosing specific values or graphing the function.
Affordances of Technology for Supporting Learning
Computing & Automating - This tool enables the user to manipulate the coefficient values within the terms of the polynomial and automatically changes the graph accordingly.
5. How does the technology fit or interact with the social context of learning?
This tool can be used by students independently or in pairs, but I would not suggest using it for anything larger than a group of two. Students exploring individually may need to be assigned a talking partner that they can bounce their ideas off of after having experimented with the tool for a while. This use of this tool to develop a conceptual understanding of graphic changes could foster good discussion between a pairing of students to compare and contrast their ideas and ultimately come to a conclusion or generalization that is more sound than if they were working solely independently.
6. What do teachers and learners need to know?
Users do not need much technical prowess to successfully use this tool in the classroom. Teachers may want to demonstrate the type of changes that occur while using one of the sliders, but would likely wish to leave the actual exploration to the individuals.
How it Supports & Supplements PBL/PrBL
This tool is very helpful for me in the classroom as it allows for my students to work on a more inquiry driven exploration, similar to the style they use when working on a problem or project. This type of exploration allows for the student to reach their own conclusions about the concept and then check with others for validation. I particularly like this tool because of the automatic changes that it produces within each graph as the coefficient values are being changed, as it frees up the students to think solely about the shifts that are occurring and not about what value they should try next. This also helps students to better analyze graphs of data in later projects as they are more familiar with the type of graphic changes that occur within polynomial functions. | 677.169 | 1 |
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A collection of common ACT Math Test problems.
Each problem set has a linked video explanation along with practice problems and an answer key. This FREE VERSION has four Sample Problems.
This unique document has a video explanation of a Common ACT Math Test Question. In addition, there is a set of practice problems for each of the video explanations. The questions in the practice sets present the student with different types of questioning methods to challenge the student to think about what is being asked to do.
Please consider sharing this product with all your colleagues that might have an interest in providing their students with practice for the upcoming ACT Test.
If you find any part of this product that could be improved, please email me at algebraconquered.dukelee@gmail.com. | 677.169 | 1 |
Mit opencourseware calculus single variable
Lagrange multipliers (3 variables) Instructor: Joel Lewis View the complete course: License: Creative Commons BY-NC-SA. The ratio test is best used when you have certain elements in the sum. The best way to get a feel for this is to build a set of sheets containing examples of tests. View the complete course at: License: Creative Commons BY-NC-SA More information at More.
Mit opencourseware calculus single variable
Highlights of Calculus. MIT Professor Gilbert Strang has created a series of videos to show ways in which calculus is important in our lives. The videos, which.
Coursera / k ə r ˈ s ɛ r ə / is a venture-backed, for-profit, educational technology company that offers massive open online courses (MOOCs). Coursera works with. The ratio test is best used when you have certain elements in the sum. The best way to get a feel for this is to build a set of sheets containing examples of tests. | 677.169 | 1 |
Monday, March 11, 2013
Once again, I will be presenting function diagrams to a math circle for teachers, this time for the American Institute of Mathematics Circle for Teachers. This will happen on March 19, at 340 Portage Ave. Palo Alto, from 5:00 to 8:00 p.m. There is no charge, and in fact a free dinner is provided.
Here is what I hope to cover, at most. Most likely, I will not emphasize the upper-level topics, because of time, and because the attendees are mostly middle school teachers.
Function Diagrams: A Visual Tool for Secondary Math
Function diagrams use parallel x and y axes. They complement Cartesian graphs and offer an alternate approach to concepts in basic algebra: operations with signed numbers; linear functions; solving inequalities; solving systems of linear equations. They are also helpful in precalculus and calculus: definition of functions; domain and range; rate of change; composition; identity and inverse functions; iterating linear functions; sequences; the chain rule. Add this tool to your repertoire!
I'd love to see you there, but if you can't make it, there's plenty about this on my Web site | 677.169 | 1 |
Book Details...
1001 Algebra II Practice Problems for Dummies
1001 Algebra II Practice Problems for Dummies : Practice makes perfect - and helps deepen your understanding of algebra II by solving problems. 1001 Algebra II Practice Problems For Dummies takes you beyond the instruction and guidance offered in Algebra II For Dummies, giving you 1001 opportunities to practice solving problems from the major topics in algebra II. Plus, an online component provides you with a collection of algebra problems presented in multiple choice format to further help you test your skills as you go. | 677.169 | 1 |
Strengthen your Algebra Skills for the new school year!
Algebra 1 Online Back-to-School Program
Don't miss your chance to solve Algebra problems in a flash from the comfort of your home, hotel or even the sunny beach!
eTutorWorld's personalized online back to school Algebra1 Program helps students understand complex algebra concepts quickly. Expert algebra tutors develop sessions that are fun, interactive and effective.
Below is a suggested schedule for the Algebra back-to-school program. A student is expected to attend three regular tutoring sessions per week. However, it will vary for most students, depending on their schedules, math ability and daily progress.
WEEK1:
•DIAGNOSTIC TEST & EVALUATION
Order of Operations, Evaluating expressions
Operations on Integers and Rational Numbers
Distributive Property, Simplifying variable expressions
Solving one and two step equations
Solving multi step equations
WEEK2:
· Inequalities and their graphs
· Solving one step, two step and multi step inequalities
· Proportions and Percentages
· Distance, rate word problems
· Time, Work word problems
WEEK 3:
· Exponents
· Properties of Exponents
· Scientific Notation
· Operations with numbers in scientific notation
· Introduction to polynomials, like terms, degree and number of terms
· Operations on polynomials
WEEK 4:
· Factoring out the G.C.F
· Factoring trinomials with a≡1
· Factoring trinomials with a≠1
· Factoring by grouping
WEEK 5:
· Review
End-of-Course Test
Middle and high school math tutors have the experience of teaching students of all academic levels and abilities. This eventually helps every child master algebra topics. The unique 'Learning by Design' teaching methodology used, makes sure that learning is complete and exact.
The Algebra1 online back-to-school program starts with a diagnostic test to assess your abilities and areas for improvement.
Our back to school programsare supported by a post-session-practice worksheet that helps students retain taught concepts. Finally an end-of-course Algebra assessment is taken to measure your accomplishments during the course.
Become an A+ grader with flexible Algebra1 Learning:
Explore multiple options you can have on our back-to-school programs. Learn how to tackle those challenging Algebra problems during our Algebra back to school Program, a 15 hour (approximately) course at an affordable price. After every session, get a fun worksheet to reinforce concepts learnt. You could also learn multiple subjects using a single pack. Talk to our representative about all that you need and how we can help you improve, progress and eventually conquer Algebra.
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Algebra Homework Help
250 Words
Algebra is the branch of mathematics which deals with utilization of equations in order to find out an unknown variable with the help of it. Algebra is quite a broad category which includes in itself a lot of other sub-disciplines like the number theory, analysis and geometry. A simple definition of algebra can be the study of the symbols of the mathematics and the rules that can be used for the purpose of manipulation of these symbols. Broadly algebra is divided into two parts; algebra part 1 and algebra part 2. Most of the students take up algebra in their higher education and are given algebra assignments and homeworks. Very often students find it really difficult to solve the problems and are unable to find the algebra homework answers. In such cases, students look for online help with algebra questions from a good and reliable online helping website. NeedAssignmentHelp is one such online help providing website and is known to provide algebra homework help.
Algebra terminology
Every discipline consists of its own terminologies and likewise, below mentioned is an algebra terminology list which is essential to learn before learning about algebra.
Absolute value – this algebraic term means the distance of a value from the origin to the number line.
Algebra – the study of the arithmetic in general form is often termed as algebra. Unknown numerals can be found out with the help of equations.
Equation- the statement of mathematics that consists of an equal sign and is used for the purpose of equating different statements.
Coefficient – the number which is present in front of a variable in an equation or an expression is termed as the coefficient.
Exponent – this algebraic term is actually the power to which a number is raised.
Algebraic expression – the term algebraic expression mainly refers to the operations between one or more variables and constants.
Importance of algebra in daily life
Formulas and equations act and serve as a very important part in our daily life, although the importance of algebra in daily life is unsung and not everybody knows about it, but most of the mathematician and people involved in mathematics have the knowledge of the importance of algebra terminology and different kinds of constants and variables in algebra. Driving a car, calculating the distance or even working on the volume of any substance in a container is simply algebra. We do not realize it but we use algebra at every point our lives. In fact, in many other professional areas, algebra is employed. Having knowledge of algebra will help you in excelling in your own field too. If you wish to work on an excel spreadsheet, even there you need to have the knowledge of algebra for planning out the design and other things in it. If you are practising any field of sciences even then algebra will be required and hence, algebra is the discipline which is inevitable and needs to be incorporated in almost all fields of study and even our day to day life.
Algebra 1 homework answers
At NeedAssignmentHelp you can get online help with algebra and hence, it also provides you with algebra 1 homework answers. Algebra 1 is basically the beginners' algebra and it includes the following parts: -
Linear equations within one variable.
Linear inequalities within one variable.
Inequalities and the equations that consist of the expressions that is in absolute value.
System of the linear equations.
Polynomials and factoring of the polynomials.
Quadratic equations
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Algebra 2 homework help
NeedAssignmentHelp provides the students all kinds of academic help and they also provide the students with the algebra 2 homework answers. Algebra 2 is just the extension of the algebra part 1 and includes the following things in it: -
Rational expressions which possess which possess the fractions in the polynomials.
Transformations on different functions.
The Inverse functions
The algorithms.
The one to one functions in them
The exponential functions
The well-qualified experts of NeedAssignmentHelp ease the work and studies of the students by providing algebra homework help. They are also well experienced in providing algebra 2 homework answers.
Polynomials
One of the very important algebra terminology is polynomial. When a sum of a finite number of nonzero terms is present in an expression, then this expression is termed as polynomials. Each of the term of the polynomial consists of a product of one finite number of variables and one constant. The finite number of variable is generally raised to a certain power which is essentially a whole number. The expression of a polynomial can be rewritten again in the form of a polynomial and this can be done by using commutative, distributive and the associative laws of mathematics. Another very important algebraic term which should be mentioned here is the polynomial function. The function that uses polynomials for its definition and fails to define itself without a polynomial is generally termed as the polynomial function. The experts of NeedAssignmentHelp are well experienced in polynomials and hence, provide online help with algebra in even this field.
Different areas of algebra
Algebra is a vast field and incorporates in itself a number of different kinds of sub-disciplines. And the experts of NeedAssignmentHelp provide online help with algebra in every form. As mentioned above NeedAssignmentHelp provides both algebra homework 1 answers as well as algebra homework 2 answers. These were two broad branches of algebra, some of the specialized branches of algebra are mentioned below: -
The elementary algebra – it is the basic level of algebra which is generally taught to the students of the elementary class.
The abstract algebra – the branch of algebra in which the algebraic structures such as the rings, groups and fields are generally studied.
The linear algebra – this particular branch of algebra includes the properties of matrices, vectors and linear equations.
Boolean algebra – this branch of algebra basically functions by abstracting the truth values with the computation.
Computer algebra – it basically implements the methods of algebras for deducing algorithms and different computer programs.
Commutative algebra – the study of different kinds of commutative rings is included in the commutative algebra.
Problems faced by students in algebra
Algebra is a bit difficult subject as it involves in itself a lot of different kinds of operations and functions. A lot of students find it difficult to study and hence, they are unable to do the assignments and look for online help with algebra. Some of the reasons due to which students fail to do assignments may include lack of the understanding of university guidelines, lack of understanding of the concept, lack of time etc. In such conditions, they look for algebra homework help. NeedAssignmentHelp is one such online help providing website which provides the students with all kind of algebra homework help.
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Synopses & Reviews
Publisher Comments
This concise, self-contained textbook gives an in-depth look at problem-solving from a mathematician's point-of-view. Each chapter builds off the previous one, while introducing a variety of methods that could be used when approaching any given problem. Creative thinking is the key to solving mathematical problems, and this book outlines the tools necessary to improve the reader's technique. The text is divided into twelve chapters, each providing corresponding hints, explanations, and finalization of solutions for the problems in the given chapter. For the reader's convenience, each exercise is marked with the required background level. This book implements a variety of strategies that can be used to solve mathematical problems in fields such as analysis, calculus, linear and multilinear algebra and combinatorics. It includes applications to mathematical physics, geometry, and other branches of mathematics. Also provided within the text are real-life problems in engineering and technology. Thinking in Problems is intended for advanced undergraduate and graduate students in the classroom or as a self-study guide. Prerequisites include linear algebra and analysis.
Synopsis
This in-depth guide to creative problem-solving techniques in mathematics shows how to use a range of valuable methods, including calculus and combinatorics, to tackle thorny academic obstacles in other fields such as engineering and applied technology. | 677.169 | 1 |
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Included in this zip folder are 8 SMART Notebook files. 2 are instruction files and 6 are assignments. A brief description of each:
Instruction file #1 is a 35 slide file. There are 17 formative assessment questions over function values and inverse function values. The idea of limiting a domain for reverse mapping is introduced in non-trig functions as well as trig functions.
The instruction file is a 33 slide intro to inverse trig functions. A graph of y=cosx is provided. Points are extracted from the graph. Then the x and y coordinates are switched and the inverse of y=cosx is graphed. The student learns the range of Arccos must be limited in order for it to be a function. In the end it is revealed that in standard Mathematics the range of Arccos is {0,pi] unless otherwise noted.
Assignment A is a 24 question short answer file. The emphasis is finding function mapping and inverse function mapping given:
1. Mapping ovals
2. Graphs
3. Algebraic functions
Sometimes the domain needs to be limited and sometimes it doesn't in inverse mapping. It is on 1 page for easy printing and coded ffor SMART Response.
Assignment B is a 32 question file. Highly accurate graphs of y=cosx and y=tanx are given. The student uses these graphs to find values such as f(150) and f(-225)^2. Also the student is asked to find values such as arccos(1) and actan(sqrt3) on a limited domain. It is on 1 page for easy printing and coded for SMART Response.
Assignment #1 is a 3 section 45 question assignment. The graphs of y=Arccosx, y=Arcsinx and y=Arctanx are provided. The student must answer the following about each graph:
1. Right end behavior
2. Left end behavior
3. The domain
4. The range
5. Types of intervals
6. Is the domain all Real Numbers?
7. Is the range all Real Numbers?
8. Is it bounded below?
9. Is it bounded above?
10. Is it continuous?
11. Does it increase over the entire domain?
12. Is it an even function?
13. Is it an odd function?
14. Does it have an asymptote?
15. Is its inverse a function?
It is on 1 page for easy printing and coded for SMART Response.
Assignment #2 is a 24 question multiple choice file. The emphasis is identifying the rotational quadrants on a limited domain and solving trig equations on a limited domain. It is on 1 page for easy printing and coded for SMART Response.
Assignment #3 is a 15 question multiple choice assignment. The emphasis is solving a trig equation on a limited domain. It is on 1 page for easy printing and coded for SMART Response.
Assignment #4 is a 15 question multiple answer file. The domain of each equation is [0,360). The student must identify all the solutions on the domain. This assignment can be used at Socrative.com and the share number is included in the file. It is on 1 page for easy printing and coded for SMART Response. | 677.169 | 1 |
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Algebra and Trigonometry.
Solve by grouping factor calculator, abstract sample of an investigatory project, fifth grade integer problems solving, mathematics for dummies +software, mcdougal littell geometry answers textbook answers, mcdougal littell grade 9 activation codes. How to do differential equations on the ti i89, printable table of sqyare roots and cubic square roots, algebra problems using squares On the Definitions of the Trigonometric Functions. We don't offer credit or certification for using OCW. Modify, remix, and reuse (just remember to cite OCW as the source.) I happened to debate with a friend during college days whether advanced mathematics is necessary for any veteran programmer. He said that programmers need only basic mathematical knowledge from high school or fresh year college math, no more no less, and that almost all of programming tasks can be achieved without even need for advanced math Lesson Masters (The University of Chicago School Mathematics Project: Functions, Statistics, and Trigonometry: Integrated Mathematics). All of them can be solved with just one Universal STD formula. Mrunal.org/aptitude NCERT Class 9 Chap 6, 7, 8 and 10. Basics explained in NCERT Class 8 Chap 3, 11. Understand basics from NCERT Class 10 Chap 8 and 9. Then exam-oriented concepts from your Quantitative Aptitude book A New Trigonometry for Schools: w. ans. If you enter a value like 60 or pi/3, the program, regardless of what mode you are in, gives you the exact values for sine, cosine, and tangent, assuming that 60 is degrees and pi/3 is radians. If you enter sin(45 in mode degree, or sin(pi/4 in mode radian, the program will give you the values for the inverse sin/cos/tan in terms of degrees or radians/pi, depending of what mode you have Algebra Examples Trigonometry 3.. Be sure to use the correct order, which is as follows: [angle, 90°, angle, leg, leg, hypotenuse]. Click Show to double-check that you have entered your information into the appropriate place. [/frame]Sine – Type the measure of the angle inside the parenthesis online.
