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Your example is fine. Both $\langle(-1)^n:n\in\Bbb N\rangle$ and $\langle(-1)^{n+1}:n\in\Bbb N\rangle$ have $1$ as a limit point, but $\langle(-1)^n+(-1)^{n+1}:n\in\Bbb N\rangle=\langle 0:n\in\Bbb N\rangle$ converges to $0$ and so does not have $1+1=2$ as a limit point.
It doesn’t matter what subsequence of $\langle a_n:n\in\Bbb N\rangle$ has $c$ as limit or what subsequence of $\langle b_n:n\in\Bbb N\rangle$ has $d$ as a limit; all that matters is whether some subsequence of $\langle a_n+b_n:n\in\Bbb N\rangle$ has $c+d$ as a limit. In your example that’s not the case, so yours is a genuine counterexample to the conjecture.
@Rolando posted what looked like a proof of this conjecture, and then deleted it after you posted your answer. It looked something like this: Let $\epsilon >0$ be given. Since $a_n \to c$ and $b_n \to d$, there are $N_1, N_2$ such that for $l \ge N_1$, $k \ge N_2$ $|a_l - c| \le \epsilon/2$ and $|b_k - d| \le \epsilon/2$. Let $m = \max\{N_1, N_2\}$ and therefore we have for $n \ge m$ we have $|a_n - c + b_n - d| \le \epsilon$. Is the problem with his proof that he is not considering subsequences, and that he interpreted the limit points as limits? – Zvpunry Aug 14 '12 at 22:14 | {
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# Sequence of functions that fails certain conditions of Arzela-Ascoli theorem
For a closed, bounded interval $[a,b]$, let $\{ f_{n}\}$ be a sequence in $C[a,b]$. If $\{f_{n}\}$ is equicontinuous, does $\{f_{n}\}$ necessarily have a uniformly convergent subsequence?
I would think not, because according to the Arzela-Ascoli Theorem, $\{f_{n} \}$ also needs to be uniformly bounded. Is this all that needs to be violated in order for an equicontinuous sequence of continuous functions on a compact interval to not have a uniformly convergent subsequence?
And if so, what is an example of a sequence that illustrates this, and how to show it does not have a uniformly convergent subsequence? Thank you.
Take $f_n(x) = n$ for $x \in [0,1]$. These functions are all constant, so clearly equicontinuous, but $\| f_n - f_m \|_\infty = \lvert n - m \rvert \ge 1$ for $n \neq m$ so no subsequence can converge since no subsequence is Cauchy.
• Yes of course. If it was uniformly bounded, then we could apply Arzela Ascoli and find a convergent subsequence. A sequence of functions $f_n$ is uniformly bounded if and only if the sequence of real numbers $\| f_n\|_\infty$ is bounded. Here that clearly is not the case because $\| f_n \|_\infty = n$. – User8128 Apr 11 '16 at 23:29 | {
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By … By graphing two functions, then, we can more easily compare their characteristics. In this section, 8th grade and high school students will have to find the missing values of x and f(x). In $f\left(x\right)=mx+b$, the b acts as the vertical shift, moving the graph up and down without affecting the slope of the line. This is also expected from the negative constant rate of change in the equation for the function. In linear algebra, mathematical analysis, and functional analysis, a linear function is a … The equation is in standard form (A = -1, B = 1, C = 3). There is a special linear function called the "Identity Function": f (x) = x. The x-intercept is the point at which the graph of a linear function crosses the x-axis. A table of values might look as below. The vertical line test indicates that this graph represents a function. The independent variable is x and the dependent variable is y. a is the constant term or the y intercept. Graph linear functions. Two competing telephone companies offer different payment plans. A function may be transformed by a shift up, down, left, or right. 3.4 Graphing Linear Equations There are two common procedures that are used to draw the line represented by a linear equation. The graph of this function is a line with slope − and y-intercept −. A linear function has one independent variable and one dependent variable. Twitter. A similar word to linear function is linear correlation. For distinguishing such a linear function from the other concept, the term affine function is often used. This tells us that for each vertical decrease in the “rise” of $–2$ units, the “run” increases by 3 units in the horizontal direction. When we’re comparing two lines, if their slopes are equal they are parallel, and if they are in … By using this website, you agree to our Cookie Policy. Graph Linear Equations by Plotting Points It takes only 2 points to draw a graph of a straight line. The first characteristic is its y-intercept which is the | {
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points to draw a graph of a straight line. The first characteristic is its y-intercept which is the point at which the input value is zero. Graph $f\left(x\right)=-\frac{2}{3}x+5$ using the y-intercept and slope. Selbst 1 Selbst 2 Selbst 3 Yes. +drag: Hold down the key, then drag the described object. GRAPHING LINEAR RELATIONS. dillinghamt. Evaluate the function at x = 0 to find the y-intercept. The second is by using the y-intercept and slope. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step. (Note: A vertical line parallel to the y-axis does not have a y-intercept. Do all linear functions have y-intercepts? Using slope and intercepts in context Get 3 of 4 questions to level up! The simplest way is to find the intercept values for both the x-axis and the y-axis. To draw the graph we need coordinates. http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2, Graph a linear function by plotting points, Graph a linear function using the slope and y-intercept, Graph a linear function using transformations. We were also able to see the points of the function as well as the initial value from a graph. Learn More at mathantics.comVisit http://www.mathantics.com for more Free math videos and additional subscription based content! How to graph Linear Functions by finding the X-Intercept and Y-Intercept of the Function? The graph of a linear relation can be found by plotting at least two points. Graph $f\left(x\right)=4+2x$, using transformations. Spell. -x + y = 3. The first one is called the slope-intercept method and involves using the slope and intercept given in the equation. In Activity 1 the learners should enter the expressions one by one into the graphing calculator and classify the functions according to the shape of the graph. Write. linear functions by the shape of their graphs and by noting differences in their expressions. Graph 3x - 2y = 8. And here is its graph: It makes a 45° (its slope is 1) It is | {
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their expressions. Graph 3x - 2y = 8. And here is its graph: It makes a 45° (its slope is 1) It is called "Identity" because what comes out is identical to what goes in: In. We can begin graphing by plotting the point (0, 1) We know that the slope is rise over run, $m=\frac{\text{rise}}{\text{run}}$. A function may also be transformed using a reflection, stretch, or compression. Plot the points and graph the linear function. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. In the equation $f\left(x\right)=mx+b$, $m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$. PLAY. This is why we performed the compression first. The y-intercept is the point on the graph when x = 0. f(x)=b. of f is the To find the y-intercept, we can set $x=0$ in the equation. Gravity. The graph of a linear function is a line. The slope of a linear function is equal to the ratio of the change in outputs to the change in inputs. This inequality notation means that we should plot the graph for values of x between and including -3 and 3. Use the resulting output values to identify coordinate pairs. The a represents the gradient of the line, which gives the rate of change of the dependent variable. In order to write the linear function in the form of y=mx+b, we will need to determine the line's: 1. slope (m) 2. y-intercept (b) We can tell from the graph that the slope of the line is negative because the line goes down and to the right. Using vertical stretches or compressions along with vertical shifts is another way to look at identifying different types of linear functions. The graph below is of the function $f\left(x\right)=-\frac{2}{3}x+5$. Linear Parent Graph And Transformations. So, for this definition, the above function is linear only when c = 0, that is when the line passes through the origin. Linear functions are functions that produce a | {
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= 0, that is when the line passes through the origin. Linear functions are functions that produce a straight line graph.. The graph of f is a line with slope m and y intercept b. The graph of f is a line with slope m and y intercept Graphing Linear Equations Find the Equation of a Line. Horizontal lines are written in the form, $f(x)=b$. Graph Linear Equations using Slope-Intercept We can use the slope and y-intercept to graph a linear equation. Graph Linear Equations in Two Variables Learning Objectives. We were also able to see the points of the function as well as the initial value from a graph. The graph of a linear function is a line. f(0). The graph of the function is a line as expected for a linear function. Evaluate the function at each input value and use the output value to identify coordinate pairs. ++drag: Hold down both the key and the key, then drag the described object. For example, following order of operations, let the input be 2. The equation for the function shows that $m=\frac{1}{2}$ so the identity function is vertically compressed by $\frac{1}{2}$. Linear functions are those whose graph is a straight line. The steepness of a hill is called a slope. Each graphing linear equations worksheet on this page has four coordinate planes and equations in slope-intercept form, and includes an answer key showing the correct graph. Graphing a Linear Function Using y-intercept and Slope. Graphing Linear Functions. In this non-linear system, users are free to take whatever path through the material best serves their needs. There are three basic methods of graphing linear functions: Keep in mind that a vertical line is the only line that is not a function.). Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. Then just draw a line that passes through both of these points. The following diagrams show the different methods to graph a linear equation. Linear functions are typically written in the | {
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show the different methods to graph a linear equation. Linear functions are typically written in the form f(x) = ax + b. Furthermore, the domain and range consists of all real numbers. A table of values might look as below. Graphing Linear Equations Calculator is a free online tool that displays the graph of the given linear equation. Notice that adding a value of b to the equation of $f\left(x\right)=x$ shifts the graph of f a total of b units up if b is positive and |b| units down if b is negative. How many solutions does this linear system have? Regardless of whether a table is given to you, you should consider using one to ensure you’re correctly graphing linear and quadratic functions. Is the Function Linear or Nonlinear | Table. Learn More at mathantics.comVisit http://www.mathantics.com for more Free math videos and additional subscription based content! Often, the number in front of x is already a fraction, so you won't have to convert it. A y-intercept is a y-value at which a graph crosses the y-axis. Solving Systems of Linear Equations: Graphing. The equation for a linear function is: y = mx + b, Where: m = the slope ,; x = the input variable (the “x” always has an exponent of 1, so these functions are always first degree polynomial.). The slopes are represented as fractions in the level 2 worksheets. In Linear Functions, we saw that that the graph of a linear function is a straight line. The equation can be written in standard form, so the function is linear. eval(ez_write_tag([[728,90],'analyzemath_com-medrectangle-4','ezslot_5',344,'0','0'])); Any function of the form The first is by plotting points and then drawing a line through the points. Furthermore, the domain and range consists of all real numbers. Properties. Graphing linear functions (2.0 MiB, 1,144 hits) Slope Determine slope in slope-intercept form (160.4 KiB, 766 hits) Determine slope from given graph (2.1 MiB, 834 hits) Find the integer of unknown coordinate (273.6 KiB, 858 hits) Find the | {