A = 1 (section 65) No.5 may be written in the form: These alternative forms are very useful. Again, if (7) be divided by (8) tan" A = 1 - cos 2A 1 + cos 2A If 2A be replaced by a, formulae (4). (5) and (6) may be written in the forms cos 6 = 2 cos? ~ - 1 (12) 84 Similar formulae may be found for tangents. Formula (11) above may be written in the form: It is frequently used in navigation. (1 - cos 6) is called the versed sine of 6 and (1 - sin 6) is called the coversed sine of6. ~ (1 - cos 6) is called the haversine, i.e. half the versed sine. 85 The preceding formulae are collected here for future reference A brief course in trigonometry, by David Raymond Curtiss and Elton James Moulton. It's an in-joke; polynomials foiled all the writers in math class. Ned's Declassified School Survival Guide: Ned hates Math Trigonometry Formula Guide (Quick Review Notes). As was described for a plane triangle, the known values involving a spherical triangle are substituted in the analogous spherical trigonometry formulas, such as the laws of sines and cosines, and the resulting equations are then solved for the unknown quantities epub.
Elements of trigonometry (Vol-1): plane and spherical
I would say Ross is the more elementary. An Elementary Introduction to Mathematical Finance, 2nd ed. Cambridge. 2003. 0521814294 Don't let the title fool you. The book requires a knowledge of calculus and some mathematical maturity. I do not claim that the next book is useful for investing. It is purely philosophical and could be viewed as the Zen meditation guide that accompanies Random Walk (the preceding book) Trigonometry Student's Solutions Manual: Graphs & Models. The Arab civilisation traditionally marks its beginning from the year 622 CE the date when Muhammad, threatened with assassination, fled from Mecca to Medina where Muhammad and his followers found safety and respect Trigonometry Slide Rule Mechanics Trigonometry and Graphs The Slide Rule Mechani. The trigonometric reciprocals are mainly useful for simplifying integration and trigonometric identities. They are known as cosecant, secant and cotangent, and are defined like this: So far, I have only talked about trigonometry as it concerns right-angled triangles and circles College Algebra and Trigonometry, Books a la Carte Edition (4th Edition). Anyways, jokes apart, the lesson here is, "adapt." Here too, Geometry+Trigonometry have been given emphasis like never before Prentice Hall Mathematics, Algebra 2, Grab & Go Chapter 13, Periodic Functions and Trigonometry. Remember: Find the height of the graph at the x-axis and then look for the first time that the graph returns to that height. In this case, the answer is $$ \pi $$ or just $$ \pi $$. Remember: The formula for the period only cares about the coefficient, $$ \color{red}{a} $$ in front of the x. The formula for the period is the coefficient is 1 as you can see by the 'hidden' 1: Remember: The formula for the period only cares about the coefficient, $$ \color{red}{a} $$ in front of the x Solutions Manual Odd-numbered Exercises for Raymond A. Barnett's Analytic Trigonometry with Applications Fourth Edition. We permit you to post or send to us comments, information, ideas, concepts, reviews, or techniques and other material relating to Products and Services (each a "Response"). pdf.
An Elementary Treatise On Plane & Spherical Trigonometry: With Their Applications To Navigation, Surveying, Heights & Distances, And Spherical ... Of Bowditch's Navigator, And The...
Freshman mathematics,
Plane Trigonometry for Colleges and Secondary Schools
Spherical Trigonometry, for the Use of Colleges and Schools
Homework Helpers: Trigonometry
The literate culture of Indian science goes back to at least the fifth century B Trigonometry - 6th edition. The larger of these PCQ is called a major segment and the smaller, PBQ, is called a minor segment. A sector of a circle is that part of the circle which is bounded by two radii and the arc intercepted between them Logarithmic tables of numbers and trigonometrical functions,. In trigonometry, identities are very important and help us tremendously. They can allow us to put a function in terms of one trigonometric function into terms of another. The most basic, and likely most common, identity is the Pythagorean identity, which states: While these three equations appear different, they are equal to one another Plane & Spherical Trigonometry With Tables, 4th Edition,. Teachers are advised to ensure learners have covered all of the question content before using such questions. You are welcome to copy the worksheets and lesson plans here for classroom use. These math problems have been written by teachers from all over the world! And they've been tested in classrooms. "Great Maths Teaching Ideas" is owned by Emeny Learning Limited Algebra & Trigonometry With Analytic Geometry - Student Solution Manual, 11TH EDITION. Probability quizzes year 8, algebraic processes in factorization of quadratic equation, cross multiplying +frations, howdo i find the y intercept of (-4,-10), unit 6 grade 5 adding and subtracting with unlike fractions, Adding integers worksheets College Algebra & Trigonometry by Ratti,J. S.; McWaters,Marcus S.. [2010,2nd Edition.] Hardcover. Select your own background from your photos or use one of the several built-in themes! MyCalculator also supports complex numbers, scientific and engineering notation, trig and hyperbolic trig functions A system of geometry and trigonometry : with a treatise on surveying ... in which, also, the principles of rectangular surveying ... are fully ... tables, and the necessary explanations. It was like telling Hal, "Hal I can't let you do that." The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides relate to the angle under consideration. Click on the correct side of the triangle on the diagram below. There are three formulae involved in trigonometry: Which formula you use will depend on the information given in the question Algebra and Trigonometry online. Find an angle t that is coterminal to 560o such that 0 <= t < 360o. Note that 560 degrees = 360 degrees + 200 degrees which is greater than 360 degrees Trigonometry: Graphing Calculator Manual. Ad College Algebra & Trigonometry by Ratti,J. S.; McWaters,Marcus S.. [2010,2nd Edition.] Hardcover. It is not always apparent that the three angles to specify a rotation are not independent of each other and must be applied in a certain order. For example imagine that we are aiming a dish at a satellite. The azimuth and elevation are independent of each other, for example we can aim south and then elevate up by the required inclination, or we can set the elevation and then turn and point toward the south Trigonometry Formula Guide (Quick Review Notes). It's six amazing and slick utilities can help you decide the undecided - whatever that might be Trigonometry, Surveying and Navigation. We again use the cot formula, getting Az = 146.46°. The calculator gives a negative angle less than 90°, so we subtract it from 180° to get the angle we want. The law of sines then gives c, which is 90° - altitude, as 78.2854°. Smith, Plane Trigonometry and Tables, 4th ed. (Boston: Ginn and Company, 1943). A standard high-school trigonometry text of 1950 download Algebra and Trigonometry pdf. | 677.169 | 1 |
I will use both of these books. They are both available in paperback, and the total cost of the two books is well under $100.
Intended Student Learning Outcomes
To complete this course students will demonstrate their ability to:
1. Understand standard Mathematics notation used in the field of Computing.
2. Recognize the difference between a proof and a counter example.
3. Able to formulate elementary proofs using mathematical induction.
4. Recognize comparative magnitudes of functions such as log(n), n2, 2n.
5. Ability to read and understand some elementary logical proofs.
Grading
Grades will be made up of, midterm quizzes and a final. Four Three in class midterm quizzes, each worth 20%, total: 60%
Final exam: 40%
NOTE: A minimum of 50% must be obtained on the final exam to pass the course. <
The quizzes will be scheduled as follows:
Quiz 1: Week 3: Tuesday, September 27. Quiz 2: Week 6: Tuesday, October 18. Cancelled due to instructor error.
Quiz 3: Week 9: Tuesday, November 8.
Quiz 4: Week 12: Tuesday, November 29.
Please make every effort to be present for the midterm quizzes. It may be that you will be forced to miss a quiz for health, or other legitimate reasons. With my permission, you may a quiz. miss up to two of the quizzes. If you miss a quiz then the marking scheme will be revised for you as follows:
1 missed - 2 quizzes 25%, and 50% final.
Calculator Policy
Calculators will not be needed nor will they be permitted at any of the quizzes or the final exam.
Grading Method
All components of this course will receive numerical percentage marks. The final grade you receive for the course will be derived by converting your numerical course average to a letter grade according to the Queen's grade conversion scale.
Numeric Range
Letter Grade
GPA
90-100
A+
4.3
85-89
A
4.0
80-84
A-
3.7
77-79
B+
3.3
73-76
B
3.0
70-72
B-
2.7
67-69
C+
2.3
63-66
C
2.0
60-62
C-
1.7
57-59
D+
1.3
53-56
D
1.0
50-52
D-
0.7
0-49
F
0
Location and Timing of Final Examinations
As noted in Academic Regulation 8.2.1, "the final examination in any class offered in a term or
session (including Summer Term) must be written on the campus on which it was taken, at the
end of the appropriate term or session at the time scheduled by the Examinations Office."
The exam period is listed in the key dates prior to the start of the academic year in the
Faculty of Arts and Science Academic Calendar and on the Office of the University Registrar's webpage. A detailed exam schedule for the Fall Term is posted before the Thanksgiving holiday;
for the Winter Term it is posted the Friday before Reading Week, and for the Summer Term the
window of dates is noted on the Arts and Science Online syllabus prior to the start of the course. Students should delay finalizing any travel plans until after the examination schedule
has been posted. Exams will not be moved or deferred to accommodate employment,
travel /holiday plans or flight reservations.
Homework
Homework will be assigned weekly. This work will not be collected for grading, rather, solutions to homework will be done in class. There will be four midterm quizzes that will be directly based on the homework assignments. Please see the grading scheme above.
Calendar Description of CISC-102
This course is a direct prerequisite to
CISC-203/3.0 (Discrete Mathematics for Computing II)
CISC-204/3.0 (Logic in Computing)
and a co- or pre-requisite to CISC-121/3.0.
This course is required in all Computing programs except COMA.
Course Syllabus
Mathematics plays an important role in many aspects of computer science. This course sets the stage for the type of mathematics that computer scientists rely on to produce effective software solutions. This course can be viewed as a language course, that is, you will be learning the language of mathematics. I will follow two books that cover similar material in distinctly different ways. Schaum's Notes (SN) are an excellent resource for a well organized source of course material. Discrete Mathematics Elementary and Beyond (DMEB) provides colour and motivation for the same material.
The course will consist of the following elements:
Notation and definitions and notational conventions: Using the language learning analogy this is equivalent to learning vocabulary and grammar and colloquialisms. SN will be the main source for this material.
Tricks and techniques: Sticking with the language learning analogy, this is equivalent to learning writing styles, problem solving methods. SN does a good job of presenting this. However, DMEB is better at providing lots of insight from experts. SN is a great guide for students, whereas DMEB comes straight from the experts in a more informal but also more insightful way.
Practice, practice, practice: This is the key to success. Doing exercises is the only way to absorb the material properly. You can't learn to play a sport, play an instrument, or how to be a good writer solely by reading a book. This material is no different.
Outline and Schedule
Topics. Chapter numbers are from SN. Topics from DMEB will be selected as the course progresses.
Sets (Chapter 1)
Relations (Chapter 2)
Functions (Chapter 3)
Logic (Chapter 4)
Counting Techniques (Chapter 5 and 6)
Integers and Induction (Chapter 11)
Patterns of Proof (PDF Handout) (Chapter 4)
The topics covered this term will be similar to last term (Winter 2016), but may differ slightly at times. You can see a fairly detailed record on last term's web page:
The following table will be updated as the term progresses.
Tuesday, September 27
Quiz #1 based on Homework 1 and 2Quiz #3 based on Homework 6, 7 and 8Tuesday, November 29
Quiz #4 based on Homework 9 and 10. and 11Accessibility Statement
Queen's is committed to an inclusive campus community with accessible goods, services, and facilities that respect the dignity and independence of persons with disabilities. This webpage is available in an accessible format or with appropriate communication supports upon request.
Please contact:
The Equity Office
B513 Mackintosh-Corry Hall
Phone: (613) 533-2563
Fax: (613) 533-2031
E-mail: equity@queensu.ca
Academic Integrity
Academic integrity is constituted by the five core fundamental values of honesty, trust, fairness, respect and responsibility (see These values are central to the building, nurturing and sustaining of an academic community in which all members of the community will thrive. Adherence to the values expressed through academic integrity forms a foundation for the "freedom of inquiry and exchange of ideas" essential to the intellectual life of the University (see the Senate Report on Principles and Priorities). Students are responsible for familiarizing themselves with the regulations concerning academic integrity and for ensuring that their assignments conform to the principles of academic integrity. Information on academic integrity is available in the Arts and Science Calendar (see Academic Regulation 1 ), on the Arts and Science website (see Academic Integrity ), and from the instructor of this course. Departures from academic integrity include plagiarism, use of unauthorized materials, facilitation, forgery and falsification, and are antithetical to the development of an academic community at Queen's. Given the seriousness of these matters, actions which contravene the regulation on academic integrity carry sanctions that can range from a warning or the loss of grades on an assignment to the failure of a course to a requirement to withdraw from the university.
Accommodation Statement
Queen's University is committed to achieving full accessibility for persons with disabilities.
Part of this commitment includes arranging academic accommodations for students with
disabilities to ensure they have an equitable opportunity to participate in all of their
academic activities. If you are a student with a disability and think you may need accommodations,
you are strongly encouraged to contact Student Wellness Services (SWS) and register as early as possible. For more information, including important deadlines, please visit the Student Wellness
website at: | 677.169 | 1 |
ISBN-10: 0321570944
ISBN-13: 9780321570949
Edition: 3rd to follow the sequence of topics in the text, this manual is an easy-to-follow, step-by-step guide on how to use the TI-83/84 Plus and TI-89 graphing calculators. It provides worked-out examples to help students fully understand and use their graphing calculator | 677.169 | 1 |
Ideal for the 1-term course, A Journey into Partial Differential Equations provides a solid introduction to PDEs for the undergraduate math, engineering, or physics student. Discussing underlying physics, concepts, and methodologies, the text focuses on the classical trinity of equations: the wave equation, heat/diffusion equation, and Laplace's equation. Bray provides careful treatment of the separation of variables and the Fourier method, motivated by the geometrical notion of symmetries and places emphasis on both the qualitative and quantitative methods, as well as geometrical perspectives. With hundred of exercises and a wealth of figures, A Journey into Partial Differential Equations proves to be the model book for the PDE course.
Dr. Bray is currently a professor and department chair at Missouri State University. He has published numerous research articles primarily in the realm of harmonic analysis and edited two international conference proceedings volumes.
The following instructor resources are available to qualified instructors for download | 677.169 | 1 |
This is a lucidly written fine-tuned introduction to discrete mathematics. It is eminently suited for students persuing BCA, MCA and B.E./B.Tech courses. Considering the importance of the subject, quite a number of universities have sought to introduce discrete mathematics as a core subject in the engineering curriculum. CONTENTS: Preface. Acknowledgments. A Note to the Reader. Foundations. Elementary Combinatorics. Recurrence Relations. Relations and Digraphs. Graphs. Boolean Algebras. Network Flows. Representation and Manipulation of Imprecision. Bibliography. Index. Printed Pages: 768. Bookseller Inventory # 11604
Used book in excellent condition and WITHOUT any highlights & underlines. may not be Accessories included/different cover . shipping takes 4-5 business days; standard shipping takes 8-10 business days, .Prompt Customer Service! international shipping | 677.169 | 1 |
Graphing Review Sheet / Study Guide
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This is a basic graphing review sheet which covers information such as making a chart, independent and dependent variables, and graphing basic charts. It is geared for graphing in the science classroom. | 677.169 | 1 |
Evaluación de la competencia argumentativa en matemáticas
Evaluación de la competencia argumentativa en matemáticas
- Download this document for free, or read online. Document in PDF available to download.
5 Ciencias naturales y matemáticas - Science
Este trabajo de grado constituye una propuesta teórico descriptiva, que busca dar a conocer una propuesta para evaluar la competencia argumentativa en geometría analítica, en ella se busca hacer claridad sobre los tipos de ejercicios pertinentes para evaluar dicha competencia, así como el diseño de pruebas escritas y la definición de los criterios valorativos a tener en cuenta en estas.