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graph (2.1 MiB, 834 hits) Find the integer of unknown coordinate (273.6 KiB, 858 hits) Find the fraction of unknown coordinate (418.5 KiB, 891 hits) Linear inequalities Graph of linear inequality (2.8 MiB, 929 hits) Facebook. The first characteristic is its y-intercept which is the point at which the input value is zero. Linear functions are typically written in the form f(x) = ax + b. In mathematics, the term linear function refers to two distinct but related notions: In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. Interactive, free online graphing calculator from GeoGebra: graph functions, plot data, drag sliders, and much more! Interpret solutions to linear equations and inequalities graphically. The function $y=x$ compressed by a factor of $\frac{1}{2}$. y = f(x) = a + bx. Google+. ; b = where the line intersects the y-axis. Now we have to determine the slope of the line. The function $y=\frac{1}{2}x$ shifted down 3 units. The equation for a linear function is: y = mx + b, Where: m = the slope , x = the input variable (the “x” always has an exponent of 1, so these functions are always first degree polynomial.). Because the slope is positive, we know the graph will slant upward from left to right. 1. The order of the transformations follows the order of operations. Evaluating the function for an input value of 2 yields an output value of 4 which is represented by the point (2, 4). $f\left(x\right)=\frac{1}{2}x+1$. Another option for graphing is to use transformations on the identity function $f\left(x\right)=x$. A linear function has the following form y = f (x) = a + bx A linear function has one independent variable and one dependent variable. The graph of the linear equation will always result in a straight line. The input values and corresponding output values form coordinate pairs. A linear function is a polynomial function in which the variable x has degree | {
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form coordinate pairs. A linear function is a polynomial function in which the variable x has degree at most one: = +.Such a function is called linear because its graph, the set of all points (, ()) in the Cartesian plane, is a line.The coefficient a is called the slope of the function and of the line (see below).. We then plot the coordinate pairs on a grid. Free graph paper is available. The graph of a linear function is a straight line, while the graph of a nonlinear function is a curve. Students learn that the linear equation y = x, or the diagonal line that passes through the origin, is called the parent graph for the family of linear equations. Vertically stretch or compress the graph by a factor. Write the equation of a line parallel or perpendicular to a given line. Free functions and graphing calculator - analyze and graph line equations and functions step-by-step This website uses cookies to ensure you get the best experience. That line is the solution of the equation and its visual representation. Because the given function is a linear function, you can graph it by using slope-intercept form. In Linear Functions, we saw that that the graph of a linear function is a straight line.We were also able to see the points of the function as well as the initial value from a graph. Explore math with our beautiful, free online graphing calculator. The equation for the function also shows that $b=-3$, so the identity function is vertically shifted down 3 units. We will choose 0, 3, and 6. Another way to think about the slope is by dividing the vertical difference, or rise, between any two points by the horizontal difference, or run. It looks like the y-intercept (b) of the graph is 2, as represented by point (0,2). f(a) is called a function, where a … Match. We repeat until we have multiple points, and then we draw a line through the points as shown below. Learn. In general, a linear function28 is a function that can be written in the form f(x) = mx + b | {
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Learn. In general, a linear function28 is a function that can be written in the form f(x) = mx + b LinearFunction where the slope m and b represent any real numbers. Begin by choosing input values. In general, a linear function Any function that can be written in the form f ( x ) = m x + b is a function that can be written in the form f ( x ) = m x + b L i n e a r F u n c t i o n where the slope m and b represent any real … Learn more Accept. Because y = f(x), we can use y and f(x) interchangeably, and ordered pair solutions on the graph (x, y) can be written in the form (x, f(x)). eval(ez_write_tag([[728,90],'analyzemath_com-medrectangle-3','ezslot_4',320,'0','0'])); Determine the x intercept, set f(x) = 0 and The slopes in level 1 worksheets are in the form of integers. Graph a straight line by finding its x - and y-intercepts. The, of this function is the set of all real numbers. By using this website, you agree to our Cookie Policy. The graph crosses the y-axis at (0, 1). Graphing Linear Functions. There are three basic methods of graphing linear functions. No. Recall that the slope is the rate of change of the function. 3. Vertical stretches and compressions and reflections on the function $f\left(x\right)=x$. This is also known as the “slope.” The b represents the y-axis intercept. These points may be chosen as the x and y intercepts of the graph for example. We can now graph the function by first plotting the y-intercept. Relating linear contexts to graph features Get 5 of 7 questions to level up! Method 1: Graphing Linear Functions in Standard Form 1. This means the larger the absolute value of m, the steeper the slope. Linear functions are those whose graph is a straight line. It is generally a polynomial function whose degree is utmost 1 or 0. Evaluating the function for an input value of 1 yields an output value of 2 which is represented by the point (1, 2). The third is applying transformations to the identity function $f\left(x\right)=x$. Write the | {
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2). The third is applying transformations to the identity function $f\left(x\right)=x$. Write the equation for a linear function from the graph of a line. b. Solver to Analyze and Graph a Linear Function. Show Step-by-step Solutions. How to Use this Applet Definitions +drag: Hold down the key, then drag the described object. This website uses cookies to ensure you get the best experience. Use $\frac{\text{rise}}{\text{run}}$ to determine at least two more points on the line. Its graph is a horizontal line at y = b. The graph of a linear function is always a line. Graphing Linear Function: Type 2 - Level 1. Evaluate the function at each input value. Created by. We previously saw that that the graph of a linear function is a straight line. The graph slants downward from left to right which means it has a negative slope as expected. Evaluate when . Linear functions word problem: fuel (Opens a modal) Practice. We know that the linear equation is defined as an algebraic equation in which each term should have an exponents value of 1. Recall that the set of all solutions to a linear equation can be represented on a rectangular coordinate plane using a straight line through at least two points; this line is called its graph. Graphing Linear Functions. For example, Plot the graph of y = 2x – 1 for -3 ≤ x ≤ 3. Identify and graph a linear function using the slope and y-intercept. When m is negative, there is also a vertical reflection of the graph. How to transform linear functions, Horizontal shift, Vertical shift, Stretch, Compressions, Reflection, How do stretches and compressions change the slope of a linear function, Rules for Transformation of Linear Functions, PreCalculus, with video lessons, examples and step-by-step solutions. Learn more Accept. Solution : y = x + 3. In addition, the graph has a downward slant which indicates a negative slope. Note: A function f (x) = b, where b is a constant real number is called a constant function. This function includes a | {
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f (x) = b, where b is a constant real number is called a constant function. This function includes a fraction with a denominator of 3 so let’s choose multiples of 3 as input values. 8 Linear Equations Worksheets. Usage To plot a function just type it into the function box. A graphing calculator can be used to verify that your answers "make sense" or "look right". solve for x. Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections. Write the equation in standard form. The slope of a linear function corresponds to the number in … For example, given the function $f\left(x\right)=2x$, we might use the input values 1 and 2. Evaluate the function at an input value of zero to find the. Graphs of linear functions may be transformed by shifting the graph up, down, left, or right as well as using stretches, compressions, and reflections. This graph illustrates vertical shifts of the function $f\left(x\right)=x$. The slope is defined as the ratio of the vertical change between two points, the rise, to the horizontal change between the same two points, the run. The Slider Area. In this video we look at graphing equations using a table of values Find the slopes and the x- and y-intercepts of the following lines. Solve a system of linear equations. The slope-intercept form gives you the y- intercept at (0, –2). Did you have an idea for improving this content? Graph horizontal and vertical lines. When it comes to graphing linear equations, there are a few simple ways to do it. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. The equation, written in this way, is called the slope-intercept form. According to the equation for the function, the slope of the line is $-\frac{2}{3}$. set of all real numbers. Flashcards. If variable x is a constant x=c, that will represent a line paralel to y-axis. Reddit. Two points that are especially useful | {
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x=c, that will represent a line paralel to y-axis. Reddit. Two points that are especially useful for sketching the graph of a line are found with the intercepts. Given a real-world situation that can be modeled by a linear function or a graph of a linear function, the student will determine and represent the reasonable domain and range of the linear function … Function Grapher is a full featured Graphing Utility that supports graphing two functions together. Book The a represents the gradient of the line, which gives the rate of change of the dependent variable. Select two options. how to graph linear equations using the slope and y-intercept. Graph a linear function: a step by step tutorial with examples and detailed solutions. This inequality notation means that we should plot the graph for values of x between and including -3 and 3. Choosing three points is often advisable because if all three points do not fall on the same line, we know we made an error. Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. We generate these coordinates by substituting values into the linear equation. Test. In general we should evaluate the function at a minimum of two inputs in order to find at least two points on the graph of the function. When the function is evaluated at a given input, the corresponding output is calculated by following the order of operations. If we have two points: A=(x1,y1) B=(x2,y2) A slope (a) is calculated by the formula: a=y2−y1x2−x1 If the slope is equal to number 0, then the line will be paralel with x – axis. Functions: Hull: First graph: f(x) Derivative Integral From ... Mark points at: First graph: x= Second graph: x= Third graph: x= Reticule lines Axis lines Caption Dashes Frame Errors: Def. This is also known as the “slope.” The b represents the y-axis intercept. $\begin{array}{l}f\text{(2)}=\frac{\text{1}}{\text{2}}\text{(2)}-\text{3}\hfill \\ =\text{1}-\text{3}\hfill \\ | {