También se describen las acciones que un estudiante realiza en el momento de argumentar y que se debe tener en cuenta para la evaluación de dicha competencia por medio de una prueba escrita; así como la importancia que tiene la argumentación en la enseñanza de la matemática, Abstract: This work degree is descriptive theoretical approaches that intend to show a proposal to assess the argumentative competence in analytic geometry. It seeks to
make clarity about the types of relevant exercises to evaluate that competence, and the design of written tests, and defining the assessment criteria to be considered in these tests.
It also describes the actions that a student performs in the moment to argue and what must be taken into account in the evaluation of the competition through a written test, and the importance that has the argumentation in teaching
mathematics.
* Images in this website could be protected under copyright laws. They are picked automatically from bing. Please use the contact form to claim a copyright infriction and the item will be removed immediatelly. | 677.169 | 1 |
This booklet is designed as a complicated undergraduate or a first-year graduate direction for college students from quite a few disciplines like utilized arithmetic, physics, engineering. It has developed whereas educating classes on partial differential equations over the last decade on the Politecnico of Milan. the most goal of those classes was once twofold: at the one hand, to coach the scholars to understand the interaction among idea and modelling in difficulties bobbing up within the technologies and nonetheless to offer them a superb history for numerical equipment, corresponding to finite modifications and finite components.
Achieve a high-quality figuring out of the rules of trigonometry and the way those recommendations observe to actual existence with McKeague/Turner's TRIGONOMETRY. This book's confirmed process offers modern innovations in short, conceivable sections utilizing present, exact examples and fascinating functions. pleasing illustrations drawn from Lance Armstrong's biking good fortune, the Ferris wheel, or even the human cannonball convey trigonometry in motion. distinctive ancient Vignettes supply a desirable glimpse at what number of the relevant principles in trigonometry started.
This booklet collects the paintings of 35 teachers who proportion their suggestions and insights approximately instructing discrete arithmetic. no matter if you educate on the university or highschool point; no matter if your scholars are from arithmetic, computing device technological know-how, or engineering; no matter if you emphasize common sense, evidence, counting, graph thought, or purposes, you'll find assets during this ebook to complement your discrete arithmetic course.
19 classroom-tested initiatives, 11 heritage modules drawing on unique resources, and 5 articles handle a variety of issues reflecting the breadth of content material and viewers in a number of discrete arithmetic classes. A sampling of subject matters contains development a geodesic dome, operating with excerpts of Pascal's "Treatise at the Arithmetical Triangle," using discrete arithmetic to biology and chemistry, and utilizing good judgment in encouraging scholars to build proofs.
About the Editor
Brian Hopkins obtained his BS in arithmetic and a BA in philosophy from the college of Texas in 1990, and earned his PhD in arithmetic from the college of Washington in 1997. He teaches at Saint Peter's collage, a Jesuit liberal arts establishment in Jersey urban, the place he has led a number of undergraduate study tasks and has been the recipient of the Varrichio Award for educating Excellence (awarded through the SPC Pi Mu Epsilon bankruptcy) twice, in 2004 and 2007. Brian is the writer, with Carl Swenson, of having began with the TI-92 in Calculus (1998, John Wiley & Sons). additionally, he has released numerous examine articles in combinatorics and, with Robin J. Wilson, gained the MAA's 2005 George Pólya Award for great expository writing within the university arithmetic magazine for "The fact approximately Königsberg." Brian works with secondary college lecturers in specialist improvement tasks with a number of enterprises together with the Institute for complex Study's Park urban arithmetic Institute, the Northwest Math interplay, the hot Jersey expert improvement and Outreach workforce, the Institute for brand new Jersey arithmetic lecturers, and the Pikes top Math instructor Circle Academy. he's a member of the Mathematical organization of the US, the yankee Mathematical Society, the nationwide Council of academics of arithmetic, and the nationwide organization of Recording Arts and Sciences. Brian performs piano, sings with Cantori manhattan, and enjoys big apple urban together with his accomplice Michael.
Michael Sullivan and Kathleen Miranda have written a latest calculus textbook that teachers will appreciate and scholars can use. constant in its use of language and notation, Sullivan/Miranda's Calculus deals transparent and exact arithmetic at a suitable point of rigor. The authors aid scholars research calculus conceptually, whereas additionally emphasizing computational and problem-solving abilities. The booklet incorporates a big choice of difficulties together with attractive problem difficulties and utilized routines that version the actual sciences, lifestyles sciences, economics, and different disciplines. Algebra-weak scholars will take advantage of marginal annotations that support develop algebraic realizing, the various references to study fabric, and wide perform routines. powerful media choices comprise interactive figures and on-line homework. Sullivan/Miranda's Calculus has been outfitted with today's teachers and scholars in brain.
Encompassing the entire significant themes scholars will come upon in classes at the topic, the authors train either the underlying mathematical foundations and the way those rules are carried out in perform. They illustrate all of the thoughts with either labored examples and lots of workouts, and, moreover, offer software program in order that scholars can try numerical equipment and so hone their talents in analyzing the consequences. accordingly, this can make a fantastic textbook for all these coming to the topic for the 1st time. Authors' notice: an issue lately discovered with the software program is because of a computer virus in formulation One, the 3rd social gathering advertisement software program package deal that used to be used for the advance of the interface. It happens while the date, forex, and so on. layout is determined to a non-United States model. Please test surroundings your machine date/currency choice to the U.S. choice . the hot model of formulation One, while prepared, may be published on WWW.
Alan Tucker's most recent factor of Applied Combinatorics builds at the past versions with extra extensive research of desktops with a purpose to aid increase talent in easy discrete math challenge fixing. As essentially the most well-known ebook in combinatorial difficulties, this version explains how you can cause and version combinatorically whereas stressing the systematic research of alternative chances, exploration of the logical constitution of an issue, and ingenuity.
World-famous mathematician John H. Conway dependent this vintage textual content on a 1966 path he taught at Cambridge college. aimed toward graduate scholars of arithmetic, it is going to additionally end up a priceless advisor to researchers mathematicians.
His issues conceal Moore's conception of experiments, Kleene's concept of standard occasions and expressions, Kleene algebras, the differential calculus of occasions, elements and the issue matrix, and the idea of operators. extra topics contain occasion periods and operator sessions, a few regulator algebras, context-free languages, communicative usual algebra, axiomatic questions, the power of classical axioms, and logical difficulties. entire strategies to difficulties look on the finish.
Analytic Combinatorics is a self-contained remedy of the math underlying the research of discrete constructions, which has emerged over the last a number of a long time as a vital instrument within the figuring out of houses of computing device courses and clinical versions with purposes in physics, biology and chemistry. Thorough remedy of a giant variety of classical functions is a vital point of the presentation. Written through the leaders within the box of analytic combinatorics, this article is bound to develop into the definitive reference at the subject. The textual content is complemented with routines, examples, appendices and notes to assist realizing consequently, it may be used because the foundation for a sophisticated undergraduate or a graduate path at the topic, or for self-study.
This publication defines units of orthogonal polynomials and derives a couple of homes chuffed by way of one of these set. It keeps by means of describing the classical orthogonal polynomials and the extra houses they have.The first bankruptcy defines the orthogonality situation for 2 services. It then supplies an iterative strategy to provide a collection of polynomials that are orthogonal to each other after which describes a couple of homes happy by way of any set of orthogonal polynomials. The classical orthogonal polynomials come up whilst the burden functionality within the orthogonality situation has a selected shape. those polynomials have one more set of homes and specifically fulfill a moment order differential equation.Each next bankruptcy investigates the homes of a selected polynomial set ranging from its differential equation. | 677.169 | 1 |
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Extending coordinate geometry principles into 3D
Description
This article focuses on applying coordinate geometry in three dimensions, and the underlying mathematics for applying coordinate geometry in chemistry. It was written by Paul Yates and is from the Royal Society of Chemistry magazine Education in Chemistry.
This article focuses on applying coordinate geometry in three dimensions, and the underlying mathematics for applying coordinate geometry in chemistry. It was written by Paul Yates and is from the Royal Society of Chemistry magazine Education in Chemistry.
ADDITIONAL INFORMATION
The Education in Chemistry app is available on iOS, Android and in a web-reader version. The app is free to download and all issues of the magazine are free to download throughout 2015.
In addition to bimonthly issues, app readers will also have access to themed supplements on emerging topics in chemical education.
The app is currently available on iOS in the EU, US, Canada, Australia and South Africa. On Android it is available in the EU, Canada, Australia and South Africa. It will be available everywhere else later in 2016. | 677.169 | 1 |
Welcome to Mrs. Lyle's 8th Grade Math
Please click here for the tutoring times of all the 8th grade math teachers.
Schedule:
1st Period-8th Math
2nd Period-Conference
3rd Period-8th Math
4th Period-8th Math
5th Period-8th Math
6th Period-PLC
7th Period-8th Math
8th grade math is a very important year. About 80% of what you will learn this year can be found on entrance exams as in SAT, ACT, MCAT, LSAT, GRE, etc. Therefore, you will be expected to pass the STAAR math test in order to advance to 9th grade. This assessment will be given the last week in March.
Topics that you will learn this year include, real number operations, geometry, measurement (formulas), equations, multiple representations of a linear function, statistics, and personal finance. | 677.169 | 1 |
Honors Geometry
Mr. Hartzer (Ms. Khalil) / Western International High School
Textbook
Geometry (Michigan Student Edition), Glencoe/McGraw-Hill, 2006
Course Description
Geometry is the mathematical study of shapes and the relationships between them. This course focuses on the basic principles of geometry and trigonometry, with the goal of aiding student mastery in the application of these principles in the real world. With the aid of geometry, students will come to understand how real world objects can be seen in terms of their basic lines, angle, and other relationships.
For the second half of the year, the topics to be covered include: Basic trigonometry, the laws of sine and cosine, the properties and area of quadrilaterals, circles, rhombi, rectangles, and other basic shapes, and the volume of solids.
For this honors course, we anticipate completing the textbook by the end of the year. In addition to the final comprehensive exam at the end of each semester, there will be an exam after each of the thirteen chapters. In general, there will be a warm-up exercise and a homework assignment each day. Students are responsible for completing all assignments in a timely fashion.
Classroom Guidelines
These are the basic guidelines of this classroom while class is in session:
No food or drinks in the classroom. This refers to open containers, and includes anything intended to be eaten, including candy, or drunk, including water.
Students are to be seated at (not on) a desk unless instructed to do otherwise. Under no circumstance is a student to leave the classroom without explicit permission from the teacher.
Students are expected to bring all materials to class. This includes their book, appropriate writing implements, paper, and their binder. Pencils are very strongly preferred over pens.
All persons in the class are expected to treat each other with respect. Speak in turn, when called on, or when presenting; do not speak over each other. Discrimination, teasing, and violent acts or threats are never appropriate and may result in particularly strong disciplinary action.
Electronics which serve to distract students from their task of learning geometry are not allowed. There is to be no texting, no playing of games, no phone calls, and so forth, without explicit prior permission from the teacher.
NOTE: Students may play music for their own entertainment through earbuds during individual work time as long as the volume is low enough not to distract others and as long as it is an aid to concentration. Students must silence these devices during lectures, presentations, and when otherwise instructed to do so. Abuse of this privilege will result in its loss, either for individual students or for the entire class.
The purpose of these guidelines is to foster an environment where students can learn in a comfortable space without distractions. It is hoped, and has been observed, that students will generally realize the value of these guidelines and strive to observe them.
Students will be expected to adhere to guidelines. Students who choose to ignore these guidelines could face disciplinary action, including:
In-class detention: Students will spend a portion of their lunch period in the classroom, performing basic tasks.
A phone call to the parent or guardian.
Commencement of potential suspension proceedings through the school administration.
The nature and degree of the consequence is at the discretion of the teacher.
90-100%: A
80-89%: B
70-79%: C
60-69%: D
0-59%: F
Grades will be based on test scores, homework, in-class assignments, participation, the completion of the binder, and extra credit assignments. Progress reports will be given on a regular basis so that students can keep track of their grade; these progress reports are intended to reflect a snapshot of the student's current standing, and are meant to encourage students to make up lapses before they're too far behind. Students in this course are expected to maintain at least a C-average.
Materials
Students are expected to maintain a binder for this class. The binder shall consist of five clearly separated and labeled sections: Warm-up activities, class notes, homework, quizzes/tests, and group/in-class work. This binder will be collected on a regular basis for review.
Students will be provided a calculator for in-class use. For this course, students will need access to a scientific calculator that provides basic trigonometric functions; during class, an appropriate calculator will be provided. For homework, students should note that such calculators are readily available online or as free cell phone apps, if an actual dedicated calculator is not available to them.
Students have been assigned books for use in both class and home, and are responsible for these books.
Tardiness and Truancy Policy
For full mastery of any subject, it is important that students are present both physically and mentally as much as possible. This means that students should come to class on time and prepared to learn. Attendance and tardiness will be tracked and tallied, and problems which are identified can result in disciplinary action, including discussion with parents. Students should also recall that positive class participation, including coming to class on time, can have a positive impact on their grade.
However, it is acknowledged that events may lead to absences. Students are encouraged to submit appropriate notes to the teacher when absences do occur. There is a class binder for students to refer to in order to catch up on missed assignments. Additionally, assignments will be posted daily on Hero's Garden (parents are also welcome to visit this site if they'd like to be involved with their student's progress).
Final Note to Students
Assignments are most effective when they're completed shortly after the corresponding lecture and group work. They should not be seen as a chore, but rather as an opportunity to practice and master important skills.
It is my goal that work be of a level that's suitably difficult for an honors course; if you feel that an assignment is too onerous or too simple, please consider discussing this with me one-on-one. I especially welcome students who seek out an additional challenge, beyond the course requirements; course requirements should be seen as minimum expectations, and I challenge each of you to reach beyond those minimums to rise to your own level of success.
All the flowers of all the tomorrows are in the seeds of today. – Swedish Proverb | 677.169 | 1 |
Holt math book homework help
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Math for Strategists
Abstract
Great strategists rely heavily on numbers as they go about their work. Offers an overview of the high- and low-brow quantitative tools that students encounter during the Strategy course. The class explores high-brow tools in detail; the focus here is on low-brow calculations. Such calculations come up often in class but because they seem so simple, they get little airtime or explanation. From past class experience, roughly 20% of the students in each section come into the course with the intuition and experience to do these simple calculations themselves. The other 80% understand the calculations after they see them and grasp their value, but don't spot the opportunities to do the math themselves, before class.
More from these Authors | 677.169 | 1 |
Description: The author has been guided by the conviction that it is much more important for the beginner to understand clearly what the processes of the Calculus mean, and what it can do for him, than to acquire facility in performing its operations or a wide acquaintance with them.
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Synopses & Reviews
Publisher Comments
This book arose out of the authors' desire to present Lebesgue integration and Fourier series on an undergraduate level, since most undergraduate texts do not cover this material or do so in a cursory way. The result is a clear, concise, well-organized introduction to such topics as the Riemann integral, measurable sets, properties of measurable sets, measurable functions, the Lebesgue integral, convergence and the Lebesgue integral, pointwise convergence of Fourier series and other subjects.
The authors not only cover these topics in a useful and thorough way, they have taken pains to motivate the student by keeping the goals of the theory always in sight, justifying each step of the development in terms of those goals. In addition, whenever possible, new concepts are related to concepts already in the student's repertoire.
Finally, to enable readers to test their grasp of the material, the text is supplemented by numerous examples and exercises. Mathematics students as well as students of engineering and science will find here a superb treatment, carefully thought out and well presented , that is ideal for a one semester course. The only prerequisite is a basic knowledge of advanced calculus, including the notions of compactness, continuity, uniform convergence and Riemann integration.
Synopsis
This clear and concise introductory treatment for undergraduates covers the Riemann integral, measurable sets and their properties, measurable functions, the Lebesgue integral and convergence, pointwise conversion of the Fourier series, and other subjects. Numerous examples and exercises supplement the text. Basic knowledge of advanced calculus is the sole prerequisite. 1978 edition.