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\\ =\text{1}-\text{3}\hfill \\ =-\text{2}\hfill \end{array}$. Subtract x from each side. Find a point on the graph we drew in Example: Graphing by Using the y-intercept and Slope that has a negative x-value. Recognize the standard form of a linear function. Graphing a Linear Equation by Plotting Three Ordered Pairs. In Activity 1 the learners should enter the expressions one by one into the graphing calculator and classify the functions according to the shape of the graph. A linear equation is drawn as a straight line on a set of axes. But if it isn't, convert it by simply placing the value of m over 1. You need only two points to graph a linear function. First, graph y = x. Students also learn the different types of transformations of the linear parent graph. A Review of Graphing Lines. These unique features make Virtual Nerd a viable alternative to private tutoring. Although this may not be the easiest way to graph this type of function, it is still important to practice each method. What is Meant by Graphing Linear Equations? Possible answers include $\left(-3,7\right)$, $\left(-6,9\right)$, or $\left(-9,11\right)$. We were also able to see the points of the function as well as the initial value from a graph. Graph a straight line by finding three ordered pairs that are solutions to the linear equation. Examples: 1. To graph, choose three values of x, and use them to generate ordered pairs. The functions whose graph is a line are generally called linear functions in the context of calculus. The slope of a line is a number that describes steepnessand direction of the line. By the end of this section, you will be able to: Plot points in a rectangular coordinate system; Graph a linear equation by plotting points; Graph vertical and horizontal lines; Find the x- and y-intercepts; Graph a line using the intercepts ; Before you get started, take this readiness quiz. In the equation $f\left(x\right)=mx$, the m is acting as the vertical stretch or compression of the | {
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In the equation $f\left(x\right)=mx$, the m is acting as the vertical stretch or compression of the identity function. In Example: Graphing by Using Transformations, could we have sketched the graph by reversing the order of the transformations? You can move the graph of a linear function around the coordinate grid using transformations. The first … Graphing Linear Equations. It has the unique feature that you can save your work as a URL (website link). Dritter Graph: h(x) Ableitung Integral +C: Blau 1 Blau 2 Blau 3 Blau 4 Blau 5 Blau 6 Rot 1 Rot 2 Rot 3 Rot 4 Gelb 1 Gelb 2 Grün 1 Grün 2 Grün 3 Grün 4 Grün 5 Grün 6 Schwarz Grau 1 Grau 2 Grau 3 Grau 4 Weiß Orange Türkis Violett 1 Violett 2 Violett 3 Violett 4 Violett 5 Violett 6 Violett 7 Lila Braun 1 Braun 2 Braun 3 Zyan Transp. The slope of a linear function will be the same between any two points. The slope is $\frac{1}{2}$. By graphing two functions, then, we can more easily compare their characteristics. Graph $f\left(x\right)=-\frac{2}{3}x+5$ by plotting points. Linear equations word problems: tables Get 3 of 4 questions to level up! Plot the coordinate pairs and draw a line through the points. It will be very difficult to succeed in Calculus without being able to solve and manipulate linear equations. Method 1: Graphing Linear Functions in Standard Form 1. Introduction to Linear Relationships: IM 8.3.5. Example 1 Graph the linear function f given by f (x) = 2 x + 4 Solution to Example 1. Graph $f\left(x\right)=\frac{1}{2}x - 3$ using transformations. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. Graphing Linear Function: Type 1 - Level 2. Although the linear functions are also represented in terms of calculus as well as linear algebra. Linear functions are functions that produce a straight line graph. Starting from our y-intercept (0, 1), we can rise 1 and then run 2 or run 2 and then rise 1. Notice that multiplying the equation | {
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(0, 1), we can rise 1 and then run 2 or run 2 and then rise 1. Notice that multiplying the equation $f\left(x\right)=x$ by m stretches the graph of f by a factor of m units if m > 1 and compresses the graph of f by a factor of m units if 0 < m < 1. Tell whether each function is linear. 2. What is a Linear Function? f (x) = m x + b, where m is not equal to 0 is called a linear function. These pdf worksheets provide ample practice in plotting the graph of linear functions. For example, Plot the graph of y = 2x – 1 for -3 ≤ x ≤ 3. Free linear equation calculator - solve linear equations step-by-step. STUDY. Complete the function table, plot the points and graph the linear function. The other characteristic of the linear function is its slope, m, which is a measure of its steepness. We’d love your input. First, graph the identity function, and show the vertical compression. Graphing a Linear Function Using y-intercept and Slope. After studying this section, you will be able to: 1. From the initial value (0, 5) we move down 2 units and to the right 3 units. We encountered both the y-intercept and the slope in Linear Functions. Linear Function Graph. (See Getting Help in Stage 1.) In mathematics, a graphing linear equation represents the graph of the linear equation. y = mx + b y = -2x + 3/2. Knowing an ordered pair written in function notation is necessary too. The first characteristic is its y-intercept which is the point at which the input value is zero. Another way to graph linear functions is by using specific characteristics of the function rather than plotting points. Let's try starting from a graph and writing the equation that goes with it. We can extend the line to the left and right by repeating, and then draw a line through the points. From our example, we have $m=\frac{1}{2}$, which means that the rise is 1 and the run is 2. Linear equations word problems: graphs Get 3 of 4 questions to level up! How to Use the Graphing Linear Equations Calculator? To find | {
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Get 3 of 4 questions to level up! How to Use the Graphing Linear Equations Calculator? To find points of a function, we can choose input values, evaluate the function at these input values, and calculate output values. m = -2 and b = -1/3 m = -2 and b = -2/3. Since the slope is 3=3/1, you move up 3 units and over 1 unit to arrive at the point (1, 1). If you have difficulties with this material, please contact your instructor. This website uses cookies to ensure you get the best experience. Convert m into a fraction. If so, graph the function. A linear function is a function which forms a straight line in a graph. Determine the y intercept, set x = 0 to find Draw Function Graphs Mathematics / Analysis - Plotter - Calculator 4.0. Draw a line which passes through the points. BYJU’S online graphing linear equations calculator tool makes the calculation faster and it displays the graph in a fraction of seconds. 8th grade students learn to distinguish between linear and nonlinear functions by observing the graphs. The output value when x = 0 is 5, so the graph will cross the y-axis at (0, 5). The y-intercept and slope of a line may be used to write the equation of a line. Examine the input(x) and output(y) values of the table inthese linear function worksheets for grade 8. Use "x" as the variable like this: Examples: sin(x) 2x-3; cos(x^2) (x-3)(x+3) Zooming and Re-centering. The same goes for the steepness of a line. Now we know the slope and the y-intercept. A linear function has the following form. Regardless of whether a table is given to you, you should consider using one to ensure you’re correctly graphing linear and quadratic functions. To zoom, use the zoom slider. Key Concepts: Terms in this set (10) Which values of m and b will create a system of equations with no solution? You can move the graph of a linear function around the coordinate grid using transformations. All linear functions cross the y-axis and therefore have y-intercepts. GeoGebra Classroom | {
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to graph linear functions is by specific! 3 units ] f ( linear function graph ) and complete the function. ) by first plotting the graph f. The a represents the gradient of the table inthese linear function. ) have a y-intercept is point! Equation in which each term should have an exponents value of m, the corresponding output form. ) =\frac { 1 } { 2 } x+1 [ /latex ] the! Also a vertical line parallel to the ratio of the equation is defined as an equation... In outputs to the linear equation calculator - Analyze and graph the identity function '': (... In this section, 8th grade and high school students will have convert. Work as a URL ( website link ) the x and y intercepts of the equation be. =\Frac { 1 } { 3 } x+5 [ /latex ] using transformations contexts to graph linear functions identify graph! The graphs your work as a straight line Cookie linear function graph steepness of a line parallel to the at. Function box y-intercept of the function table, plot points, visualize algebraic equations, are... Or the y intercept, set x = 0 to find f ( )... Function rather than plotting points slope-intercept we can more easily compare their characteristics ( y ) values x! Types of transformations of the line to the left and right by,... To: 1 although this may not be the easiest way to graph linear functions are typically written Standard. This may not be the easiest way to graph this Type of function, you will able! { 4 } x+6 [ /latex ] to look at identifying different types of transformations of the diagrams. Inthese linear function from the negative constant rate of change of the function tables the equations two! 3, and more supports graphing two functions, we can now the... Always result in a graph, or compression fraction, so the function is a are., 1 ) by reversing the order of the function tables visualize algebraic equations, sliders. Along with vertical shifts is another way to graph features Get 5 of 7 questions to level!. Book we previously saw that that the | {
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# Probability of finding a molecule in the ground vibrational level
I wanted to estimate the probability of finding a molecule in the ground vibrational level using the Boltzmann distribution:
$$p_i = \frac{e^{-\epsilon_i/kT}}{\sum_{i=0}^{N}e^{-\epsilon_i/kT}}$$
Using the quantum harmonic oscillator as a model for the energy
$$\epsilon_i = h\nu (i+1/2) =/i=0/=\frac{h\nu}{2}$$
In the Boltzmann distribution, we have the state of interest divided by the sum of all possible states. But how should I treat the denominator?
Searching a bit I found that the analytical expression for this geometric series is ($i$ not imaginary number)
$$\sum_{i=0}^{N}e^{-i h\nu/kT} = \frac{1}{1 - e^{h\nu/kT}}$$
However, is this using a shifted energy scale for the harmonic potential? In that the vibrational energies are $0$, $h\nu$, $2h\nu$, ..., and not $\frac{1}{2}h\nu$, $\frac{3}{2}h\nu$, $\frac{5}{2}h\nu$, ...? Should I make sure I use the same energy scale for the nominator and denominator in the Boltzmann distribution?
Doing what porphyrin suggested, I get
$$\sum_{i=0}^{\infty} e^{-h\nu(i+\frac{1}{2})/kT} = e^{-h\nu/kT} \sum_{i=0}^{\infty} e^{- ih\nu/kT}$$
Expanding the four first terms
$$e^{-h\nu/kT} \sum_{i=0}^{\infty} e^{- ih\nu/kT} = (e^{-h\nu/kT} \cdot 1) + (e^{-h\nu/kT} \cdot e^{-h\nu/kT}) + (e^{-h\nu/kT} \cdot e^{-2h\nu/kT}) + (e^{-h\nu/kT} \cdot e^{-3h\nu/kT}) \\ = e^{-h\nu/kT} + e^{-2h\nu/kT} + e^{-3h\nu/kT} + e^{-4h\nu/kT} = \sum_{1}^{\infty}e^{-n\cdot h\nu/kT}$$
which has an analytical expression for the converged value, right? | {
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which has an analytical expression for the converged value, right?
• The denominator should be calculated exactly. Ever heard of a geometric progression and its sum? Oct 17 '16 at 12:30
• Ah, yes. Looking at it, it should converge.
– Yoda
Oct 17 '16 at 12:32
• The summation is called the partition function and it always extends over all possible levels $n=0 ..,\infty$. You are assuming a harmonic oscillator which has an infinite number of levels. Its easier if the 1/2 is separated out first and treated separately, this just adds a constant to the energy as you realise. Expand the summation as $1+e^{-a}+e^{-2a}+....$ ($a=h\nu /(kT)$ which converges to the result you quote. Oct 17 '16 at 12:54
You're on the right track.
Also, using $i$ as an index can be confusing some times because it can be confused with the imaginary number; however, here it should not present a problem. As a matter of habbit however, I like to use $j$ or $n$ or something else..there are only so many letters in the alphabet.
The sum in the denominator is called the partition function, and has the form
$$Z = \sum_{j}e^{-\frac{\epsilon_j}{kT}}$$
For the harmonic, oscillator $\epsilon_j = (\frac{1}{2}+j)\hbar \omega$ for $j \in \{ 0,1,2.. \}$
Note that $\epsilon_0 \neq 0$ there exists a zero point energy.
Let's write out a few terms $$Z = e^{-\frac{\hbar \omega/2}{kT}} + e^{-\frac{\hbar \omega3/2}{kT}} + e^{-\frac{\hbar \omega5/2}{kT}} +.....$$
factoring out $e^{-\frac{\hbar \omega/2}{kT}}$
$$Z = e^{-\frac{\hbar \omega/2}{kT}} \left( 1+ e^{-\frac{\hbar \omega}{kT}} + e^{-\frac{2\hbar \omega}{kT}} +.....\right)$$
The sum in the bracket takes the form of a geometric series whose sum converges as shown below $$1+x+x^2+... = \frac{1}{(1-x)}$$ herein, $x \equiv e^{-\frac{\hbar\omega}{kT}}$
Putting all of this together
$$Z = \frac{e^{-\frac{\hbar \omega/2}{kT}}}{(1-e^{-\frac{\hbar\omega}{kT}})}$$ | {
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$$Z = \frac{e^{-\frac{\hbar \omega/2}{kT}}}{(1-e^{-\frac{\hbar\omega}{kT}})}$$
Now, $$p_0 = \frac{e^{-\epsilon_0/kT}}{Z} = \frac{e^{-\frac{\hbar \omega/2}{kT}}}{Z} = \frac{e^{-\frac{\hbar \omega/2}{kT}}}{\frac{e^{-\frac{\hbar \omega/2}{kT}}}{(1-e^{-\frac{\hbar\omega}{kT}})}} = (1-e^{-\frac{\hbar\omega}{kT}})$$
• Great answer. But I strongly disagree with not using i as an index variable. The possible confusion with the imaginary unit only arises from typographic sloppiness. Variables should be italic and mathematical constants should be upright set. This includes e.g. the euler number, but no one would think that you exponented the variable e. From context it is clear that you exponented the euler number. The same holds true for sum symbols, it would make no sense to write the imaginary unit underneath them and even with typographic sloppiness it is clear from context that you mean the variable i. Oct 18 '16 at 13:04 | {
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# Dimension of Image/Kernel of Linear Transformation
Consider the vector spaces $P_n$ (polynomials of degree no more than n). The differentiation D gives a linear transformation from $P_n$ to $P_{n-1}$. What is the dimension of the image and the kernel of D? Is D a valid linear transformation from $P_n$ to $P_n$? If it is, what is the dimension of the kernel and image in this case? It seems like this uses the rank-nullity theorem, but I'm not sure. Thanks
The transformation is a linear transformation. In order for a transformation $T$ to be a linear transformation, it has to implement 2 conditions: $$(1) \hspace{0.2cm} T(v + w) = T(v) + T(w) \\ (2) \hspace{0.2cm} T(\alpha v) = \alpha T(v)$$ for any $v, w \in V$.
The differentiation transformation $D$ satisfies those constraints because of the derivatives' properties.
As for dimensions:
$Im(D) = P_{n-1}$, because for any $p^{\prime}(x) \in P_{n-1}$ you can find some $p(x) \in P_n$ s.t. $D(p(x)) = p^{\prime}(x)$, namely $\int f^{\prime}(x) dx$. Therefore $dim(Im(D)) = dim(P_{n-1}) = n$.
By rank nullity theorem, $dim(Im(D)) + dim(Ker(D)) = dim(P_n)$, and therefore $n + dim(Ker(D)) = n+1$ and $dim(Ker(D)) = 1$
• So what would be the dimension of Im(D)? Would it just be n? – JanoyCresva Apr 15 '17 at 19:47
• Had a typo, edited. The dimension of $Im(D)$ is $n$ as explained in my answer – AsafHaas Apr 15 '17 at 19:49
• Ok I got it. I'm still slightly confused about the reasoning on why it is a valid linear transformation, though – JanoyCresva Apr 15 '17 at 19:53
• Edited my answer to contain an explanation of this as well. – AsafHaas Apr 15 '17 at 20:03
Yes, it's a valid linear transformation. If $f, g \in P_n$, $c \in \mathbb{R}$ or whatever field you're working over, $D(f + cg) = D(f) + cD(g)$.
$D$ is surjective. For any $f \in P_{n-1}$, you can integrate, $\int f \in P_n$, and then $D(\int f) = f$.
The kernel has dimension 1. You could do this by rank nullity, or just note that if $D(f) = 0$, $f$ is constant. | {
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The mapping $D$ is a linear transformation:
Proof: Let $f,g \in P_n$, let $a,b \in \mathbb{R}$
$$(af+bg)'(c) = \lim\limits_{x\to c} \frac{(af+bg)(x) - (af + bg)(c)}{x-c}$$ $$= \lim\limits_{x\to c} \frac{af(x)+bg(x) - af(c) - bg(c)}{x-c}$$ $$= \lim\limits_{x\to c} \frac{af(x) - af(c) +bg(x) - bg(c)}{x-c}$$ $$= \lim\limits_{x\to c} \frac{af(x) - af(c) }{x-c} + \lim\limits_{x\to c} \frac{ bg(x) - bg(c)}{x-c}$$ $$= a\lim\limits_{x\to c} \frac{f(x) - af(c) }{x-c} + b\lim\limits_{x\to c} \frac{ g(x) - g(c)}{x-c}$$ $$=af'(c) + bf'(c)$$
Hence: $D(af + bg) = aD(f) + bD(g) \quad \triangle$
$Im(D) = P_{n-1}$
Proof:
Let $Q \in Im(D)$ Then, there is a $P \in P_n$ such that $D(P_n) = Q$. Because differentiation is an operation that reduces the exponent of a power function by $1$ (for example polynomials), $Q \in P_{n-1}$. Hence, $Im(D) \subset P_{n-1}$
Now, let $Q \in P_{n-1}$. Then, there is a $P$ in $P_n$ such that $D(P) = Q$. Simply take $$P = \int Q dx$$ Hence, $Q \in Im(D)$
We conclude that $P_{n-1} \subset Im(D)$
obtaining $P_{n-1} = Im(D) \quad \triangle$
From this, it follows that $dim(Im(D)) = n$, and by rank nullity theorem $dim(ker(D)) = (n+1)-n = 1$
One can also prove directly the kernel has dimension 1 (which is, in fact, easier), by showing $ker(D) = P_0$ ($P_0$ might be bad notation but I mean the constant polynomials)
Proof:
$P \in ker(D) \iff D(P) = 0\iff P \text{ constant}$ We deduce that $ker(D) = P_0 \quad \triangle$
Alternatively, by rank nullity theorem, it would follow that $dim(Im(D)) = n$ | {
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# Does this double integral exists?
I've calculated $$\iint_D\frac{y}{x^2+y^2}\,dA$$ where D is bounded by $$y=x$$, $$y=2x$$, $$x=2$$, in this way: $$\iint_D\frac{y}{x^2+y^2}\,dA=\int_0^2\int_x^{2x}\frac{y}{x^2+y^2}\,dy\,dx=\cdots = \ln \frac{5}{2}.$$ However, I wonder if the fact that the $$\frac{y}{x^2+y^2}$$ is not bounded on D invalidates all the calculations and in fact the double integral does not exist. All the theorems that I consult have as a hypothesis that the integrand is bounded on the region of integration and hence my doubt regarding this double integral. Can anyone help me? Does this integral exist?
• You didn't bother to show the computation, but probably it's valid. The function you integrate is positive and for such you can carelessly change the order of integration due to Tonelli's theorem. Also, a function doesn't have to be bounded for an integral to exist, think for simplicity about one-dimensional integral $\int_0^1 x^{-1/2}\,dx$. Jun 10 '20 at 21:29
• Why is the fraction not bounded on the region $D$? Jun 10 '20 at 23:27
Changing to polar coordinates reveals why the integral is finite even though the integrand is $$\mathcal{O}(r^{-1})$$ as $$r \to 0$$.
For a quick check, note that the integrand is nonnegative and $$D$$ is a subset of the sector
$$S=\{(r,\theta): 0 \leqslant r \leqslant 2\sqrt{5},\, \frac{\pi}{4} \leqslant \theta \leqslant \arctan (2) \}$$
Thus,
$$\int\int_D \frac{y}{x^2+ y^2} dA \leqslant \int_{\frac{\pi}{4}}^{\arctan(2)}\int_0^{2\sqrt{5}} \frac{r \sin \theta}{r^2} r \, dr \, d\theta = 2\sqrt{5}\int_{\frac{\pi}{4}}^{\arctan(2)}\sin \theta \, d\theta,$$
where the integral on the RHS is finite since the sine function is bounded.