Table of Contents
Chapter 1. The Riemann Integral
1. Definition of the Riemann Integral
2. Properties of the Riemann Integral
3. Examples
4. Drawbacks of the Riemann Integral
5. Exercises
Chapter 2. Measurable Sets
6. Introduction
7. Outer Measure
8. Measurable Sets
9. Exercises
Chapter 3. Properties of Measurable Sets
10. Countable Additivity
11. Summary
12. Borel Sets and the Cantor Set
13. Necessary and Sufficient Conditions for a Set to be Measurable
14. Lebesgue Measure for Bounded Sets
15. Lebesgue Measure for Unbounded Sets
16. Exercises
Chapter 4. Measurable Functions
17. Definition of Measurable Functions
18. Preservation of Measurability for Functions
19. Simple Functions
20. Exercises
Chapter 5. The Lebesgue Integral
21. The Lebesgue Integral for Bounded Measurable Functions
22. Simple Functions
23. Integrability of Bounded Measurable Functions
24. Elementary Properties of the Integral for Bounded Functions
25. The Lebesgue Integral for Unbounded Functions
26. Exercises
Chapter 6. Convergence and The Lebesgue Integral
27. Examples
28. Convergence Theorems
29. A Necessary and Sufficient Condition for Riemann Integrability
30. Egoroff's and Lusin's Theorems and an Alternative Proof of the Lebesgue Dominated Convergence Theorem
31. Exercises
Chapter 7. Function Spaces and £ superscript 2
32. Linear Spaces
33. The Space £ superscript 2
34. Exercises
Chapter 8. The £ superscript 2 Theory of Fourier Series
35. Definition and Examples
36. Elementary Properties
37. £ superscript 2 Convergence of Fourier Series
38. Exercises
Chapter 9. Pointwise Convergence of Fourier Series
39. An Application: Vibrating Strings
40. Some Bad Examples and Good Theorems
41. More Convergence Theorems
42. Exercises
Appendix
Logic and Sets
Open and Closed Sets
Bounded Sets of Real Numbers
Countable and Uncountable Sets (and discussion of the Axiom of Choice) | 677.169 | 1 |
Algebraic Reasoning - Basic
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Replacing a number in an equation with a space, square, shape, or symbol. Solving for a shape or symbol. Solving for x. 1 step and 1 variable equations for addition, subtraction, multiplication, and division. Multiplication questions solving for x using a dot or brackets instead of the multiplication symbol. Suggested for grade 4. | 677.169 | 1 |
3 000 solved problems in calculus schaum's SCHAUM'S OUTLINE OF THEORY ANB PROBLEMS OF Calculus FINITE ENCES and DIFFERENCE EQUAT w e BY MURRAY R. SPIEGEL, Ph.D. Professor of Mathematics Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the
This site is intended as a resource for university students in the mathematical sciences. Books are recommended on the basis of readability and other pedagogical value. Schaum's Outline of Numerical Analysis [Francis Scheid] on Amazon.com. *FREE* shipping on qualifying offers. If you want top grades and thorough …
Schaum's Outline of Partial Differential. The following outline is presented as an overview of and topical guide to discrete mathematics: Discrete mathematics is the study of mathematical structures that are | 677.169 | 1 |
1.2: Solving linear systems via Gauss-Jordan elimination: echelon and
reduced echelon
forms of a matrix, identifying dependent and independent variables, recognizing
when a
system is consistent/inconsistent.
1.3: Relationship between the number of nonzero rows, the number of leading
1's, and the
number of columns in an augmented matrix in reduced echelon form.
Homogeneous linear
systems. The number of possible solutions to (for example)
1. a general linear system.
2. an m*n system with m < n.
3. a homogeneous system.
1.6: Properties of matrix addition, multiplication, and multiplication by
scalars. The matrix
transpose and its properties. What is the transpose of a product? What is a
symmetric matrix?
Also vector norm (length) in terms of the dot product.
1.7: Linear combinations, linear dependence/independence: determinination of
whether
a given set of vectors is linearly dependent/independent. Non-singular matrices
(remember,
only square matrices can be singular or non-singular!); conditions equivalent to
nonsingularity
of
1. Ax = θ has only the trivial solution x = θ
2. columns of A are linearly independent
3.
Ax = b has a unique solution.
4. A is invertible.
5. A is row equivalent to the identity, In.
1.9: Matrix inverses: existence of inverses (see above, e.g., A is invertible
<-> A is nonsingular),
using inverses to solve systems, computing inverses by row reduction, formula
for
inverse of 2*2 matrix, algebraic properties of inverses (e.g., what is the
inverse of a product
of two matrices?) Remember, only square matrices can be invertible.
Notes:
•Only sets of vectors can be linearly dependent/independent. It does not make
sense to
speak of matrices or systems of equations being linearly dependent/independent.
•Only systems of linear equations can be consistent/inconsistent. It does not
make sense
to speak of matrices or sets of vectors as consistent/inconsistent.
•Only squares matrices can be singular/non-singular. It does not make sense
to speak of
a system of equations or set of vectors as being singular/non-singular. The same
applies for invertibility. | 677.169 | 1 |
Applied Differential Equation (MTME1043)
Pre-requisite(s)
Recommended Book(s)
Reference Book(s)
Advanced Engineering Mathematics, Erwin Kreyszig,8th Edition
Theory And Problems Of Laplace Transforms, Schaum's Outline Series
Course Objectives
The successful completion of this course would help students in achieving the following objectives: • Model the problems arising in different areas of science and engineering in the form of differential equations • Solve the linear 1st order differential equations that appear in circuit analysis, electronics, motion, electric machines etc. • Solve second order differential equations using different techniques • Apply 2nd order differential equations to the variety of theoretical problems • Understand the meaning, use and applications of the partial differential equations
Course Learning Outcomes (CLO)
CLO-1: Use their knowledge of calculus to solve the 1st order ordinary differential equations. (C3)
CLO-2: Use various techniques to solve higher order ordinary differential. (C3)
CLO-3: Apply their knowledge of differential equations to solve the problems arising in different areas of science and engineering. (C3)
CLO-4: Understand the meaning, use and applications of the partial differential equations. (C2)
Apply their knowledge of differential equations to solve the problems arising in different areas of science and engineering. | 677.169 | 1 |
Description
All aspects of engineering require the use of mathematics to analyze and design physical systems. This section provides a brief review of the basic concepts in algebra, geometry, trigonometry, statistics, and calculus. The material presented here is not complete and is not intended to be a resource for learning these topics for the first time. The study of algebra involves examining the basic properties of numbers. Algebra is founded on several basic laws. The areas and volumes of common geometric shapes are frequently needed in the solution of engineering problems. Statistics is the area of mathematics that describes the characteristics of a population by use of parameters that characterize the population. | 677.169 | 1 |
[SR: 4659982], Paperback, [EAN: 9780986874307], 4JC Tutoring, 4JC Tutoring, Book, [PU: 4JC Tutoring], 4JC Tutoring, LOOK objective of this book: see through examples and remember, do through practice and understand. --- Content --- ThisTeaching through examples is the most distinctive strength of this book. Very little text reading is required, as the relevant examples work as the fundamental teaching tool. The practice drills make the learning internalized and complete."- John R. Pierce, M.S., CAGS, Education Consultant, MA"This book is a mind-opening way of learning: understanding through examples. It is a strong foundation to success: strengthening math skills through practice."- Dr. T. Fung , Senior Mathematical Consultant, Univ. of Calgary, AB ..... LOOK INSIDE the book at 4jc-tutoring.com ....., 8975347011, Education & Teaching, 69820, Higher & Continuing Education, 5267708011, Schools & Teaching, 5267723011, Studying & Workbooks, 5267710011, Test Preparation, 1000, Subjects, 283155, Books
LOOK ob used books,books Books, 4JC Tutoring
The unique strengths of this book:
1
2. Instead of focusing too much on strategies that teach guessing and eliminating, yet lightly touch upon the problem-solving skills, I focus on teaching the important and foundational math skills that give students confidence in solving problems and answering correctly
3. "Practice makes perfect," is crucial to math review, hence the well thought-out structure and exhaustive practices and solutions in every chapter. The best strategies to solve problems are also taught and demonstrated in the lessons and practice solutions.
A wise proverb says, "I see, I remember; I do, I understand." This is the objective of this book: see through examples and remember, do through practice and understand.
--- Content ---
This
Teaching through examples is the most distinctive strength of this book. Very little text reading is required, as the relevant examples work as the fundamental teaching tool. The practice drills make the learning internalized and complete.
" - John R. Pierce, M.S., CAGS, Education Consultant, MA
"This book is a mind-opening way of learning: understanding through examples. It is a strong foundation to success: strengthening math skills through practice." - Dr. T. Fung , Senior Mathematical Consultant, Univ. of Calgary, AB | 677.169 | 1 |
Precalculus With Limits | Download eBook PDF/EPUB
Access cards can be packaged with most any textbook, please see your textbook rep or contact WebAssign.
Precalculus with Limits: A Graphing Approach. Authors. easier and help students succeed. tool offers instructors the option to assign homework and tests.It is a set of templates to help you take notes,. and keep track of homework assignments.
Save time with ready-to-use assignments built by subject matter experts specifically for this textbook.Precalculus with Limits Precalculus with Limits A Graphing Approach a Graphing Approach by Ron. offers instructors the option to assign homework and.
Precalculus Review / Calculus Preview at Cool math .com
More Graphing | CourseNotes
McGuffey Longevity Award for CALCULUS: AN APPLIED APPROACH,. with Limits: A Graphing Approach.
Precalculus With Limits A Graphing Approach 5th Edition
This Precalculus review (Calculus preview) lesson explains limits at infinity on the graphs of rational functions and introduces the concepts of limits from the right.
Precalculus with limits: a graphing approach. About WorldCat Help. Analytical geometry in three dimensions --Limits and introduction to calculus.Holt precalculus a graphing approach homework help annotated bibliography apa for dummies Limits a graphing calculator and.
Precalculus a graphing approach answers
The MathGraphs.com website is designed to provide students using textbooks authored by Ron Larson with enlarged printable versions of.
Larson Precalculus With Limits A Graphing Approach | Fast
Limits and an Introduction to Calculus. Precalculus With Limits: Graphing Approach.
If you are in need of technical support, have a question about advertising opportunities, or have a general question, please contact us by phone or submit a message through the form below.Browse the instructional videos by chapter, section and video title in the navigation above.Larson Precalculus with Limits A Graphing Approach, Larson Precalculus with.Instructional Videos by Dana Mosely. The articles are coordinated to the topics of Larson Calculus.Use the Textbook Edition Upgrade Tool to automatically update all of your assignments from the previous edition to corresponding questions in this textbook.Precalculus review and Calculus preview. Graphing. Slopes of Lines. More on Limits.
Holt Precalculus: A Graphing Approach: Graphing Calculator
Master It Tutorials (MI) show how to solve a similar problem in multiple steps by providing direction along with derivation so students understand the concepts and reasoning behind the problem solving.
Wiley: Precalculus: With Limits, 2nd Edition - Cynthia Y
Read It links under each question quickly jump to the corresponding section of a complete, interactive eBook that lets students highlight and take notes as they read. | 677.169 | 1 |
Educational Homepage of Jerome Dancis
My interview
by the newsletter of the Baltimore Curriculum Project contain
many of my views on Math education. For the summary
"spotlighted" in the Feb. 2006 newsletter. The
last question was "What can be done to improve K-12 math education
in Maryland?"; my nine answers follow in the interview.
Teaching Word Problems and Fractions
Fractions - "Toward
Understanding and Remembering -- How to do Hand Calculations
with Fractions" This paper presents a method for teaching
calculations with fractions in a manner that is easily understood,
easily remembered, and which will considerably reduce the need
for memorization and homework. Also, it will reduce students making
"dumb" mistakes.
College Class Notes:
Linear Equations - Linear Operator
and Linear Algebra Theory for Linear Differential Equations.
(A second year calculus course.) These notes may be used
for both a discovery method and a lecture method of instruction.
Commentaries on education and mis-education.
Masses of college students are relegated to remedial Algebra I and Arithmetic classes.
"More than one in four remedial students work on elementary
and middle school arithmetic. Math is where students often lose
confidence and give up." (New York Times)
Masses of high school students are not able to read and comprehend their
textbooks.
College professors are distressed by the low level of understanding
of Algebra and Arithmetic by masses of students as they enter college;
even students, who have taken calculus in high school. This prompted
the local college math professors' professional association [ the
MD/DC/VA section of the MAA] to issue its statement "ON MATHEMATICS PREPAREDNES [NOT]" College
math professors decreed: "Students should be able to perform Algebra and
Arithmetic calculations, without the assistance of calculators."
This is the opposite of the MD HSA on [pretend] Algebra, which effective
mandates the exclusive use of calculators for Arithmetic and Algebra.
A fun, critical video on how two popular Reform math textbook series: TERC and EverydayMath(used
in Montgomery County, MD) muck up the teaching of Arithmetic.
MD's Pretend Algebra and Pretentious Data Analysis
Exam
Our MD college faculty "Petition to
Upgrade Maryland's Mathematics Standards" was signed by
50 math and engineering college professors. One of
its main points is that "the State of Maryland's mathematics standards
neglect the math skills [like arithmetic] and conceptual understanding
that are essential for real algebra." It also notes:
"Teaching to such a low standard will increase the already high number
of students taking remedial math [that is, real Algebra] in college."
The implementation, of the MD HSA Algebra (and Data
Analysis) test as an end-of-year exam and and as a graduation
requirement, will likely result in a major dumbing down of math
education in Maryland. Read: Beware
the MD Algebra Test - (Presented at the Nov., 2002 MD-DC-VA
Sectional Meeting of the MAA.) Montgomery County Public
Schools may be taking the lead:
"State
and county math standards hurt student performance"
in Silver Chips, (Dec. 18, 2003), the student newspaper of
Blair High School (Silver Spring, MD) Two
quotes: "MCPS [Montgomery County Public Schools] mandated changes to
the Algebra I curriculum to align the course with the tested material
[MD's Algebra Exam]. 'We don't think the material is what they need
to know to be successful [in Algebra II and Precalculus],' said Blair
H.S. algebra lead teacher Maria Costello".
"Changes in the curriculum are cited as a main cause for students' deficiencies
in basic algebra, which are manifesting themselves in higher
level math courses that require an understanding of concepts taught
in Algebra I. 'Our Algebra II students are worse than ever. Our Pre-Calculus
students are worse than ever. It's falling apart as we go up the
ladder,' said Costello."
Stop the Maryland's (MD) Pretend Algebra and Pretentious Data Analysis exam,
which is part of the Maryland High School Assessments (HSA)
program and the pending proposal by the Maryland State Board
of Education (MSBE) to require that students pass the HSAs as
a graduation requirement.
1. Pretend "Highly Qualified" Math Teachers
The MD standards for middle school math teachers does not guarantee math
teacher fluency in the many middle school math topics on MD's Pretend Algebra
and Pretentious Data Analysis exam. It was supposed to be the
No Child Left Behind Act (NCLB) to the rescue. NCLB mandates that states
set standards for "highly qualified" middle school teachers. But
Maryland is setting the bar for middle school math teachers below
its bar for students. Some details in:
2. Much wrong Algebra and
misleading Data Analysis. The MD Pretend Algebra
and Pretentious Data Analysis exam includes much misleading and
wrong Algebra and Data Analysis. Some examples in Beware the MD Algebra Test (Items
4 and 6 and end of #3)
3. Adding 3
+ 5 and solving 2x = 200 without a calculator
not required. MD's Pretend Algebra Exam
Versus Students Learning
Simple Math The overuse of trusty calculators
on the exam will lead to their overuse in MD high schools and middle
schools which in turn, will result in more students needing to
take remedial math when they enter college as well as high school
graduates not fluent in Arithmetic. Relatedly, "With 'Pretend' Testing, a Poor Imitation of Preparing Students"by Karin Chenoweth in her Dec. 25, 2003 Homeroom column
in Montgomery Extra Section of Washington Post (
She writes: "I now call the test a 'pretend algebra' exam
and fear that it will undermine mathematics instruction throughout
the state."
4. If Johnny can't read,
then he can't pass MD Algebra exam. The reading
level of the students, demanded by the MD Pretend Algebra and
Pretentious Data Analysis exam is higher than the math level required
of the students. Large numbers of students should not be
flunking a math exam because of reading.
Two reading problems masquerading as an Algebra
problem and a data analysis problem are listed in When It Comes To Math, Words Count,
Washington Post. Outlook Sunday September
8, 2002.
5. Math Level on HSA Algebra Exam NOT
a Step Up
The math level required of the students is not
clearly a step up from that of MD Functional Math exam.
It should be. If one looks at just the level of the math done
by the students, then the MD Pretend Algebra Exam is a step to the
side. The exam sidesteps the math; this includes, but is not
limited to leaving the bulk of the math to the hand calculators. Also
the exam includes misleading and wrong Algebra and Data Analysis, which
is counterproductive. Students no longer need to be able to add
3+5 without a calculator.
6. The California Path to Remedial Math.
The math section, of the Maryland State Dept. of Education (MSDE),
has been colonized by the MATH Reform movement, which is lead by
the National Council of Teachers of Mathematics (NCTM). During the
period, when "Reform" instruction was becoming official state policy
and was being increasingly used in the California schools, with "pretend"
Algebra replacing real Algebra, the remediation rate in mathematics
(Algebra) skyrocketed from 23% in 1989 to 54%
in 1998 for freshmen in the Cal. State University System.