$$\int \frac{ydy}{x^2+y^2} = \frac{1}{2} \int \frac{d(y^2+x^2)}{x^2+y^2} =\frac{1}{2} \ln(x^2+y^2) + C$$ So $$\int_0^2\int_x^{2x}\frac{y}{x^2+y^2}\,dy\,dx= \frac{1}{2} \int_{0}^{2}\ln(x^2+y^2)|_{x}^{2x} dx$$ | {
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2) The direction of the relationship, which can be positive or negative based on the sign of the correlation coefficient. correlation is rather strong, so the correlation coefficient should be closer to 1. Correlation Coefficient WorksheetName: Calculator steps for creating a scatter plot: Stat. Students estimate the correct r value given a scatter plots and some reasonable choices to interpret positive and negative slope and strength or weakness of the correlation coefficient of a li Title: Regression Worksheet Answers Author: Plano ISD Though simple, it is quite helpful in understanding the relations between at least two variables. Continue on the following page. ��� N _rels/.rels �(� ���j�0@���ѽQ���N/c���[IL��j���]�aG��ӓ�zs�Fu��]��U �� ��^�[��x ����1x�p����f��#I)ʃ�Y���������*D��i")��c$���qU���~3��1��jH[{�=E����~ S~ A ~9u.~p. Excel provides two worksheet functions for calculating correlation â and, they do exactly the same thing in exactly the same way! Excel CORREL function. Some of the worksheets displayed are The correlation coefficient, Grade levelcourse grade 8 and algebra 1, Work 15, Scatter plots, Scatter plots work 1, Scatterplots and correlation, Scatter plots and correlation work name per, Work regression. The correlation coefficient r is given by: r =. Four things must be reported to describe a relationship: 1) The strength of the relationship given by the correlation coefficient. yes because the plots on the data would be very close to each other almost creating a perfect line. O-Md . Test for the significance of the correlation at the .05 level of significance. Compute the correlation coefficient . Scatter Plot And Correlation Coefficient Quiz. Those are the two main correlation functions. The correlation coe cient ris given by: r= n P (xy) ( P x)( P y) q n P x2( P x)2. q n P y ( P y)2. Instructions: Click and drag the points around the screen and examine the effect on the correlation coefficient and on the line of best fit. 7. Based on the scatter | {
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the effect on the correlation coefficient and on the line of best fit. 7. Based on the scatter plot, predict the sign of the linear correlation coefficient. Name:!!_____! The others are RSQ, COVARIANCE.P, and COVARIANCE.S. Compute the linear correlation coefficient and compare its sign to your answer to part (b). I -J . Y1. Calculate the correlation coefficient of the data. To compute a correlation coefficient by hand, you'd have to use this lengthy formula. To download/print, click on pop-out icon or print icon to worksheet to print or download. Students estimate the correct r value given a scatter plots and some reasonable choices to interpret positive and negative slope and strength or weakness of the correlation coefficient ⦠weak positive b. 4. Explain your answer. Why Excel offers both CORREL and PEARSON is unclear, but there you have it. A specific value of the y-variable given a specific value of the x-variable b. 6. ~ S:J- Y /0"1.09 . Ahead of discussing Linear Regression And Correlation Coefficient Worksheet, you should realize that Knowledge will be your answer to an even better tomorrow, plus discovering wonât just stop right after the university bell rings.Which being claimed, we all offer you a selection of very simple nonetheless educational posts and web themes designed well suited for every academic purpose. In this worksheet, we will practice calculating and using Pearsonâs correlation coefficient r to describe the strength and direction of a linear relationship. Spearmanâs correlation coefficient can be calculated whether the data are quantitative or descriptive. I~ ~ \b;l.i. Correlation Worksheet (5 points) Dr Sarah L. Napper 3. (no.of pairs) n r 3 0.997 4 0.950 5 0.878 6 0.811 7 0.755 8 0.707 2nd y = Choose first type of graph. A specific value of the x-variable given a specific value of the y-variable c. U~r ~F~ a... cl . Use the following GeoGebra file and student worksheet to learn how the line of best fit and the correlation | {
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following GeoGebra file and student worksheet to learn how the line of best fit and the correlation coefficient are affected by changes in the data. 2 .81 9.2; Since the coefficient of is greater than 0, 10. To find correlation coefficient in Excel, leverage the CORREL or PEARSON function and get the result in a fraction of a second. (yx 5 0.5) (Fake) ⦠Suppose that there are nordered pairs (x;y) that make up a sample from a population. Mathematically, the strength and direction of a linear relationship between two variables is. o��vG D word/_rels/document.xml.rels �(� ���N�0��H�C�;qR���i/�W�ěۑ����5I��z�q��̧���͏�/��5:ci��tid��}�7�,r(��ѐ�=8�Y__�ޠH�\��. 6. You can & download or print using the browser document reader options. Edit â put xâs in L1 and yâs in L1. Find the correlation coefficient between the Average Number of Assignments in Class and the Class Absences. If ris close to 1, we say that the variables are positively correlated. 8. Y-vars. It shows that more the pages are viewed, the more money they spend. Calculate the product-moment correlation coefficient for this data set, giving your answer correct to three decimal places. bb( S'x. Found worksheet you are looking for? represented by the correlation coefficient. If you're seeing this message, it means we're having trouble loading external resources on our website. Calculate the correlation coefficient of the data. "�2� �Ϝ��%\lz�tR��hkދ�S��I��v���'��Vf�n%���.�6UՖ�bʝ�g"xe�%Gak�� s����oC�;@�f�DpT|!r^�0{+X� ��8̾�4d�ީ,5>��b����f-oA�@��aD�[���1J�&��sQt�6F�Gq����a&�@�B�����@�bӸ�oC�d�� �� PK ! Correlation coefficient is obtained as 0.72. Worksheet for Correlation and Regression (February 1, 2013) Part 1. n. P (xy) â (. 'DaR\., -h'O-e,. The correlation coefficient is used to determine: a. f?��3-���]�Tꓸ2�j)�,l0/%��b� 6) r .644 - stays the same 7) r .644 - stays the same . 1; Because the slope of the linear regression equation of best fit is positive (0.5), the correlation coefficient must be positive. | {
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regression equation of best fit is positive (0.5), the correlation coefficient must be positive. 9\~..,f{j ~d. correlation coefficient are. Based on the correlation coefficient, is there a correlation between the heights of daughters and mothers? Predictthe!type!(positive,!negative,!no)!and!strength!of!correlation! Here are data from four students on their Quiz 1 scores and their Quiz 5 scores and a graph where we connected the points by a line. Enter. Correlation Coefficient With Answers - Displaying top 8 worksheets found for this concept. 1; The correlation coefficient for the plot must be negative. Correlation!Coefficient!&Linear!of!Best!Fit!HW! Displaying top 8 worksheets found for - Scatter Plot And Correlation Coefficient Quiz. �Oj� �A word/document.xml�Y�n�6��?\��h,���a�A&�A�Ad�@W#Qa�$H�w��@�s��^ʒm9�����ƲH���s$�W�?$1L��L���t�����ӏ�hCx@b��ЙQ��}�ū���&�@��L�C'2Fz����&Dw�+�Eh:�H\�̧n&T����n�O*�S�q�S§D;\rMHʱ3*!o��M���� �%1������] #�N��W@,YoN���j�q�&g�����1r\GL.��T4�J��C��&q�\&{G���L�/K�M�s�$�3����#ba� ��%��0��IҬ��{�8��:�o眷J�r�ƶC�����GN^��ގ���H����.�\(r##t��`��aŹ��^%dV��z�t�ǃ���7N�t�l��hpؽX4�ѐ��Yy�>CmO#�E�N�^��uy����-P��[���� �֜$8��ފ7ğų�. Explain your answer. Designed for the new GCSE (AQA Higher), this worksheet is for practising equating coefficients in identities. 5. Showing top 8 worksheets in the category - Correlation Coefficient With Answers. Worksheet focuses on matching scatter plots with the correct correlation coefficient. It also tells us if the correlation is _____ or _____. Calculator steps for finding ârâ and graphing: Stat. PK ! Correlation Coefficient With Answers - Displaying top 8 worksheets found for this concept. weak postive b. Calc #4 (LinReg) Vars. The good news is - there is a value called the _____ that helps us determine the _____ of a correlation. may be required to answer a question such as: Estimate the correlation coefficient from a particular scatter plot? Showing top 8 worksheets in the category - Product Moment Correlation | {
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a particular scatter plot? Showing top 8 worksheets in the category - Product Moment Correlation Coefficent. Based on the correlation coefficient, is there a correlation between the heights of daughters and mothers? Worksheet will open in a new window. I * f. 1"?>I.b95i -O.42fl &4 *' 3. Unit 4 Worksheet #1 Intro to correlation As you can see â it is sometimes tricky to decide if a correlation is strong, moderate, or weak. While the LSRL is different, the correlation coefficient remains the same. 11 r. r= ~-~ (x . The CORREL function returns the Pearson correlation coefficient for two sets of values. Answers are provided, plus an editable version. the acceptable alpha level of 0.05, meaning the correlation is statistically significant. This correlation coefficient indicates that there is a stronger relationship between the number of pages viewed on the website and the amount spent. a. Compute the correlation between age in months and number of words known. b. Use the following data set to answer the questions. For example, for n =5, r =0.878 means that there is only a 5% chance of getting a result of 0.878 or greater if there is no correlation between the variables. Based on the computed value of r, what can you say about the association between the temperature and the number of soft drinks sold. i -x)(:J. i - 5) -.--L. Besides whole-class teaching, teachers can consider different grouping strategies â such Worksheet: Spearmanâs Rank Correlation Coefficient Mathematics In this worksheet, we will learn about Spearmanâs rank correltion coefficient. Such a value, therefore, indicates the likely existence of a relationship between the variables. If you must quickly visualize the connection between the 2 variables, draw a linear regression chart. ����ϬA0c��ƣ�Ţ��uM� �3����:*eJ��z�Tx���p�R�@����SB�mO(�w�p$����>�2��M��w������\�l�I���9�i�\e��k7�m!�+�*i+Z(��?�az���i�;!n*�'h|���} �;V0���y'Hn-�}�Fk� F�İw�D(AH���4�g�C/�����'��A���@ �|/���ͩH�z�$��c��G�������\$A���� �� PK | {
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�;V0���y'Hn-�}�Fk� F�İw�D(AH���4�g�C/�����'��A���@ �|/���ͩH�z�$��c��G�������\$A���� �� PK ! Ti 83/84: Linear Regression & Correlation Coefficient Linear Regression, r and r 2 This video demonstrates how to generate the least squares regression line for a set of (x, y) data, how to make a scatter plot of the data with the line shown, and how to predict values of y using the regression line on the Ti-83/84 series graphing calculator. 1. For the sample data $\begin{array}{c|c c c c c} x &1 &3 &4 &6 &8 \\ \hline y &4 &1 &3 &-1 &0\\ \end{array}$ Draw the scatter plot. This will always be a number between -1 and 1 (inclusive). Explain your answer. If you're behind a web filter, please make sure that the domains ⦠Scatterplots and correlation coefficients are two closely related concepts. Print The Correlation Coefficient: Definition, Formula & Example Worksheet 1. Worksheet focuses on matching scatter plots with the correct correlation coefficient. English Unseen Comprehension Passage With Mcq For Class8. Match correlation coefficients to scatterplots to build a deeper intuition behind correlation coefficients. Some of the worksheets for this concept are The correlation coefficient, Grade levelcourse grade 8 and algebra 1, Work 15, Scatter plots, Scatter plots work 1, Scatterplots and correlation, Scatter plots and correlation work name per, Work regression. c. Recall what you learned in Ch. Do this one manually. Some of the worksheets for this concept are The correlation coefficient, Grade levelcourse grade 8 and algebra 1, Work 15, Scatter plots, Scatter plots work 1, Scatterplots and correlation, Scatter plots and correlation work name per, Work regression. Hello Math Teachers! �cG�� � [Content_Types].xml �(� ���j�0E����Ѷ�J�(��ɢ�eh��5�E�����lj)%�Co�̽�H2�h��U��5)&�ɬT�H���-~dQ@a�����m ����f4�8�MHY��8Y Z��:0Tɭ�i��D�- Consider the following hypothetical data set. Linear Regression and Correlation Coefficient Worksheet with Worksheet 05 More Linear Regression. | {
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Linear Regression and Correlation Coefficient Worksheet with Worksheet 05 More Linear Regression. (strong,!weak)!for!the!following! The meaning of a p-value in a Pearson correlation coefficient is to determine whether the correlation between the chosen variables is significant by comparing the p-value to the significance level. A more challenging question can be reserved for others: Interpret the correlation coefficient in the context of the variables? CORRELATION & REGRESSION MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, select the best answer. Enter. You are interested in whether there is an association between age and postmate cognitions (measured using the ⦠Suppose that there are n ordered pairs (x, y) that make up a sample from a population. ���z���ʼn�, � �/�|f\Z���?6�!Y�_�o�]A� �� PK ! RSQ calculates the coefficient of determination [â¦] Yes , how I know this is because the plots on the data would be very close to eachother , ⦠The heights of daughters and mothers called the _____ of a second the following multiple-choice questions, the... The data would be very close to each other almost creating a scatter and... Reported to describe the strength and direction of a second a fraction of a linear relationship between number. This worksheet, we say that the variables and the Class Absences of is greater than 0,.! Variables are positively correlated value called the _____ of a linear relationship between variables... By hand, you 'd have to use this lengthy Formula is obtained as 0.72 of pages viewed the! Positively correlated the browser document reader options for others: Interpret the coefficient. There you have it question can be reserved for others: Interpret the correlation coefficient WorksheetName: steps! It also tells us if the correlation coefficient r is given by the correlation coefficient is. * f. 1 ''? > I.b95i -O.42fl & 4 * ' 3 a number between and...! of! correlation With Answers unclear, but there you have it, and.... | {
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# Find value of $a_1$ such that $a_{101}=5075$
Let $$\{a_n\}$$ be a sequence of real numbers where $$a_{n+1}=n^2-a_n,\, n=\{1,2,3,...\}$$
Find value of $$a_1$$ such that $$a_{101}=5075$$.
I have
$$a_2=1^2-a_1$$
$$a_3=2^2-a_2=2^2-1^2+a_1$$
$$a_4=3^2-2^2+1^2-a_1$$
$$a_5=4^2-3^2+2^2-1^2+a_1$$
$$\vdots$$
$$a_{101}=100^2-99^2+98^2-97^2+\ldots+2^2-1^2+a_1.$$
Therefore, $$a_{101}=\sum_{i=1}^{50}(2i)^2-\sum_{i=1}^{50}(2i-1)^2.$$
Thus, $$5075=\sum_{i=1}^{50}(4i^2-4i^2+4i-1)+a_1,$$
and, $$a_1=5075-4\sum_{i=1}^{50}(i)+\sum_{i=1}^{50}(1).$$
Hence, $$a_1=5075-4(\frac{50}{2})(51)+50=25,$$
and $$a_1=25$$.
Is it correct? Do you have another way? Please check my solution, thank you.
Yes, your solution is correct. Another method of solution is to note that if $$a_{n+1} = n^2 - a_n,$$ we want to find some (possibly constant) function of $$n$$ such that $$a_{n+1} - f(n+1) = -(a_n - f(n)).$$ This of course implies $$f(n+1) + f(n) = n^2.$$ A quadratic polynomial should do the trick: suppose $$f(n) = an^2 + bn + c,$$ so that $$n^2 = f(n+1) + f(n) = 2a n^2 + 2(a+b)n + (a+b+2c).$$ Equating coefficients in $$n$$ gives $$a = 1/2$$, $$b = -1/2$$, $$c = 0$$, hence $$f(n) = \frac{n^2 - n}{2}.$$ It follows that if $$b_n = a_n - f(n) = a_n - \frac{n^2-n}{2},$$ then $$b_{n+1} = - b_n.$$ This gives us $$b_1 = b_{101}$$ which in terms of $$a_n$$, is $$a_1 = a_1 - \frac{1^2 - 1}{2} = a_{101} - \frac{(101)^2 - 101}{2} = 5075 - 5050 = 25.$$ This solution seems to come out of nowhere, but it is motivated by the idea that if we can transform the given recurrence into a corresponding recurrence for a sequence that is much simpler to determine, we can use this to recover information about the original sequence.
What if at the point where:
$$a_{101} = 100^2 - 99^2 + 98^2 - 97^2 \cdots 2^2 - 1^2 + a_1$$
You opted for a clever factorization of squares:
$$a_{101} = (100 - 99)(100 + 99) + (98 - 97)(98 + 97) \cdots (2 - 1)(2 + 1) + a_1$$
$$a_{101} = 199 + 195 + 191 \cdots 3 + a_1$$ | {
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$$a_{101} = 199 + 195 + 191 \cdots 3 + a_1$$
Therefore those numbers become a simple arithmetic series with the first term as 3 and the common ratio as 4.
But how many terms exactly?
$$3 + 4(n - 1) = 199$$
$$4(n - 1) = 196$$
$$n - 1 = 49$$
$$n = 50$$
Therefore with the knowledge of evaluating the sum this comes down to:
$$a_{101} = 25(199 + 3) + a_1$$
$$a_{101} = 5050 + a_1$$
And by substituting the choice for a_101:
$$5075 = 5050 + a_1$$
$$a_1 = 25$$
Thanks to @Zera for recommending the edit. I'm learning how to use these markup languages
• Question to @Zera, is my reasoning the same as yours. I feel like it is – Nεo Pλατo Dec 21 '19 at 16:59
In the most simple way:
$$A_{n+1}+A_n=n^2~~~(1)$$ is a non-homogeneous recurrence equation. Its homogeneous part $$A_{n+1}+A_n=0 \implies A_{n+1}=-A_n \implies A_n= (-1)^n S~~~(2)$$ In the RHS of (1) being $$n^2)$$, we can take $$A_n =P n^2+Q n+R$$; inserting this in (1), we get $$2P=1,P+Q=0,P+Q+2R=0 \implies P=1/2, Q=1/2, R=0.$$ Then the total finally solution of (1) is $$A_n=\frac{n(n-1)}{2}+ (-1)^n S.$$ Given that $$A_{101}=5075$$, finally we get $$A_n=\frac{n(n-1)}{2}+(-1)^{n+1}~ 25.$$
Another way :
$$a_{n+1}=n^2-a_n=n^2-((n-1)^2-a_{n-1})=a_{n-1}+2n-1$$
$$a_{n-1}=a_{n-3}+2(n-1)-1=a_{n-3}+2(n-2)+1$$
$$a_{n+1}=a_{n+1-2r}+\underbrace{2n-1+2n-3+\cdots}_{r\text{ terms}}$$
Here $$n+1=101,n+1-2r=1$$
You could also have notice the pattern $$a_n=(-1)^{n+1}a_1+\frac{n(n-1) }{2}$$ | {
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1. ## Vector Intersection
Hi, I have a mechanics question here I can't quite get.
A destroyer sights a ship at a point with position vector 600(3i + j)m relative to it and moving with velocity 5j m/s. The destroyer alters course so that it moves with speed v m/s in the direction of the vector 4i + 3j. Find v so that the destroyer intercepts the ship and the time to the interception.
Any help would be greatly appreciated.
Thanks
2. Hello steve989
Originally Posted by steve989
Hi, I have a mechanics question here I can't quite get.
A destroyer sights a ship at a point with position vector 600(3i + j)m relative to it and moving with velocity 5j m/s. The destroyer alters course so that it moves with speed v m/s in the direction of the vector 4i + 3j. Find v so that the destroyer intercepts the ship and the time to the interception.
Any help would be greatly appreciated.
Thanks
Denote the velocity of the destroyer by $\vec{v_D}$ and the velocity of the ship by $\vec{v_S}$. Then:
$\vec{v_D}=\frac{v}{5}\Big(4\vec i + 3 \vec j\Big)$, since this vector has magnitude $v$.
and
$\vec{v_S} = 5\vec j$
So the velocity of the ship relative to the destroyer is
$\vec{v_S}-\vec{v_D} = -\frac{4v}{5}\vec i +\Big(5-\frac{3v}{5}\Big)\vec j$
Therefore after time $t$, the position of the ship relative to the destroyer is:
$600(3\vec i + \vec j) -\frac{4vt}{5}\vec i +\Big(5-\frac{3v}{5}\Big)t\vec j$
$=0\vec i + 0\vec j$ when the destroyer intercepts the ship
Solving for $v$ and $t$ gives:
$t = 390$ and $v = \frac{75}{13}$
But check my working!
3. Hello, steve989!
A destroyer sights a ship at a point with position vector $1800i + 600j$ relative to it
and moving with velocity $5j$ m/s.
The destroyer alters course so that it moves with speed $v$ m/s
in the direction of the vector $4i+3j$ | {
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Find $v$ so that the destroyer intercepts the ship and the time to the interception.
Code:
|
| o P
| o |
| o | 5t
| o |
| o |
| o * S
| o |
| * |
| v * | | 600
| * |3 |
| * θ | |
D* - - - + - - - - - - *
: 4 :
: - - - -1800 - - - - :
The destroyer is at: $D(0,0)$
The ship is at: $S(1800, 600)$
The ship is moving north at 5 m/s.
In the next $t$ seconds, it moves $5t$ m to point $P.$
. . Its position vector is: . $\vec S \:=\:\langle 1800,\:600+5t\rangle$
The destroyer heads in direction $4i+3j$
Let $\theta$ represent its direction, where: $\cos\theta \,=\,\tfrac{4}{5},\;\sin\theta\,=\,\tfrac{3}{5}$
. . Its position vector is: . $\vec D \;=\;\langle (v\cos\theta)t,\:(v\sin\theta)t\rangle \;=\;\left\langle\tfrac{4}{5}vt,\:\tfrac{3}{5}vt\r ight\rangle$
The destroyer intercepts the ship at point $P.$
This happens when: $\vec D \,=\,\vec S$
We have: . $\begin{Bmatrix}\frac{4}{5}vt &=& 1800 & [1] \\ \\[-3mm] \frac{3}{5}vt &=& 600 + 5t & [2] \end{Bmatrix}$
Divide [2] by [1]: . $\frac{600+5t}{1800} \:=\:\frac{\frac{3}{5}vt}{\frac{4}{5}vt} \quad\Rightarrow\quad\frac{120+t}{360} \:=\:\frac{3}{4}$
. . $480 + 4t \:=\:1080 \quad\Rightarrow\quad 4t \:=\:600 \quad\Rightarrow\quad t \:=\:150$
Therefore, the destroyer intercepts the ship in 150 seconds.
Substitute into [1]: . $\tfrac{4}{5}v(150) \:=\:1800 \quad\Rightarrow\quad v \:=\:15$
Therefore, the destroyer must travel at 15 m/s.
4. Thanks for both of your replies! I've gone though them as well as I can.
Grandad - I tried your way and think I see how you're thinking there, I got a different formula for one stage though. | {
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Therefore after time , the position of the ship relative to the destroyer is:
when the destroyer intercepts the ship
For that bit; wouldn't it be $600(3i + j) + (4/5)vti - (5 - (3/5)v)tj$ ? The signs switch as they are on different sides of the equation I assumed, but I may of miss-understood since I got a completely different answer anyway. Sorry about the layout too, I'm useless with this maths typing.
Soroban - I do like that approach and the answers seem very reasonable, I missed the fact that the ship was moving directly north. I do have one question though for both you and grandad; why does the velocity vector of the destroyer end up with $v/5$ as the scalar? I thought it was just $v$, I think I'm going to get a simple answer to that question though.
5. Hello everyone
I agree with Soroban's answer: I got a sign wrong in my working. My equation for the relative velocity was correct, though. The principle is:
The velocity of A relative to B is the velocity of A minus the velocity of B.