Absurd.
7. Cover-up for the Algebra the students
are not learning. As stated, the official
title "MD HS A on Functions, Algebra, Data Analysis and Probability"
is such a gross exaggeration as to highly misleading to parents and
students. Needed a "Truth in labeling" law for state exams.
At least rename it the MD Math Exam.
Related reading: Arithmetic problems are
also Masquerading as Algebra on
the MD Algebra test, in Free For All in Washington Post Saturday, September
7, 2002. "State
math test dumbs down curriculum", Viewpoint, Prince George's
Journal, July 3, 2001 | 677.169 | 1 |
Departments
Mathematics
The mathematics program at RHS provides engaging instruction to enable students to think critically and creatively through problem solving, and compute fluently. The mathematics program will assist students in achieving their highest potential, as they develop the ability to interpret, communicate and solve real-world problems.
We encourage all students to purchase a TI-84 graphing calculator. The department has a small number of these calculators available for student use.
Introductory Math provides learners with an opportunity to review and study foundational topics for higher-level mathematics. Topics include: simplifying expressions and solving one-variable equations and inequalities; one-variable statistics; different representation of functions; linear functions; the Pythagorean theorem; volume; solving systems of linear equations; graphing line of best fit; and operations with polynomials. Students will solve relevant and authentic problems using manipulates
FOUNDATIONS OF MATH I (MATH IA) (ELECTIVE CREDIT)
This course should be paired with Math IB (21032X0B)
The purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. In conjunction with Math IB, this
MATH IB
Foundations of Math IA; Foundations of Math IAMATH I
Mastery of the middle school mathematics curriculumFOUNDATIONS OF MATH II (ELECTIVE CREDIT)
Marginal proficiency in Math I
Foundations of Math II provides learners with an opportunity to review and study foundational topics for higher-level mathematics. The topics covered will be based on student needs and will be aligned with Math II. Students will solve relevant and authentic problems using manipulatives and appropriate technology.
MATH II higher degree polynomials. New methods for solving quadratic and exponential equations are developed. The characteristics of advanced types of functions are investigated (including power, inverse variation, radical, absolut
MATH II (HONORS) more sophisticated polynomials. New methods for solving quadratic and exponential equations are developed. The characteristics of more advanced types of functions are investigated (including power, inverse variation, radica
FOUNDATIONS OF MATH III (ELECTIVE CREDIT)
Marginal proficiency in Math II
Foundations of Math III provides learners with an opportunity to review and study foundational topics for higher-level mathematics. The topics covered will be based on student needs and will be aligned with Math III. Students will solve relevant and authentic problems using manipulatives and appropriate technology.
MATH III
MathMATH III (HONORS)
Honors MathESSENTIALS FOR COLLEGE MATH (SREB)
Marginal proficiency in Math III
Concepts explored in this course include exponentials, quadratics, equations, measurement, number operations, systems, linear functions, and statistics. Emphasis is on understanding mathematics concepts rather than just memorizing procedures. Students will learn the context behind procedures: for example, why they should use a certain formula or method to solve a problem. This equips them with higher-order thinking skills enabling them to apply math skills, functions, and concepts in different s
ADVANCED FUNCTIONS AND MODELING
Math III
Advanced Functions and Modeling provides students an in-depth study of modeling and applying functions, probability, statistics, trigonometry, financial literacy. Appropriate technology, from manipulatives to calculators and application software, are used regularly for instruction and assessment. Advanced Functions and Modeling is not an honors level course. This course is accepted as the fourth math for admission to UNC System institutions.
PRECALCULUS (HONORS)
Honors Math III
The Precalculus curriculum includes a complete study of trigonometry, as well as advanced algebra topics, analytic geometry, sequences and series, data analysis, vectors, and limits. Applications and modeling are included throughout the course of study. Appropriate technology, from manipulatives to calculators and application software, is used for instruction and assessment. This course is accepted as the fourth math for admission to UNC System institutions.
The AP Statistics curriculum is divided into four major themes: exploratory analysis, planning a study, probability, and statistical inference. This is a college-level course. Use of computers and graphing calculators play an important role in this course. For each session of classroom instruction, the student is expected to spend, as a minimum, an equal amount of time outside the classroom for review, written assignments, and preparation. It is expected that students enrolled in this course wil
ADVANCED PLACEMENT CALCULUS: AB
Mastery of the Precalculus curriculum
The AP Calculus curriculum includes limits, continuity, derivatives with applications, and elementary integration with applications. This is a college-level course. Use of computers and graphing calculators play an important role in this course. For each session of classroom instruction the student is expected to spend, as a minimum, an equal amount of time outside the classroom for review, written assignments, and preparation. It is expected that students enrolled in this course will take the C
ADVANCED PLACEMENT CALCULUS: BC
AP Calculus AB
The BC level of AP Calculus revisits some topics introduced in the AB course. Topics include differentials, integrals, infinite series, and differential equations. In addition, the curriculum for this course includes convergence and divergence of sequences and series, parametric representation of curves, polar curves, and additional integration techniques. This is a college-level course. Use of computers and graphing calculators play an important role in this course. For each session of classroo
INTRODUCTION TO COLLEGE MATHEMATICS (HONORS)
Advanced Functions and Modeling
The ICM curriculum includes data analysis; applications of functions, matrices, and a continuation of trigonometry; vectors, limits and their applications; and the mathematics of networks, social choice, and decision-making. Applications and modeling are included throughout the course of study. Appropriate technology, from manipulatives to calculators and application software, is used for instruction and assessment.
SAT/ACT PREPARATION (ELECTIVE CREDIT)
Math III
This course helps students prepare to take the reading and math portions of the SAT and the ACT. Verbal preparation focuses on reading comprehension, vocabulary development, critical thinking, and analogies. Math preparation focuses on arithmetic, algebra, and geometry skills to answer the high-level questions that appear on the tests. This course does not replace core English or math courses. | 677.169 | 1 |
04866468Introductory Complex Analysis (Dover Books on Mathematics)
Introductory Complex Analysis is a scaled-down version of A. I. Markushevich's masterly three-volume "Theory of Functions of a Complex Variable." Dr. Richard Silverman, the editor and translator of the original, has prepared this shorter version expressly to meet the needs of a one-year graduate or undergraduate course in complex analysis. In his selection and adaptation of the more elementary topics from the original larger work, he was guided by a brief course prepared by Markushevich himself. The book begins with fundamentals, with a definition of complex numbers, their geometric representation, their algebra, powers and roots of complex numbers, set theory as applied to complex analysis, and complex functions and sequences. The notions of proper and improper complex numbers and of infinity are fully and clearly explained, as is stereographic projection. Individual chapters then cover limits and continuity, differentiation of analytic functions, polynomials and rational functions, Mobius transformations with their circle-preserving property, exponentials and logarithms, complex integrals and the Cauchy theorem , complex series and uniform convergence, power series, Laurent series and singular points, the residue theorem and its implications, harmonic functions (a subject too often slighted in first courses in complex analysis), partial fraction expansions, conformal mapping, and analytic continuation. Elementary functions are given a more detailed treatment than is usual for a book at this level. Also, there is an extended discussion of the Schwarz-Christolfel transformation, which is particularly important for applications. There is a great abundance of worked-out examples, and over three hundred problems (some with hints and answers), making this an excellent textbook for classroom use as well as for independent study. A noteworthy feature is the fact that the parentage of this volume makes it possible for the student to pursue various advanced topics in more detail in the three-volume original, without the problem of having to adjust to a new terminology and notation . In this way, IntroductoryComplex Analysis serves as an introduction not only to the whole field of complex analysis, but also to the magnum opus of an important contemporary Russian mathematician | 677.169 | 1 |
Saxon Math Course 3: Graphing Calculator Activities
The activities in this book are designed for whole-class instruction that leads into individual and group practice. However students could do most of each activity individually, following the demonstration steps and completing the practice problems. | 677.169 | 1 |
Über dieses Buch
Beschreibung:
Written with a strong pedagogical focus, this second edition of the book continues to provide an exhaustive presentation of the fundamental concepts of discrete mathematical structures and their applications in computer science and mathematics. It aims to develop the ability of the students to apply mathematical thought in order to solve computation-related problems. The book is intended not only for the undergraduate and postgraduate students of mathematics but also, most importantly, for the students of Computer Science & Engineering and Computer Applications. The introductory chapter presents an overview of the foundations of the subject, consisting of topics such as logic, set theory, relations, functions, algebraic structures, and graphs. The subsequent chapters provide detailed coverage of each of these topics as well as major areas of discrete mathematics such as combinatorics, lattices and Boolean algebras. Major applications such as computer models and computation, coding theory, cryptography and databases are dealt with in the final chapters of the book. In addition to this, a new chapter on matrices is included in this edition of the book, which forms a part of MCA course curriculum. The book is replete with features which enable the building of a firm foundation of the underlying principles of the subject and also provide adequate scope for testing the comprehension acquired by the students. Each chapter contains numerous worked-out examples within the main discussion as well as several chapter-end Supplementary Examples for revision. The Self-Test and Exercises at the end of each chapter provide large numbers of objective type questions and problems respectively. Answers to objective type questions and hints to exercises are also provided. All these pedagogic features, together with thorough coverage of the subject matter, make this book a readable text for beginners as well as advanced learners of the subject. Contents: Preface ? Preface to the First Edition 1. Foundations 2. Predicate Calculus 3. Combinatorics 4. More on Sets 5. Relations and Functions 6. Recurrence Relations 7. Algebraic Structures 8. Lattices 9. Boolean Algebras 10. Graphs 11. Trees 12. Models of Computers and Computation 13. Additional Topics Printed Pages: 880. Buchnummer des Verkäufers 94646
Über diesen Titel:
Inhaltsangabe: The introductory chapter presents an overview of the foundations of the subject, including topics such as logic, set theory, relations, functions, algebraic structures, and graphs. The subsequent chapters provide detailed coverage of each of these topics, as well as major areas of discrete mathematics, such as combinatorics, lattices and Boolean algebras. Major applications such as computer models and computation, coding theory, cryptography and databases are dealt with in the final chapters. In addition, a new chapter on matrices has been included. The book includes features to help build a firm foundation in the principles of the subject and to provide adequate scope for testing comprehension. Each chapter contains numerous worked-out examples. There are several chapter-end Supplementary Examples for revision. The Self-Test and Exercises at the end of each chapter provide objective type questions and problems respectively. Answers to objective type questions and hints to exercises are also provided. The book is intended undergraduate and postgraduate students of mathematics and also for students of computer science and engineering, and computer applications.
About the Author:
N. Chandrasekaran, Ph.D., has been Professor of Mathematics at St. Joseph's College, Tiruchirappalli. Prof. Chandrasekaran has many years of experience in teaching mathematics at the undergraduate and postgraduate levels in reputed institutions. He is the co-author of Theory of Computer Science: Automata, Languages and Computation, published by PHI Learning, Delhi, India. M. Umaparvathi, M.Phil., has been Professor of Mathematics at Seethalakshmi Ramaswami College, Tiruchirappalli, India. Prof. Umaparvathi has a decade of experience in teaching mathematics at the undergraduate and postgraduate levels.
Buchbeschreibung Eurospan Jul 2015, 2015. Taschenbuch. Buchzustand: Neu. Neuware - Englisch. Artikel-Nr. 9788120350977 | 677.169 | 1 |
Math homework help logarithms
Please respond to the following: Determine whether or not matrix multiplication is commutative.In some educational institutions, pre-calculus is divided into separate algebra and trigonometry courses to provide a solid foundation for a calculus course.
Applied Math - Landlords and Logarithms | 24HourAnswers Blog
It is how many times we need to use 10 in a multiplication, to get our desired number.Considering the difficulty of the subject, students can turn in online for our Precalculus homework help in order to assess their knowledge and understanding of various topics.Support your answer using at least one (1) real-world example or scenario.
Studying with our professional online Precalculus tutors will introduce the notes of constructivism and effectiveness in the learning process.Integer Polynomial Factorization and Roots Multiplicity Questions View.Logarithms And Exponents Questions And Answers CAT Quant question from arithmetic in exponents and logarithms.These algebraic procedures are important in theory and in practice and are relevant in the study of calculus. This disc.If you post an essay for editing or other help, create a Google doc or similar.Walls-of-text are almost impossible to edit with any effectiveness.Multiplying and Dividing are all part of the same simple pattern.Algebra-equation.com offers helpful info on math homework solver for logarithms, math and elimination and other algebra subjects.
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Homework Help! We are doing Logarithms for pre-calculus
Homework with logarithms - Mathematics Stack Exchange
If you decide to answer any of these, please provide me with a.It reviews previously learned topics like trigonometry, introduces new topics like matrices and determinants, and prepares students for a formal course in calculus for the following year.The inventor of logarithms is John Napier, and was born in Scotland and followed mathematics as a hobby.
Example: Condense the following logarithmic equation into one logarithm.One technique for solving this equation is to multiply both sides by e x.
math homework solver for logarithms - Algebra-equation.com
Math Skills - Logarithms
A few of the many free sources of data that might be appropriate to use are: ht.In mathematics, the logarithm is the inverse operation to exponentiation.
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A REVIEW OF LOGARITHMS - S.O.S. Math
Read reviews, compare customer ratings, see screenshots, and learn more about Cymath - Math Problem Solver and Homework Help.Help for the Logarithm is available at assignmenthelp.net. All type of help like assignment help, project help, homework help are also available.DO NOT DO Graphing the Exponential Function. HOMEWORK: Solving Exponential Equations Handout Homework.Homework help algebra logarithms solver February 5, 2015 by Uncategorized.Demonstrate that you have already put in some thought and effort into solving the problem.Introduction to logarithms, applications, and computations using logarithms.
Module 7 Homework Assignment Use the traditional method of hypothesis testing to test the given claim about the means of two populations.
Homework - Mrs. Staples at Notre Dame College School
Logarithms And Exponents Questions And Answers
There is a total of 12 questions all pertaining to exponentials and logarithms.Choosing your topic and getting help within the Precalculus homework help is convenient for you.Discuss the steps of hypothesis testing (should be around 8steps).Choose one of the two options: 1.Create and post two functions, f(x).The area of a projected picture on a movie screen varies directly as the square of the distance from the projector to the screen.Pre-calculus is an interesting area of math for students because of its multi-purpose nature.Introduction to solving logarithms homework: In mathematics, the logarithm to a number to a given base is the exponent to which the base should be raised in order to.