So, as I said:
$\vec{v_S}-\vec{v_D} = -\frac{4v}{5}\vec i +\Big(5-\frac{3v}{5}\Big)\vec j$
not as you suggest:
Originally Posted by steve989
...
For that bit; wouldn't it be $600(3i + j) + (4/5)vti - (5 - (3/5)v)tj$ ? The signs switch as they are on different sides of the equation I assumed, but I may of miss-understood since I got a completely different answer anyway. Sorry about the layout too, I'm useless with this maths typing.
However, I then got a sign wrong when I equated the $\vec j$ component of the displacement to zero. The equations should have been:
$\vec i$ component: $1800 - \frac{4vt}{5} = 0$
$\vec j$ component: $600 + 5t -\frac{3vt}{5}=0$
which, of course, are the same as Soroban's equations [1] and [2].
Originally Posted by steve989
...I do have one question though for both you and grandad; why does the velocity vector of the destroyer end up with $v/5$ as the scalar? I thought it was just $v$, I think I'm going to get a simple answer to that question though. | {
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Because the magnitude of $4\vec i + 3\vec j$ is $5$. So $v(4\vec i + 3\vec j)$ will have magnitude $5v$.
6. Ahh I see, all makes perfect sense now, I knew there had to be simple reasoning behind that. Thanks a lot to you both. | {
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If the set $$S$$ consists of five consecutive positive integers, what is the sum of these five integers?
(1) The integer 11 is in $$S$$, but 10 is not in $$S$$.
(2) The sum of the even integers in $$S$$ is 26
[Reveal] Spoiler: OA
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Re: If the set S consists of five consecutive positive integers, what is.. [#permalink]
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nalinnair wrote:
If the set $$S$$ consists of five consecutive positive integers, what is the sum of these five integers?
(1) The integer 11 is in $$S$$, but 10 is not in $$S$$.
(2) The sum of the even integers in $$S$$ is 26
Statement 1: The integer 11 is in S, but 10 is not in S
This means the integers start from 11
The integers will be 11, 12, 13, 14, 15
We can find out the sum.
SUFFICIENT
NOTE: We do not need to find out the actual sum. | {
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NOTE: We do not need to find out the actual sum.
Statement 2: The sum of the even integers in S is 26
The only possible series is 11, 12, 13, 14, 15
We can find out the sum
SUFFICIENT
Correct Option: D
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nalinnair wrote:
If the set $$S$$ consists of five consecutive positive integers, what is the sum of these five integers?
(1) The integer 11 is in $$S$$, but 10 is not in $$S$$.
(2) The sum of the even integers in $$S$$ is 26
(1) The integer 11 is in $$S$$, but 10 is not in $$S$$.
Since number are consecutive, it there cannot be any skipping among the number. If 11 is there, then the series should start with 11
s={11,12,13,14,15}; 12 and 14 are even
SUFFICIENT
(2) The sum of the even integers in $$S$$ is 26
Now there are total of 5 integers. there are two possibilities
OEOEO OR EOEOE
OEOEO means two consecutive even ; sum of these is 26
==>E+(E+2)=26 ==>E=12, second even = 14
EOEOE means there are three consective even numbers
E+(E+2)+(E+4)=26
==>3E=20
==>E=20/3 NOT ACCEPTABLE because its not an integer value.
SO in any case both cases gives unique values
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Sorry to bring this one back. By consecutive, do we just assume they mean x, x+1, x+2 or anything that has a pattern? ie. x+2, x+5, x+8...?
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Smileyface123 wrote:
Sorry to bring this one back. By consecutive, do we just assume they mean x, x+1, x+2 or anything that has a pattern? ie. x+2, x+5, x+8...?
Since the question states 5 consecutive positive numbers, it means x, x+1, x+2....
Else, it would have been indicated about the pattern.
Hope it helps.
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Re: If the set S consists of five consecutive positive integers, what is.. [#permalink]
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24 Nov 2016, 02:28
Statement 1 : The integer 11 is in S, but 10 is not in S = It means it has to start from 11. Hence the count would start from 11,12,13,14,15 = Sufficient
Statement 2 : The sum of the even integers in SS is 26 = In this case only one set can be formed . 11,12,13,14,15 and in this sum of even numbers are 26 hence Sufficient.
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# If x is a positive integer, is x-1 a factor of 104?
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If x is a positive integer, is x – 1 a factor of 104?
(1) x is divisible by 3.
(2) 27 is divisible by x.
[Reveal] Spoiler: OA
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enigma123 wrote:
If x is a positive integer, is x – 1 a factor of 104?
(1) x is divisible by 3.
(2) 27 is divisible by x.
For me the answer is clearly B. But OA is C. Can someone please explain?
If x is a positive integer, is x – 1 a factor of 104?
(1) x is divisible by 3 --> well, this one is clearly insufficient, as x can be 3, x-1=2 and the answer would be YES but if x is 3,000 then the answer would be NO. | {
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(2) 27 is divisible by x --> factors of 27 are: 1, 3, 9, and 27. Now, if x is 3, 9, or 27 then the answer would be YES (as 2, 8, and 26 are factors of 104) BUT if x=1 then x-1=0 and zero is not a factor of ANY integer (zero is a multiple of every integer except zero itself and factor of none of the integer). Not sufficient.
(1)+(2) As from (1) x is a multiple of 3 then taking into account (2) it can only be 3, 9, or 27. For all these values x-1 is a factor of 104. Sufficient.
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22 Jan 2012, 17:08
Thanks very much buddy for shedding light on concept of ZERO.
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Hi Guys,
The question deals with the concepts of factors and multiples of a number. Its important to analyze the information given in the question first before preceding to the statements. Please find below the detailed solution:
Step-I: Understanding the Question
The question tells us that $$x$$ is a positive integer and asks us to find if $$x-1$$ is a factor of 104
Step-II: Draw Inferences from the question statement
Since $$x$$ is a +ve integer, we can write $$x>0$$. The question talks about the factors of 104. Let's list out the factors of 104.
$$104 = 13 * 2^3$$. So, factors of 104 are {1,2,4,8,13,26,52,104}, a total of 8 factors.
If $$x-1$$ is to be a factor of 104, $$2<=x<=105$$. With these constraints in mind lets move ahead to the analysis of the statements. | {
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Step-III: Analyze Statement-I independently
St-I tells us that $$x$$ is divisible by 3. This would mean that $$x$$ can take a value of any multiple of 3. Now, all the multiples of 3 are not factors of 104. So, we can't say for sure if $$x-1$$ is a factor of 104. Hence, statement-I alone is not sufficient to answer the question.
Step-IV: Analyze Statement-II independently
St-II tells us that 27 is divisible by $$x$$ i.e. $$x$$ is a factor of 27. Let's list out the factors of 27 - {1,3,9,27}. But, we know that for $$x-1$$ to be a factor of 104, $$2<=x<=105$$. We see from the values of factors of 27, $$x$$ can either be less than 2(i.e. 1) or greater than 2 (i.e. 3,9 & 27). Hence, statement-II alone is not sufficient to answer the question.
Step-V: Analyze both statements together
St-I tells us that $$x$$ is a multiple of 3 and St-II tells us that $$x$$ can take a value of {1, 3, 9, 27}. Combining these 2 statements we can eliminate $$x=1$$ from the values which $$x$$ can take. So, $$x$$={ 3, 9, 27} and $$x-1$$ = {2, 8, 26}. We observe that all the values which $$x-1$$ can take is a factor of 104. Hence, combining st-I & II is sufficient to answer our question.
Takeaway
Analyze the information given in the question statement properly before proceeding for analysis of the statements. Had we not put constraints on the values of x, we would not have been able to eliminate x=1 from st-II analysis.
Hope it helps!
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20 Sep 2016, 10:28
Let's take x=30, in this case,
1. A Will be sufficient. However 30-1 is 29 is not a factor of 104.
2. 27 is also not divisible by 30. Not sufficient. | {
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Hence, In this case is the answer E. Can anybody answer my doubt.
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21 Sep 2016, 01:47
prashantrchawla wrote:
Let's take x=30, in this case,
1. A Will be sufficient. However 30-1 is 29 is not a factor of 104.
2. 27 is also not divisible by 30. Not sufficient.
Hence, In this case is the answer E. Can anybody answer my doubt.
I am not sure what you are trying to do here.
for statement 1 : you are considering only one value of x, which is making your case sufficient. Take x =3 and x = 6, you will get 104 divisible for x-1 = 2 but not for x-1 = 5.
Hence, it is Insufficient.
Statement 2 : We are given 27 is divisible by x. It means x is a factor of 27. The factors could be 1,3,9 and 27.
Divide 104 by each of (x-1) as 0, 2,8 and 26. You will find 104 divisible by all but 0. hence, insufficient.
On combining, we know that x cannot be 0. Hence, Answer C.
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Re: If x is a positive integer, is x-1 a factor of 104? [#permalink]
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21 Sep 2016, 03:13
prashantrchawla wrote:
Let's take x=30, in this case,
1. A Will be sufficient. However 30-1 is 29 is not a factor of 104.
2. 27 is also not divisible by 30. Not sufficient.
Hence, In this case is the answer E. Can anybody answer my doubt.
Your logic there is not clear. Why do you take x as 30? You cannot arbitrarily take x to be 30 and work with this value only. Also, how is the first statement sufficient? If x is 3, then x-1=2 and the answer would be YES but if x is 3,000 then the answer would be NO.
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04 Dec 2017, 11:54
enigma123 wrote:
If x is a positive integer, is x – 1 a factor of 104?
(1) x is divisible by 3.
(2) 27 is divisible by x.
Lets look at the prime factorisation of 104: 2^3 * 13
Thus factors of 104 = 1, 2, 4, 8, 13, 26, 52, 104
We are asked whether x-1 is one of these 8 integers, OR IS x one of these: 2, 3, 5, 9, 14, 27, 53, 105
(1) x is divisible by 3, so x could be any multiple of 3 like 9 or 27 or 54. Insufficient.
(2) 27 is divisible by x, so x is a factor of 27. Now factors of 27 are: 1, 3, 9, 27.
If x is 1, then x-1 is 0 and thus NOT a factor of 104, but if x is 3 or 9 or 27, then x-1 will take values as 2 or 8 or 26 respectively, and thus BE a factor of 104.
So Insufficient.