While attending these online services, students will not just acquire additional knowledge, but they will also become more confident before their tests. | 677.169 | 1 |
Domain and Range 2 TEACHER NOTES
Transcription
1 Math Objectives Students will be able to identify the domain and range of a relation from its graph. Students will be able to write symbolic expressions to describe the domain and range of a relation. Students will be able to recognize that different relations can have the same domain or the same range. Students will create a graph with a given domain and range. Students will use appropriate tools strategically. (CCSS Mathematical Practice) Vocabulary domain range infinity discrete continuous relation function interval notation About the Lesson This lesson involves finding domain and range by changing the endpoints of a graph. As a result, students will: Manipulate given segments and make conjectures about the relationships between the lengths of the segments and the possibility of forming a triangle. Drag point P along the points in a scatter plot. From the ordered pairs, they will determine the domain and range. Determine the domain and range of several different relations by dragging a point along the graph. Compare domains and ranges that are discrete with those that are continuous over an interval. Sketch graphs given a domain and range. TI-Nspire Navigator System Send out the Domain_and_Range_2.tns file. Monitor student progress using Class Capture. Use Live Presenter to spotlight student answers. Activity Materials Tech Tips: This activity includes screen captures taken from the TI-Nspire CX handheld. It is also appropriate for use with the TI-Nspire family of products including TI- Nspire App. Slight variations to these directions may be required if using other technologies besides the handheld. Watch for additional Tech Tips throughout the activity for the specific technology you are using. Access free tutorials at Nspire software and TI- lators/pd/us/online- Learning/Tutorials Lesson Materials: Student Activity Domain_and_Range_2 _Student.pdf Domain_and_Range_2 _Student.doc TI-Nspire document Domain_and_Range_2.tns Compatible TI Technologies: TI-Nspire CX Handhelds, TI-Nspire Apps for ipad, TI-Nspire Software 2015 Texas Instruments Incorporated 1 education.ti.com
2 Discussion Points and Possible Answers TI-Nspire Navigator Opportunity: Class Capture or Live Presenter See Note 1 at the end of this lesson. Tech Tip: If students experience difficulty dragging a point, check to make sure that they have moved the cursor (arrow) until it becomes a hand ( ) getting ready to grab the point. Also, be sure that the word point appears. Then press / x to grab point P and close the hand ({). When finished moving the point, press d to release the point. Teacher Tip: Students should not begin dragging point P until they have recorded the answer to part a in question 1. If they do, the can click on the reset slider and then drag the point. For pages , there is a reset slider to remove the tracing of the domain and range. Move to page Grab and move point P to each point on the scatter plot and note the changes. a. Name the ordered pair for point P. What is the relationship between this ordered pair and the cross mark on each axis? Answer: Answers may vary for the ordered pair for point P. The cross mark on the x-axis is at the x-coordinate of the ordered pair. If using the TI-Nspire CX handheld or the TI-Nspire ipad app, it is marked in blue. The cross mark on the y- axis is at the y-coordinate of the ordered pair. If using the TI-Nspire CX handheld or the TI-Nspire ipad app, it is marked in red Texas Instruments Incorporated 2 education.ti.com
3 b. Grab and drag point P to each of the circles on the scatter plot. As you move from point to point, record the coordinates in the table. Answer: The completed table is below. x y Teacher Tip: The order in which students place the ordered pairs in the table does not matter. c. State the domain and range of the relation. Answer: Domain: { 4, 0, 2, 3, 5}; Range: { 5, 3, 2, 3, 4} Teacher Tip: It is easier to specify the set if the domain and range are listed from least to greatest, but order is not critical. The values should not be repeated. d. Where on the graph do you see the domain represented? The range represented? Answer: The domain of the graph can be seen as the cross marks on the x-axis or as the highlighted blue section on the x-axis. The range can be seen as the cross marks on the y-axis or as the highlighted red section on the y-axis. Move to page Move point P back and forth along the entire graph. a. What does the highlighted portion along the x-axis represent? Answer: The highlighted portion along the x-axis represents the domain of the graph. If you are using the TI-Nspire CX handheld or the TI-Nspire ipad App, the highlighted portion will be blue Texas Instruments Incorporated 3 education.ti.com
4 b. What does the highlighted portion along the y-axis represent? Answer: The highlighted portion along the y-axis represents the range of the graph. If you are using the TI-Nspire CX or the Nspire ipad App, the highlighted portion will be red. c. How do the domain and range in this problem differ from the domain and range in question 1? Answer: The domain and range in this graph are continuous over an interval, whereas the domain and range in question 1 are discrete. d. State the domain as an inequality and in interval notation. Answer: Domain: 4 x 6; x [ 4, 6] Teacher Tip: There may be some confusion between interval notation and an ordered pair. To alleviate this confusion, you may want to use x, which means x is an element of, before the interval notation. e. State the range as an inequality and in interval notation. Answer: Range: 4 y 5; y [ 4, 5] Teacher Tip: There may be some confusion between interval notation and an ordered pair. To alleviate this confusion, you may want to use y, which means y is an element of, before the interval notation. f. If the endpoints of the graph were open circles, how would the domain and range change? State the new domain and range as inequalities and in interval notation. Answer: The domain would be between 4 and 6, not including 4 and 6. The range would be between 4 and 5, not including 4 and 5. Domain: 4 < x < 6 Range: 4 < y < 5 x ( 4, 6) y ( 4, 5) 2015 Texas Instruments Incorporated 4 education.ti.com
5 Teacher Tip: Emphasize what open and closed circles mean on a graph, in words, and in an inequality. The endpoint may or may not be part of the domain and range. In this problem, if the circles were open, the inequality symbols would be strictly less than, not less than or equal to. Move to page Move point P back and forth along the entire graph. a. State the domain and range of the graph. Answer: Domain: 1 x < 5 or x [1, 5) Range: {1, 2, 3, 4} Teacher Tip: As point P is being dragged, make sure that the hand remains close to point P. If the hand moves too far away from point P, the students will not be able to see the entire domain. Point out all of the representations that are being seen. b. How does the domain of this graph compare to the domain of the graph in question 1? Question 2? Answer: The domain of this graph is over an interval (continuous) with two defined endpoints, whereas the domain in question 1 is discrete. The domain in question 2 is like the domain in this graph. It is over an interval with two defined endpoints. However, both endpoints are included in the domain in question 2. c. How does the range of this graph compare to the range of the graph in question 1? Question 2? Answer: The range of this graph is discrete, like the range in question 1. The range in question 2 is over an interval (continuous) with two defined endpoints. Move to page Move point P back and forth along the entire graph. a. State the domain and range as inequalities and in interval notation. Answer: Domain: x > 4 or x ( 4, ) Range: y 0 or y (, 0] 2015 Texas Instruments Incorporated 5 education.ti.com
6 b. What is different about the domain and range of this graph compared to the others? Answer: The domain of this graph is over an interval (continuous) with one defined endpoint. The graph continues infinitely in the positive direction, whereas the domain in question 1 is discrete. The domain in question 2 is like the domain in this graph. It is over an interval. However, the graph in question 2 has two defined endpoints. The range of this graph is over an interval (continuous) with one defined endpoint. The graph continues infinitely in the negative direction, whereas the range in question 1 is discrete. The range in question 2 is like the range in this graph. It is over an interval. However, the graph in question 2 has two defined endpoints. Move to page Move point P back and forth along the entire graph. a. State the domain and range as inequalities and in interval notation. Answer: Domain: < x < or x (, ) Range: y > 0 or y (0, ) b. What is different about the domain and range of this graph compared to the others? Answer: The domain of this graph is the set of all real numbers. The graph continues infinitely in both directions, whereas the domain in question 1 is discrete. The domain in question 2 is like the domain in this graph. It is over an interval. However, the graph in question 2 has two defined endpoints. The range of this graph is the set of all real numbers greater than 0. The graph continues infinitely in the positive direction, whereas the range in question 1 is discrete. The domain in question 2 is like the range in this graph. It is over an interval. However, the graph in question 2 has two defined endpoints Texas Instruments Incorporated 6 education.ti.com
7 Move to page Grab and move the endpoints of the line segment to new locations. a. What do you notice about the domain and range as you drag the endpoints? Answer: As you move the endpoints of the segment, the domain and range change as well to reflect the new domain and range. TI-Nspire Navigator Opportunity: Class Capture or Live Presenter See Note 2 at the end of this lesson. b. Move the open endpoint to ( 1, 4) and the closed endpoint to (2, 3). State the domain and range as inequalities and in interval notation. Answer: Domain: 1 < x 2 or x ( 1, 2] Move to page 6.2. Range: 3 y < 4 or y [ 3, 4) 7. Grab and move the endpoints of the line segment to satisfy each of the following conditions: a. The open endpoint is ( 4, 3) and the closed endpoint is (3, 4). State the domain and range using inequalities and interval notation Answer: Domain: 4 < x 3 or x ( 4, 3] Range: 3 < y 4 or y ( 3, 4] TI-Nspire Navigator Opportunity: Quick Poll See Note 3 at the end of this lesson Texas Instruments Incorporated 7 education.ti.com
8 b. The domain is [ 7, 7) and the range is ( 4, 5]. Write the domain and range as inequalities. State the endpoints of the line segment and indicate which endpoint is open. Answer: Domain: 7 x < 7 Range: 4 < y 5 The endpoints are ( 7, 5) and (7, 4). The open endpoint is (7, 4). Teacher Tip: Students may have difficulty naming the correct endpoints. You may need to redirect their attention to the brackets and parentheses in the domain and range to ensure the correct endpoints are named. c. Given a domain of 3 x < 8 and a range of y = 2, state the endpoints of the line segment and indicate which endpoint is open. Answer: The endpoints are ( 3, 2) and (8, 2). The open endpoint is (8, 2). d. The domain is x = 5. State the endpoints of the line segment and indicate which endpoint is open. Compare your answers with another student s. Explain how you can have different answers and still both be correct. Answer: Answers may vary, and either endpoint may be open. However, all ordered pairs should contain 5 as the x-coordinate. 8. Sketch a graph and write a description of a function or relation that satisfies the following conditions. Compare your answers with another student s. How are they alike or different? a. Domain: 3 < x 4; Range: 1 y < 5 Possible Answer: See example at right. b. Domain: (, ); Range: [2, ) Possible Answer: See example at right Texas Instruments Incorporated 8 education.ti.com
9 Wrap Up Upon completion of the discussion, the teacher should ensure that students understand: How to determine the domain and range of a relation from its graph. The connection between graphical and numerical representations. How to read and write symbolic expressions to describe the domain and range of a relation. How to sketch a graph with a given domain and range. TI-Nspire Navigator Note 1 Entire Document, Class Capture or Live Presenter: If students experience difficulty with dragging the point or understanding the results, use Class Capture to show the solution to the students. Highlight the bold areas on the axes that represent the domain and range. Note 2 Question 6a, Class Capture or Live Presenter: Use this opportunity to show different students graphs and point out how the domain and range are affected by changing the endpoints of the segment. Note 3 Question 7a, Quick Poll (Multiple Choice or Open Response): Use Quick Poll as an opportunity to assess students understanding of domain and range for a function. Have students identify the domain and range for this function. As a follow-up question, ask students what kind of line will have a domain of x = Texas Instruments Incorporated 9 education.ti.com
These Teacher Notes were enhanced to demonstrate how to integrate the Eight Essential Mathematics Teaching Practices into a lesson, with specific examples for when and how to address students questions
Math Objectives Students will describe the relationship between a central angle, the radius, and the arc length of a circle. Students will recognize that changing the radius of a circle does not affect
Math Objectives Students will discover that the zeros of the linear factors are the zeros of the polynomial function. Students will discover that the real zeros of a polynomial function are the zeros of
Math Objectives Students will identify a rotation as an isometry, also called a congruence transformation. Students will identify which properties (side length, angle measure, perimeter, area, and orientation)
Math Objectives Students will understand the role of the values of a and n in the equation r = asin(n). Students will be able to predict the number of petals and their length by examining the polar equation.
Math Objectives For power functions of the form f(x)=x n, where n is a positive integer and the domain is all real numbers, students will be able to identify which functions are invertible (odd powers)
Math Objectives Students will investigate and state the relationship between the altitude to the hypotenuse and the two segments of the hypotenuse formed by this altitude. Students will write a conditional
Math Objectives Students will recognize that the sum of two central angles that combine to make a circle is 360. Students will recognize that the ratio of the central angle to 360 determines the length
Math Objectives Students will identify the values of cotangent, secant and cosecant for various values of θ from the unit circle. Students will understand the correspondence of the graphs of cotangent,
Math Objectives Students will describe the overall pattern of a distribution by its shape, center, and spread. Students will recognize and identify properties of distributions such as symmetry, skewness,
Math Objectives Students will use symbols to represent unknowns and variables. Students will look for patterns and represent generalizations Students will represent relationships among quantities using
Math Objectives Students will represent data on a scatter plot, and they will analyze the relationship between the variables. Students will fit a linear function to a scatter plot and analyze the fit by
Math Objectives Students will describe the relationship between the unit circle and the sine and cosine functions. Students will describe the shape of the sine and cosine curves after unwrapping the unit
Math Objectives Students will identify corresponding parts of congruent triangles. Students will use appropriate notation to describe two congruent triangles. Students will reason abstractly and quantitatively
Science Objectives Students will learn how certain properties of the elements tend toward a periodic similarity when the elements are arranged in order of increasing atomic number. Students will note the
Science Objectives Students will discover that atoms are made up of particles and that the number of particles in an atom affects its properties. Students will learn that these particles protons, neutrons,
Math Objectives Students will represent bivariate categorical data with bar charts and split bar charts. Students will examine and describe patterns in bivariate categorical data by displaying frequencies
Math Objectives Students will recognize that a p-value only has meaning if the null hypothesis is true (a conditional probability). Students will interpret a p-value in given contexts as the relative frequency
These Teacher Notes were enhanced to demonstrate how to integrate the Eight Essential Mathematics Teaching Practices into a lesson, with specific examples for when and how to address students questions
Math Objectives Students will understand that normal distributions can be used to approximate binomial distributions whenever both np and n(1 p) are sufficiently large. Students will understand that when
Math Objectives Students will explain how a change in only the base (or the height) of a rectangle affects its perimeter. Students will explain how a change in only the base (or the height) of a rectangle
Math Objectives Students will be able to predict how a specific change in the value of a will affect the shape of the graph of the quadratic ax bxc. Students will be able to predict how a specific change
Math Objectives Students will describe power as the relative frequency of reject the null conclusions for a given hypothesis test. Students will recognize that when the null hypothesis is true, power is
Math Objectives Students will use a moveable line to find a line of fit for a data set. Students will interpret the slope and y-intercept of the equation of their line of fit. Students will make predictions
Science Objectives Students will observe what happens during a titration of a strong acid with a strong base, using a simulation accompanied by a molecular view and ph graph. Students will determine the
About the Lesson Students will graph a figure in the coordinate plane. They will use list operations to perform reflections, rotations, translations, and dilations on the figure, and graph the resulting
Math Objectives Students will use an interactive area model to multiply and divide positive decimals and mixed numbers. Students will use the distributive and commutative properties to multiply decimals
Science Objectives Students will observe the electron configuration of atoms and discover the relationship between the electron configuration and the position of an element in the Periodic Table. Students
Math Objectives Students will understand that a system of two linear equations in two variables can have one solution, no solution, or infinitely many solutions. Students will understand the connection
Math Objectives Students will explore and predict patterns found in repeating decimals for select fractions. Students will determine the repeating digit (or group of digits) for a repeating decimal. Students
Math Objectives Students will be able to describe the effects of the coefficients of a linear function in three variables on the graph of the function. Students will be able to identify the number of solutions
Science Objectives Students will learn that when isotopes of different atoms change chemically (in a decomposition reaction) and emit radioactivity in the process. Students will learn how the known rate
Math Objectives Students will use algebra tiles to build a geometric model of a perfect square quadratic. Students will recognize the characteristics in the algebraic expression of a perfect square quadratic.
Math Objectives Students will interpret the variables in the formula for compound interest. Students will use the formula for compound interest and understand the effects of changes in the interest rate
Math Objectives Students will understand how the Law of Sines is derived. Students will understand when the Law of Sines can be used to find missing sides and angles in a triangle. Students will be able
Math Objectives Students will investigate that if two chords of a circle intersect, the product of the segment lengths of one chord equals the product of the segment lengths of the other chord. Students
Math Objectives Students will be able to state the Law of Cosines Students will be able to apply the Law of Cosines to find missing sides and angles in a triangle Students will understand why the Law of
Math Objectives Students will interpret a confidence level as the degree of confidence in the process used to produce an interval that will contain the true mean of the population. Students will recognize
Math Objectives Students will compute unit rates. Students will find linear equations using unit rates to represent proportional relationships. Students will examine ordered pairs to confirm that linear
Math Objectives Students will show why certain segments or lines in triangles are medians, angle bisectors, altitudes, or perpendicular bisectors. Students will interpret what properties these particular
Math Objectives Students will understand the distinction between growth rate (slope) and relative growth rate (slope/y-value) of a function. Students will understand that the relative growth rate f (x)/f(x)
Math Objectives Students will discover how the coefficients of a given trinomial x + bx + c affect the factors. Students will discover how the coefficients of a given trinomial ax + bx + c affect the factors.
Science Objectives Students will describe electric current, voltage, and resistance and the units of measurements for these physical quantities. Students will construct series and parallel circuits. Students
Science Objectives Students will observe the interaction between an enzyme and its substrate. Students will observe the effect that enzymes have on chemical reactions. Students will explore the effects
About the Lesson Students use formulas to find the differences of the consecutive terms, plot a scatter plot of each sequence, and determine that sequences with common differences (called arithmetic sequences)
About the Lesson In this activity, students will first investigate linear equations that form a triangle. They will determine which vertex is a solution to a system of equations. Students will also investigate
Math Objectives Students will identify parallel lines and planes. Students will identify perpendicular lines and planes. Students will recognize skew lines. Students will discuss the intersection of planes.
Math Objectives Students will know the definitions of and identify central angles, major and minor arcs, intercepted arcs, and inscribed angles of a circle. Students will determine and apply the following
Science Objectives Students will describe the variables that affect potential energy and kinetic energy Students will explore the conversion of potential energy to kinetic energy Students will describe
About the Lesson In this activity, students are introduced to modeling linear data through an investigation of comparing grams of fat and Calories in fast food hamburgers. Students will use multiple representations
Math Objectives Generalize a variety of patterns involving side lengths and angle measurements to draw conclusions about the properties of a parallelogram. Determine additional properties of rectangles,
Math Objectives Students will explore constant, linear, quadratic, and cubic functions. The functions will be modeled from numerical data that they generate by thinking of an n n n cube being dipped in
Science Objectives Students will identify the changes that occur in cells during each phase of mitosis. Students will correlate these changes to the duration of time cells spend in each phase. While lookingMath Objectives Students will understand that if the intersection point P of two lines lies inside a circle, then the measure of the angle formed by the two secants is equal to the average of the measures
Math Objectives Students will be able to describe how the area of a parallelogram relates to the area of a rectangle with the same base and height. Students will be able to describe how the area of a triangle
About the Lesson In this activity, students explore transformations and dilations with a two-dimensional fish figure. As a results students will Describe the effects of a transformation and dilation on
Math Objectives Students will capture data to explore the relationship between the circumference and diameter of a circle. Students will use a scatter plot and a movable line to find an algebraic model
Math Objectives Students will use visual representations of fractions to estimate the area of irregular shapes. Students will use arithmetic operations with mixed numbers and decimals in a real-life context.
Science Objectives Students will know the names and functions of the major organelles found in eukaryotic cells. Students will be able to contrast different cell types (plant and animal cells), based on
Science Objectives Students will explore the relationship between a vector and its components. Students will explore how two vectors may be combined to produce a resultant vector. Students will be able
Science Objectives Students will identify patterns in data associated with the lunar phases. Students will describe how the relative positions of the Earth, the Moon, and the Sun cause lunar phases. Students
About the Lesson Students will explore a net representation for a right cylinder. The surface area will be developed from the parts of the net. As a result, students will: Construct 3-dimensional cylinders
Math Objectives Students will construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Students will informally explore the role
About the Lesson In this activity, students will investigate the total cost of a private party at each of three restaurants. Each restaurant states its pricing information differently, but all charge a
About the Lesson In this activity, students will learn to identify the domain and range of various real-world step functions. They will graphically explore numerical data points and observe step functions.
Science Objectives Students will describe and identify the survival needs for both plants and animals. Students will simulate what will happen to a population when limiting factors in its environment are
About the Lesson Students will explore the ratio of perimeter and area in twodimensional figures. As a result, students will: Find ratio of perimeter and area. Vocabulary ratio perimeter area Teacher Preparation
Lesson Overview This TI-Nspire lesson uses interactive number lines to investigate addition and subtraction of fractions that refer to the same whole (i.e. have the same denominator). Students may reason
Science Objectives Students will identify the number of significant figures in a numeric value. Students will learn how accuracy of measurement relates to the number of significant figures in a numeric
Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of
Math Objectives Students will be able to identify a restricted domain interval and use function translations and dilations to choose and position a portion of the graph accurately in the plane to match
Math Objectives Students will be able to solve a system of three equations with three unknowns using the elimination method. Students will attend to precision (CCSS Mathematical Practice). Vocabulary system
About the Lesson In this activity, students use the Cabri Jr App to create and manipulate dynamic sketches of triangles to explore and conjecture many triangle properties. As a result, students will: Discover
Science Objectives Students will understand that the chemical formula identifies each constituent element by its chemical symbol. Students will understand that a subscript after the chemical symbol indicates
About the Lesson In this activity, students will explore distances in the coordinate plane. Students will substitute the coordinates of a segment s endpoints into the distance formula and compare the results
Math Objectives Students will be able to classify expressions as the sum of cubes, difference of cubes, or neither. Students will be able to make connections between the graph of a cubic polynomial, in
Math Objectives Students will understand that a family of t-distributions is determined by the number of degrees of freedom. Students will recognize that a t-distribution with a small number of degrees
Math Objectives Students will be able to identify situations in which stratification might reduce variation. Students will recognize that some stratification designs are useful in reducing variation and
Lesson Overview This TI-Nspire lesson uses unit squares to help students investigate addition and subtraction of fractions with unlike denominators. Students should recall that only fractions with a common
Lesson Overview This TI-Nspire lesson uses a unit square to explore division of a unit fraction and a fraction in general by a whole number. The concept of dividing a quantity by a whole number, n, can
TI Nspire Introduction to the Absolute Value Function Students will explore the properties of the absolute value function via its definition. Michele Leschik 10/17/2008 Introduction to the Absolute Value
Math Objectives Students will identify and know the difference between central angles and inscribed angles of a circle. Students will identify the relationships between the measures of inscribed angles
Science Objectives Students will observe the affect of moving an object toward and away from a concave mirror. Students will graph the relationship between the object distance and the image distance from
About the Lesson In this activity, students will use the Transformational Graphing Application to stretch and translate the parabola given by y = x. As a result, students will: Determine the effects of
Science Objectives Students will develop a deeper understanding of the variables affecting the rate of photosynthesis in plants. Students will manipulate variables, such as light intensity and wavelength
About the Lesson In this activity, students will explore how the measures of angles constructed in a circle are related to the measures of the intercepted arcs. Beginning with central and inscribed angles,
Scavenger Hunt: What can you do with a TI-Nspire CX Handheld? Getting to Know the Keypad Draw arrows from these descriptions to the picture of the TI-NSpire CX shown at right. 1. The most important keys:
About the Lesson In this activity, students will explore multiplication and division of fractions using several representations to develop fluency. Students will also solve a problem involving similarity
Math Objectives Given functions f and g, the student will be able to determine the domain and range of each as well as the composite functions defined by f ( g( x )) and g( f ( x )). Students will interpret
Graphing Eponential Functions Math Objectives Students will infer why the conditions b0 and b 1 are necessary for the function to be eponential. Students will determine that for b 1 the function is increasing
Math Objectives Students will discover the determinant method for finding the areas of triangles and quadrilaterals. Vocabulary determinant scalar multiple convex/concave polygons About the Lesson This
Math Objectives Students will determine and analyze an exponential model for the data involving carbon dioxide in the atmosphere. Students will develop an improved model for this data by using the residuals | 677.169 | 1 |
Are you looking for North Carolina 7th Grade Math Test Prep Books? You can Download and Read OnlineNorth Carolina 7th Grade Math Test Prep Book for Free.
You can see the list of related books. Click on Download or Read Online button to get the full book.
Our 1st edition North Carolina 7 in each section. The material covered includes "understanding ratio concepts and using ratio reasoning to solve problems; applying and extending previous understandings of multiplication and division to divide fractions by fractions; computing fluently with multi-digit numbers and find common factors and multiples; applying and extending previous understandings of numbers to the system of rational numbers; applying and extending previous understandings of arithmetic to algebraic expressions; reasoning about and solve one-variable equations and inequalities; representing and analyzing quantitative relationships between dependent and independent variables; solving real-world and mathematical problems involving area, surface area, and volume; developing understanding of statistical variability; and summarizing and describing distributions." These standards are covered extensively by the practice problems. This book contains over 850 practice problems aligned to each Common Core State Standard. In addition the book contains an answer key to practice problems. Paperback: 317 double-sided pages Publisher: Teachers' Treasures, Inc. Language: English
Our 2nd edition North Carolina 5 and open questions in each section. The material covered includes emphasis on representing and solving problems by developing fluency with addition and subtraction of fractions, and developing understanding of the multiplication of fractions and of division of fractions in limited cases (unit fractions divided by whole numbers and whole numbers divided by unit fractions); extending division to 2-digit divisors, integrating decimal fractions into the place value system and developing understanding of operations with decimals to hundredths, and developing fluency with whole number and decimal operations; and developing understanding of volume. These standards are covered extensively by the practice problems. This book contains over 600 practice problems aligned to each Common Core State Standard. In addition the book contains an answer key to practice problems. Paperback: 226 double-sided pages Publisher: Teachers' Treasures, Inc. Language: English
Preparation for North Carolina's READY EOG Mathematics Assessments for 2015-2016! This book will develop the mathematics skills that students need, while preparing students for North Carolina's READY End-of-Grade Mathematics assessments. It provides ongoing test preparation and practice, while being focused on developing Common Core mathematics skills. listed in the Common Core State Standards - Practice sets can be used for testing, revision, and retesting Preparation for the READY End-of-Grade Mathematics Tests - Develops all the skills needed for the READY End-of-Grade tests - Provides extensive practice completing assessment questions - Rigorous questions and advanced tasks encourage in-depth understanding - Students gain experience applying mathematics skills to solve problems - Full answer key lists the Common Core skill assessed by each question Key Benefits - Develops all the mathematics skills that students need - Encourages a thorough understanding of mathematics concepts - Enhances problem-solving ability and helps students apply skills - Prepares students for the higher difficulty of the assessments - Builds confidence by helping students prepare before taking the real test - Reduces test anxiety by allowing low-stress practice - Detailed answer key allows missing skills to be identified - 60-minute practice sets allow for ongoing test preparationPreparation for North Carolina's READY End-of-Grade Assessments for 2015-2016! This practice test book is the perfect preparation tool for the READY End-of-Grade Assessments for English Language Arts/Reading. It includes six mini-tests to introduce students to the test, as well as two complete full-length practice tests. Complete Preparation for the READY English Language Arts/Reading Assessments - Begins with 6 short mini-tests to give students practice answering test questions - Mini-tests develop key Common Core skills, including close reading, using evidence from texts, and analyzing and evaluating texts - Mini-tests provide practice responding to literary and informational texts - Two complete practice tests allow students to take a test just like the actual READY ELA/Reading test - Practice tests cover all the skills assessed on the real state tests - Full answer key lists the Common Core standard assessed by each question Key Benefits of this Workbook - Builds confidence by helping students prepare before taking the real test - Develops all the English Language Arts and reading skills needed - Provides experience answering reading comprehension questions - Helps students know what to expect when taking the real tests - Reduces test anxiety by allowing low-stress practice - Detailed answer key allows missing skills to be identified expectations for their grade level. | 677.169 | 1 |
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Quantitative Skills for Edexcel A Economics
For top marks in Economics, students must get to grips with the maths. Guide your students to success with this comprehensive pack of theory notes, worked examples and practice questions to use in class or as homework. Perfectly matched to the QS annex – all the tools to tackle any calculation!
Essential exam practice
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Answers and A5 booklet included
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"The explanations and examples were very clear and would make it easy for someone to understand even if they though they were not that good at maths."
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Price is £54 (AS Only) / £69 (AS and A Level) — Includes photocopiable master with site licence; electronic copies available. Orders are normally dispatched in 2-5 working days. Click here for full details and to order. | 677.169 | 1 |
Download and read online Elementary Statistics in PDF and EPUB For algebra-based introductory statistics courses Statistics opens a window to the modern world, and this market-leading text makes it easy to understand Elementary Statistics: Picturing the World, Sixth Edition, presents a better teaching and learning experience-for you and your students. Here's how: *Personalized learning with MyStatLab(TM): the accompanying MyStatLab online course provides users with countless opportunities to practice, plus statistics-specific resources and tools that enhance their experience and comprehension.*Stepped out instruction and guided student learning through an abundance of exercises build students' knowledge and skills in statistics.Included are visual descriptions, unique chapter overviews and summaries, clear learning objectives, boxed definitions and formulas, margin features, and more. *Real-life and interactive features present statistics at work in the real world and show how statistics is relevant to students' lives. *Examples and exercises tie the material to students' lives.
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Download and read online Statistical Reasoning for Everyday Life in PDF and EPUB This is the eBook of the printed book and may not include any media, website access codes, or print supplements that may come packaged with the bound book. Statistical Reasoning for Everyday Life, Fourth Edition, provides students with a clear understanding of statistical concepts and ideas so they can become better critical thinkers and decision makers, whether they decide to start a business, plan for their financial future, or just watch the news. The authors bring statistics to life by applying statistical concepts to the real world situations, taken from news sources, the internet, and individual experiences. Note: This is the standalone book If you want the Book/Access Card you can order the ISBN below NOTE: Make sure to use the dashes shown on the Access Card Code when entering the code. Student can use the URL and phone number below to help answer their questions: 800-677-6337 0321890132 / 9780321890139 Statistical Reasoning for Everyday Life Plus NEW MyStatLab with Pearson eText -- Access Card Package 4/e Package consists of: 0321817621 / 9780321817624 Statistical Reasoning for Everyday Life Intro Stats Richard De Veaux, Paul Velleman, and David Bock wrote Intro Stats with the goal that you have as much fun reading it as they did in writing it. Maintaining a conversational, humorous, and informal writing style, this new edition engages readers from the first page. The authors focus on statistical thinking throughout the text and rely on technology for calculations. As a result, students can focus on developing their conceptual understanding. Innovative Think/Show/Tell examples provide a problem-solving framework and, more importantly, a way to think through any statistics problem and present their results. New to the Fourth Edition is a streamlined presentation that keeps students focused on what's most important, while including out helpful features. An updated organization divides chapters into sections, with specific learning objectives to keep students on track. A detailed table of contents assists with navigation through this new layout. Single-concept exercises complement the existing mid- to hard-level exercises for basic skill development Elementary Statistics Using Excel Plus NEW | 677.169 | 1 |
Abstract
In recent years, new kinds of calculators have begun to appear in secondary schools, especially in mathematics departments. These calculators are commonly called 'graphics' calculators (or some variation on the adjective), because they have a small graphics display screen rather than the one-line numerical display that we have been used to with hand-held calculators. The appearance of these devices in schools has been hastened considerably by the official approval of their use in the TEE from the 1998 examinations in all mathematics subjects and also in some science subjects. It would be a mistake, however, to make the inference that these devices are mainly useful for examinations. Rather, the official TEE sanction is an acknowledgment that examination prohibition hampers the use of appropriate technology in schools.
Although they are relatively new to WA, having first appeared in class-set quantities in schools about six or seven years ago, graphics calculators can hardly be described as a 'new' technology. For example, they have been on sale to the general public in USA since 1985, and have been in widespread use in high schools for a decade. Similarly, they have been available in schools and used in A-level examinations in the UK since the late 1980's. The machines on sale today are arguably the third generation of graphics calculators. There are four manufacturers at present, each a familiar multinational company. In alphabetical order, they are Casio, Hewlett Packard, Sharp and Texas Instruments.
The main uses of graphics calculators are for students learning and doing mathematics. However, in Western Australia, it is almost always the same students learning both science and mathematics. So, we can expect that almost all science students in upper school will have an increasing level of access to a graphics calculator in the near future, and that many students will own their own model. Even in the lower secondary school, the most likely scenario for the near future is that a graphics calculator will become standard equipment for most students, instead of the scientific calculator. Although many schools continue to ask students to purchase a scientific calculator early in secondary school, the decreasing price of graphics calculators (the least expensive now around $60 tax exempt) are likely to encourage a rethink of this practice, to reduce the risk that students will need to purchase more than one calculator over their secondary schooling years.
This paper suggests some graphics calculator capabilities that may be of interest to science teachers, with the emphasis on student learning rather than assessment. | 677.169 | 1 |
Algebra 1: Arithmetic vs Geometric Sequences Spring 2013 (Editable)
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I like to have my Algebra 1 students do both arithmetic and geometric sequences on the same day, and I wrote this worksheet to help them with this concept.
In the case of arithmetic sequences with unknown terms between two given terms, I have them express the first "unknown" term as u1 + d, the second unknown term as u1 + 2d, and so on. They can then set up a linear equation and solve it for d.
Likewise, with geometric sequences, I have them express the first unknown term as u1*r, the second as u1*r2, and so on. They enjoy working with the nth root button on their calculator.
This is intended for regular Algebra 1 students.
Students write directly on this worksheet.
Please download the pdf preview file first, so you can see exactly what's included; the product file is a word document, which you may edit for your students. | 677.169 | 1 |
Students will develop essential core mathematical skills necessary to support electrical and electronics principles and skills upon which these principles are based. It provide broad introduction to the fundamental mathematical operations such as: Indices and Logarithms, Trigonometric functions, Exponential and Logarithmic Functions, Complex number, elementary Boolean algebra needed by electrical, electronics and computer systems programs.
This learning unit is one of a group of units designed to collectively meet underpinning skill & applied knowledge essential for developing the following Core Competency –
UTE NES 008A – Provide technical leadership in the workplace
Which is contained in the National Electrotechnology Training Package UTE99
• attendance and satisfactory completion of prescribed practical exercises , which may be scheduled during the day or evening depending on the demand. • evidence of participation in and satisfactory completion of work simulation projects. • satisfactory completion of class assignment work • progressive tests • timely submission and standard presentation for all assessment material / documentation | 677.169 | 1 |
The goal of the course is the contact with modern and interesting applications
of mathematical theories.
1. Wavelets They are recently used in data compression technology (image processing,
etc.):
"Applications of wavelet theory continues to grow rapidly. Engineers,
working in everything from mathematics and physics to digital signal processing,
image compression, and speech and pattern recognition, need to understand
this exciting subject".
2. Quantum physics "Low and high energy phenomena" … (too many to list!).
( It is a different way of thinking
about "reality" worth knowing!)
3. Quantum computing The parallel computing is an old subject. The modern one is "infinitely
many parallel computing paths". It is a new emerging domain of CS, promising
to give us a clue on how we think and what conscience is!
We will learn the mathematical tools needed to be able to address these
applications: Hilbert space theory, Fourier transform and wavelets, linear
operators, generalized functions and quantum mechanics, quantum bits, Q-gates
and Q-copying.
Prerequisites Basic knowledge in linear algebra (matrices, determinants, vector spaces)
and calculus is required. An intuitive understanding of Euclidean R3 will
help with Hilbert spaces, a calculus level understanding of differential
equations would suffice and some real analysis would be helpful.
The approach will be rigorous, without being hindered by very technical
issues of analysis, focusing on the conceptual part of the mathematics,
physics or CS involved.
Optional exercises are included as a way to feel we understand the
formalism, being meant to give a feed-back on the hidden difficulties involved.
Format I will start presenting the first couple of lectures. Meanwhile students
will decide what to present and when, oriented on a subject, or a part
of the textbook, or a solution to a problem. | 677.169 | 1 |
There are 7 expectations that describe the mathematical
processes that students need to learn and apply as they work to achieve the
expectations of this course. These include expectations include:
Problem
Solving
Reasoning
and Proving
Reflecting
Selection
of Tools and Strategies
Connecting
Representing
Communicating
Student work will be selected to help students develop
proficiency with these expectations and thus, one component of assessing
student work will be the demonstration of the pertinent expectation(s)
The student mark will be determined according to the
following breakdown: | 677.169 | 1 |
General Advice
GENERAL ADVICE
Every question you have about the subject should be asked, and we will be happy to try to answer it. Don't let your questions sit around gathering dust, because later portions of the course will make use of what came earlier. The longer you wait, the harder it is to catch up. You learn the subject by solving problems. But we'd like you to do more than get correct answers to the problems. We want you to learn to write mathematics clearly. Remember that you are writing in English about mathematics, and your purpose is to present a readable (as opposed to merely legible) document. Therefore, your mathematical writing should be more than a sequence of calculations, but should consist of complete sentences, in which are embedded the mathematical expressions. (A good way to test whether you have produced a readable document is to try to read it aloud without adding words that are not on the page. If it says what you want to say, you're ok.)In grading exams in the past, we have thought that students need to understand better what we expect of them. Below are some notes that could be useful for students.
In an exam problem, we do not give you a selection of five possible answers and ask you to find the correct answer. We give you a problem, and we ask you to provide a *solution*. Some problems are explicitly "show-that" problems: they ask for an argument, an explanation, a proof. Other problems may simply ask for a computation. But even such problems should be understood as "show-that" problems. We do not want to see an answer alone; we want to see the steps of the computation that lead to the answer. If correctly written, these steps constitute a proof that the answer is correct.
Correct answers given with no justification may receive no credit.
Correct answers with incorrect justification are only *accidentally* correct; they may receive no credit.
Incorrect answers given with some correct justification may receive partial credit.
However, in a long problem, it may happen that you receive no credit for any work done after your first mistake (if you make a mistake). Therefore you should be especially careful in the early steps of a problem.
Always check your work, if you have time. Some answers are easy to check. For example, to check the proposed solution to an equation, just substitute into the equation. To check an indefinite integral, just differentiate. In the grading of problems whose answers are easily checked, possibly no partial credit will be given.
Write your solutions in the conventional fashion: left to right, top to bottom. Otherwise, the reader cannot tell how to read what you have written, and you may lose credit. Remember, we graders do not have the benefit of watching you write (or of having you stand by to explain what you have written); all we have is the finished product of you writing, and this is what you are graded on.
We graders are not mind-readers. It is not our job to *figure out* what you might mean; it is your job to *say* what you mean, in the manner of expression established in lectures and textbooks.
It is possible to write down *too much* justification for your answers. How much is enough, and how much is too much? There is no clear answer; you just have to develop a feeling by reading books and observing and questioning instructors.
If your solution contains irrelevant information, then the grader may conclude that you do not understand the problem fully, and you may lose credit.
Write your symbols clearly. A "3" should not look like a "7"; a "t" should not look like a "+".
Use mathematical symbols correctly. For example, the double-shafted arrow "=>" is a logical symbol meaning "implies", so that "A => B" means "A implies B", that is, "If A, then B," that is, "A is false, or B is true." Do *not* use this arrow to mean "therefore" or "now read the following". | 677.169 | 1 |
Mathematics
Online Lectures:
Further Reading:
Mathematical concepts and theories underpin much of the physical sciences and engineering. Yet maths is a subject that many students find challenging, and even intimidating – despite it being so central to their field of study.
Mathematical Techniques provides a complete course in mathematics, covering all the essential topics with which a physical sciences or engineering student should be familiar.
By breaking the subject into small, modular chapters, the book introduces and builds on concepts in a progressive, carefully-layered way – always with an emphasis on how to use the power of maths to best effect, rather than on theoretical proofs of the maths presented.
With a huge array of end of chapter problems, and new self-check questions, the fourth edition of Mathematical Techniques provides extensive opportunities for students to build their confidence in the best way possible: by using the maths for themselves.
Online Resource Centre
The Online Resource Centre features the following materials for all users of the book:
· Figures from the book in electronic format, ready to download
· A downloadable solutions manual, featuring worked solutions to all end of chapter problems
· Mathematica-based programs, relating to the Projects featured at the end of the book
Engineering Mathematics is the best-selling introductory mathematics text for students on science and engineering degree and pre-degree courses. Sales of previous editions stand at more than half a million copies. It is suitable for classroom use and self-study.
Its unique programmed approach takes students through the mathematics they need in a step-by-step fashion with a wealth of examples and exercises. The book is divided into two sections with the Foundation section starting at Level 0 of the IEng syllabus and the main section extending over all elements of a first year undergraduate course and into many second year courses. The book therefore suits a full range of abilities and levels of access.
The Online Personal Tutor guides students through exercises in the same step-by-step fashion as the book, with hundreds of full workings to questions.
(Note that the first half of this book is largely identical to the same authors book: Foundation MathematicsPaperback– 21 Apr 2009, so there is no reason to buy both).
A unique technique-oriented approach takes the student through the mathematics in a highly accessible way
* Comprehensive coverage of all topics required by undergraduates at advanced levels of mathematics in engineering and science
* Hundreds of worked examples and progressively more challenging exercises
* Ideal either as part of a course or for self-study | 677.169 | 1 |
Control Tutorials for MATLAB and Simulink - Introduction
MATLAB Mathematical Analysis - hitebook.net
MATLAB Mathematical Analysis PDF: MATLAB Mathematical Analysis is a reference book that presents the techniques of mathematical analysis through examples and.
MATLAB Mathematical Analysis is a reference book that presents the techniques of mathematical analysis through examples and exercises resolved with MATLAB softw.Matlab Mathematical Analysis Book also available for Read Online, mobi, docx and mobile and.Learn how MATLAB can supplement the capabilities of Microsoft Excel by providing access to thousands of pre-built mathematical and advanced analysis functions.Southern Illinois University Carbondale OpenSIUC Research Papers Graduate School Fall 11-9-2011 Finite Element Analysis: Mathematical Theory and Applications.
Relationships between mathematical, MATLAB and Excel expressions.MATLAB Mathematical Analysis is a reference book that presents the techniques of mathematical analysis through examples and exercises resolved with MATLAB.In international academia, Matlab has already been accepted as an accurate and reliable standard computational soft ware.Learn More about Mathematical Modeling with MATLAB Products.MATLAB is an integrated environment for mathematical and technical computing.The purpose is to give you examples of the mathematical analysis functions offered by MATLAB so that you can use them in your daily.EBSCOhost serves thousands of libraries with premium essays, articles and other content including Applications of Matlab in Mathematical Analysis.
Matlab Mathematical Analysis Download - Only Books
MATLAB Mathematical Analysis offers a broad. results in the form of a rational. 5707963267948966192313216916398.5129i 13.Chapter 3 Complex.
Powerful and solid tool for the implementation of statistical and mathematical analysis.
MATLAB Answers - MATLAB Central - MathWorks
Examples and Exercises on Amazon.com FREE SHIPPING on qualified orders.We will send you an email each month containing links to all the book updates you subscribe to.
MATLAB Mathematical Analysis - ebooklink.info
What Is MATLAB?
MATLAB tutorial - Mathematical sciences
A novel software is presented to connect Abaqus, a sophisticated finite element package, with Matlab, the most comprehensive program for mathematical analysis.
MATLAB Mathematical Analysis [2014 - 1337x.unblocked.bid
I use Matlab for all of my mathematical modeling and design optimization.
Enabling Matlab with an Environment Variable (Windows
All rights are reserved by the Publisher, whether the whole.Recent versions of the MATLAB mathematical analysis software package have been experiencing issues finding the correct license file.To view the rest of this content please follow the download PDF link above.Buy MATLAB Mathematical Analysis on Amazon.com FREE SHIPPING on qualified orders. | 677.169 | 1 |
Calculus: Multivariable, 6th Edition continues the effort to promote courses in which understanding and computation reinforce each other. The 6th Edition reflects the many voices of users at research universities, four-year colleges, community colleges, and secondary schools. This new edition has been streamlined to create a flexible approach to both theory and modeling. For instructors wishing to emphasize the connection between calculus and other fields, the text includes a variety of problems and examples from the physical, health, and biological sciences, engineering and economics. In addition, new problems on the mathematics of sustainability and new case studies on calculus in medicine by David E. Sloane, MD have been added. WileyPLUS sold separately from text.
New Strengthen Your Understanding problems at the end of every
section. These problems ask students to reflect on what they have
learned by deciding "What is wrong?" with a statement
and to "Give an example" of an idea.
Updated Data and Models: For example, Section 11.7 follows the
current debate on Peak Oil Production, underscoring the importance
of mathematics in understanding the world's economic and
social?problems.
Drill Exercises build student skill and confidence.
Online Problems available in WileyPLUS or WeBWorK, for example.
Many problems are randomized, providing students with expanded
opportunities for practice with immediate feedback.
Innovative and engaging problems. Under the approach called the
"Rule of Four," ideas are presented graphically,
numerically, symbolically, and verbally, thereby encouraging
students with a variety of learning styles to expand their
knowledge.
A Flexible Approach to Technology: Adaptable to courses having
various levels of computer involvement, ranging from little or none
to intensive. The book does not require any specific software or
technology, though it has been used successfully with graphing
calculators, graphing software, and computer algebra systems.
Applied Problems for instructors wishing to emphasize the
connection between calculus and other fields | 677.169 | 1 |
How to evaluate progress in problem solving by Randall I Charles(
Book
) 13
editions published
between
1987
and
1994
in
English
and held by
400 WorldCat member
libraries
worldwide
"This book puts the goals of problem-solving evaluation in sharper foucs, describes several classroom evaluation techniques,
and illustrates how these techniques might be used in practice."--Page 1
Geometry : an investigative approach by Phares G O'Daffer(
Book
) 18
editions published
between
1976
and
1992
in
English and Undetermined
and held by
306 WorldCat member
libraries
worldwide
Mathematics for elementary school teachers by Phares G O'Daffer(
Book
) 10
editions published
between
1998
and
2008
in
English and Spanish
and held by
262 WorldCat member
libraries
worldwide
Future elementary and middle school teachers need a clear, coherent presentation of the mathematical concepts, procedures,
and processes they will be called upon to teach. This text uniquely balances what they will teach (concepts and content) with
how to teach (processes and communication). As a result, students using Mathematics for Elementary School Teachers leave the
course knowing more than basic math skills; they develop a deep understanding of concepts that enables them to effectively
teach others. This Fourth Edition features an increased focus on the 'big ideas' of mathematics, as well as the individual
skills upon which those ideas are built
Problem solving : tips for teachers : selections from the Arithmetic teacher by Phares G O'Daffer(
Book
) 7
editions published
between
1988
and
1991
in
English
and held by
200 WorldCat member
libraries
worldwide
Although it is difficult to predict which mathematical ideas will be most important to the citizens of the 21st century, most
mathematics educators agree that problem solving will become an even more important skill. This booklet is designed to provide
teachers with ideas on helping students learn to: (1) use problem solving thinking skills; (2) select and use problem solving
strategies; (3) build helpful attitudes and beliefs about problem solving; (4) use related knowledge; (5) monitor and evaluate
their thinking and progress while solving problems; (6) solve problems in cooperative learning situations; and (7) find correct
answers to a variety of problems. The material in the booklet is organized into four sections which focus on developing problem
solving strategies, extending problem solving strategies, developing problem solving skills, and implementing a problem solving
program. (Tw)
Investigating school mathematics by Robert E Eicholz(
Book
) 40
editions published
between
1973
and
1976
in
English
and held by
146 WorldCat member
libraries
worldwide
Addison-Wesley pre-algebra : a transition to algebra by Phares G O'Daffer(
Book
) 16
editions published
between
1987
and
2006
in
English and Spanish
and held by
124Pre-algebra by Phares G O'Daffer(
Book
) 8
editions published
between
1987
and
1992
in
3
languages
and held by
36Elementary school mathematics by Robert E Eicholz(
Book
)
in
English
and held by
34 WorldCat member
libraries
worldwide
This 356-page, hardcover textbook is designed to introduce students in grade 5 to a variety of mathematic concepts
Mathematics in our world by Robert E Eicholz(
Book
) 1
edition published
in
1978
in
English
and held by
34 WorldCat member
libraries
worldwide
Mathematics in our world by Robert E Eicholz(
Book
) 24
editions published
between
1979
and
1983
in
English
and held by
30 WorldCat member
libraries
worldwide | 677.169 | 1 |
Algebra: Form and Function by Deborah Hughes-Hallett, William G. McCallum, Eric Connally
Algebra: Form and Function by Deborah Hughes-Hallett, William G. McCallum, Eric Connally
By Deborah Hughes-Hallett, William G. McCallum, Eric Connally
Shape is expounded to operate. An plane wing has the shape it does due to its lifting functionality. The pillars of the Parthenon and the girders of a skyscraper are formed to the aim of assisting their substantial buildings. equally, the shape of an algebraic expression or equation displays its functionality. Algebra: shape and serve as initial variation introduces every one function--linear, energy, quadratic, exponential, polynomial--and offers a research of the elemental type of expressions for that functionality. Readers are inspired to check the elemental types, see how they're built, and think about the function of every part. through the textual content, there are instruments sections positioned on the ends of chapters to assist readers gather the abilities they should practice simple algebraic manipulations large ethnic minorities, insufficient agrarian reforms and gradual business improvement sustained through international capital.
A transparent, concise, brand new, authoritative background via one of many top historians within the country.
Give Me Liberty! is the prime e-book out there since it works within the school room. A single-author ebook, supply Me Liberty! deals scholars a constant procedure, a unmarried narrative voice, and a coherent viewpoint through the textual content. Threaded in the course of the chronological narrative is the subject matter of freedom in American heritage and the numerous conflicts over its altering meanings, its limits, and its accessibility to varied social and financial teams all through American background. The 3rd variation areas American heritage extra totally in a world context. The pedagogy can be stronger within the 3rd version, with a Visions of Freedom characteristic in each one bankruptcy and extra huge end-of-chapter overview to your tests!
3x − 10 = 20 3x = 30 add 10 to both sides x = 10 divide both sides by 3. We check to see that 10 is a solution: 3(10) − 10 = 30 − 10 = 20. (b) In the expression on the left side of the equal sign, we first add 3 to x and then multiply by 2. To solve for x in the equation, we undo these operations in the reverse order. We first divide by 2 and then subtract 3 from both sides to produce equivalent equations. 2(x + 3) = 50 x + 3 = 25 divide both sides by 2 x = 22 subtract 3 from both sides. We check to see that 22 is a solution: 2(22 + 3) = 2(25) = 50.
5 times their weight in pounds, w, divided by the square of their height in inches, h. 4. The volume of a cone with radius r and height h is represented by the expression Volume of cone = 1 2 πr h. 3 Which has greater volume: a short fat cone with height 2 and radius 3 or a tall thin cone with height 5 and radius 1? The surface area of a cylinder with radius r feet and height h feet is 2πr 2 + 2πrh square feet. In Exercises 5–8, find the surface area of the cylinder with the given radius and height. | 677.169 | 1 |
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