Combining the two statements, x has to be a multiple of 3, yet a factor of 27 also. So x could be either 3 or 9 or 27. For each of these cases, x-1 will be a factor of 104, as explained in statement 2. Sufficient. Hence C answer
Re: If x is a positive integer, is x-1 a factor of 104? [#permalink] 04 Dec 2017, 11:54
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# Finding a closed formula for $1\cdot2\cdot3\cdots k +\dots + n(n+1)(n+2)\cdots(k+n-1)$
Considering the following formulae:
(i) $1+2+3+..+n = n(n+1)/2$
(ii) $1\cdot2+2\cdot3+3\cdot4+…+n(n+1) = n(n+1)(n+2)/3$
(iii) $1\cdot2\cdot3+2\cdot3\cdot4+…+n(n+1)(n+2) = n(n+1)(n+2)(n+3)/4$
Find and prove a ‘closed formula’ for the sum
$1\cdot2\cdot3\cdot…\cdot k + 2\cdot3\cdot4\cdot…\cdot(k+1) + … + n(n+1)(n+2)\cdot…\cdot (k+n-1)$
generalizing the formulae above.
I have attempted to ‘put’ the first 3 formulae together but I am getting nowhere and wondered where to even start to finding a closed formula.
#### Solutions Collecting From Web of "Finding a closed formula for $1\cdot2\cdot3\cdots k +\dots + n(n+1)(n+2)\cdots(k+n-1)$"
The pattern looks pretty clear: you have
\begin{align*} &\sum_{i=1}^ni=\frac12n(n+1)\\ &\sum_{i=1}^ni(i+1)=\frac13n(n+1)(n+2)\\ &\sum_{i=1}^ni(i+1)(i+2)=\frac14n(n+1)(n+2)(n+3)\;, \end{align*}\tag{1}
where the righthand sides are closed formulas for the lefthand sides. Now you want
$$\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)\;;$$
what’s the obvious extension of the pattern of $(1)$? Once you write it down, the proof will be by induction on $n$.
Added: The general result, of which the three in $(1)$ are special cases, is $$\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)=\frac1{k+1}n(n+1)(n+2)\dots(n+k)\;.\tag{2}$$ For $n=1$ this is $$k!=\frac1{k+1}(k+1)!\;,$$ which is certainly true. Now suppose that $(2)$ holds. Then
\begin{align*}\sum_{i=1}^{n+1}i(i+1)&(i+2)\dots(i+k-1)\\ &\overset{(1)}=(n+1)(n+2)\dots(n+k)+\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)\\ &\overset{(2)}=(n+1)(n+2)\dots(n+k)+\frac1{k+1}n(n+1)(n+2)\dots(n+k)\\ &\overset{(3)}=\left(1+\frac{n}{k+1}\right)(n+1)(n+2)\dots(n+k)\\ &=\frac{n+k+1}{k+1}(n+1)(n+2)\dots(n+k)\\ &=\frac1{k+1}(n+1)(n+2)\dots(n+k)(n+k+1)\;, \end{align*} | {
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exactly what we wanted, giving us the induction step. Here $(1)$ is just separating the last term of the summation from the first $n$, $(2)$ is applying the induction hypothesis, $(3)$ is pulling out the common factor of $(n+1)(n+2)\dots(n+k)$, and the rest is just algebra.
If you divide both sides by $k!$ you will get binomial coefficients and you are in fact trying to prove
$$\binom kk + \binom{k+1}k + \dots + \binom{k+n-1}k = \binom{k+n}{k+1}.$$
This is precisely the identity from this question.
The same argument for $k=3$ was used here.
Or you can look at your problem the other way round: If you prove this result about finite sums
$$\sum_{j=1}^n j(j+1)\dots(j+k-1)= \frac{n(n+1)\dots{n+k-1}}{k+1},$$
you also get a proof of the identity about binomial coefficients.
For a fixed non-negative $k$, let $$f(i)=\frac{1}{k+1}i(i+1)\ldots(i+k).$$
Then $$f(i)-f(i-1)=i(i+1)\ldots(i+k-1).$$
By telescoping,
$$\sum_{i=1}^ni(i+1)(i+2)\dots(i+k-1)=\sum_{i=1}^n\left(f(i)-f(i-1)\right)=f(n)-f(0)=f(n)$$
and we are done.
I asked exactly this question a couple of days ago, here:
Telescoping series of form $\sum (n+1)\cdot…\cdot(n+k)$
My favourite solution path so far is
$$n(n+1)\cdot…\cdot(n+k)/(k+1)$$ | {
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# Evaluating an integral 2.
I'm calculating the following integral: $\int \frac{1}{\sqrt{x^2+2}}dx$
I tried the following:
Performing $u$ substitution:
Let $x=\sqrt{2}\tan(u)$
$x^2=2\tan^2(u)$
$dx=\sqrt{2}(1+\tan^2(u))du$
$\int \frac{1}{\sqrt{x^2+2}}dx=\int \frac{\sqrt{2}(1+\tan^2(u))du}{\sqrt{2\tan^2(u)+2}}= \int \frac{(1+\tan^2(u))du}{\sqrt{\tan^2(u)+1}} = \int \frac{(1+\tan^2(u))du}{\sqrt{\tan^2(u)+1}}=\int \frac{1}{\cos u}du$
=$\int \frac{\cos u}{ cos^2u}du=\int \frac{\cos u}{1-\sin^2u}du$
Let $t = \sin u$
$dt=\cos u.du$
$\int \frac{dt}{1-t^2}du= \frac{1}{2}\int\frac{1}{1-t}+\frac{1}{2}\int\frac{1}{1+t}du=\frac{1}{2}\ln(\frac{1+t}{1-t})=\frac{1}{2}\ln(\frac{1+\sin u}{1-\sin u})$
$=\ln(\sqrt{\frac{1+\sin u}{1-\sin u}})=\ln\left(\sqrt{\frac{(1+\sin u)^2}{1-\sin^2u}}\right)=\ln\left(\frac{1+\sin u}{\cos u}\right)=\ln\left(\frac{1}{\cos u}+\tan u\right)$
Substituting $u$ with $\arctan\frac{x}{\sqrt{2}}$ We get:
$\ln\left(\frac{1}{\cos u}+\tan u\right)=\ln\left(\frac{x}{\sqrt{2}}+\sqrt{1+\frac{x^2}{2}}\right)$.
Although the formula says $\int \frac{1}{\sqrt{x^2+a^2}}dx=\ln(x+\sqrt{x^2+a^2})$
So the answer shoud have been $ln(x+\sqrt{x^2+2})$?
Thanks for the help!
You are absolutely correct. Note that the difference between your answer and the most common form of the answer is a constant. Note that: $$\ln(\frac{x}{\sqrt 2} + \sqrt{1+ \frac{x^2}{2}}) = \ln(\frac{1}{\sqrt 2}\left[x + \sqrt{x^2+2}\right]) = \ln(x + \sqrt{x^2+2}) - \color{red}{\ln \sqrt 2}$$
where the red part is a constant. Note that $\ln(ab) = \ln a + \ln b$.
your result is given by $$\ln(x+\sqrt{2+x^2})-\ln(\sqrt{2})$$ it differes only by a costant so the results are equal | {
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# Probability that someone will pick a red ball first?
A father and son take turns picking red and green balls from a bag. There are 2 red balls and 3 green balls. The first person to pick a red ball wins. There is no replacement. What is the probability that the father wins if he goes first?
I drew a binary tree to solve this. The father can only win the first round and the third round.
P(father wins first round) = $\frac25$
P(father wins third round) = $\frac35 * \frac24 * \frac23 = \frac15$
P(father wins first round) + P(father wins third round) = $\frac25+\frac15 =\frac35$
Is this correct?
• Yes, this is correct. Nov 27 '14 at 18:56
• By the way, you can get a multiplication dot, as in $\frac35\cdot\frac 23$, by typing \cdot.
– MJD
Nov 27 '14 at 19:21
Your proposed solution is exactly correct. Nice work.
To check your answer, you can use the same method to calculate the second player's probability of winning; it ought to be $1-\frac35 =\frac 25$. Let $P_i$ be the probability that the game ends in round $i$; you have calculated $P_1 + P_3 = \frac 35$. Then \begin{align} P_2 & = \frac35\cdot \frac 24 & = \frac 3{10}\\ P_4 & = \frac 35\cdot \frac24\cdot\frac 13\cdot \frac22 &= \frac1{10} \end{align}
So $P_2 + P_4 = \frac25$ as we expected.
Correct
An alternative approach is, out of all $\binom{5}{2}$ ways to place red (and green) balls in a line, count ways that place the second red ball when the first red ball is either the first or third ball in line.
$$\dfrac {\binom{4}{1}+\binom{2}{1}}{\binom{5}{2}}=\frac 3 5$$
Can this be interpreted as winning in first round OR not losing in second round AND winning in third round
which would be P(wr1) + P(nlr2) * P(wr3)
P(winning in round 1) = P(red ball)/P(total) = 2/5
P( not losing in round 2) = P(picking a green ball) * P(opponent picking a green ball from remaining) = 3/5 * 2/4
P( winning in round 3) = P(picking a red ball from remaining) = 2/3
Total probability = 2/5 + (3/5 * 2/4) * 2/3 | {
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Total probability = 2/5 + (3/5 * 2/4) * 2/3
Trying to create a recurrence would be:
Pw(2,3) + Pnl(2,3) * Pw(2,1)
which is Probability of choosing red with 2 red, 3 green + Probability of not loosing * Probability of winning with 2 red and 1 green.
• Use LaTeX please. Jun 21 '19 at 4:52 | {
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Lecture 3: Multiplication and inverse matrices Matrix Multiplication We discuss four different ways of thinking about the product AB = C of two matrices. Suppose $A$ is an invertable matrix. In Problem, examine the product of the two matrices to determine if each is the inverse of the other. Step by Step Explanation. Can any system be solved using the multiplication method? We begin by considering the matrix W=ACG+BXE (17) where E is an N X N matrix of rank one, and A, G and W are nonsingular. Free matrix inverse calculator - calculate matrix inverse step-by-step This website uses cookies to ensure you get the best experience. around the world. Just to provide you with the general idea, two matrices are inverses of each other if their product is the identity matrix. We use cij to denote the entry in row i and column j of matrix … Find a Linear Transformation Whose Image (Range) is a Given Subspace. Proof of the Property. Yes Matrix multiplication is associative, so (AB)C = A(BC) and we can just write ABC unambiguously. This site uses Akismet to reduce spam. But the product ab D 9 does have an inverse, which is 1 3 times 1 3. The Inverse of a Product AB For two nonzero numbers a and b, the sum a C b might or might not be invertible. If A is an M by n matrix and B is a square matrix of rank n, then rank(AB) = rank(A). Solutions depend on the size of two matrices. Let A be an m×n matrix and B be an n×lmatrix. All Rights Reserved. Inverse of a Matrix The matrix B is the inverse of matrix A if $$AB = BA = I$$. - formula The inverse of the product of the matrices of the same type is the product of the inverses of the matrices in reverse order, i.e., ( A B ) − 1 = B − 1 A − 1 Required fields are marked *. (b) If the matrix B is nonsingular, then rank(AB)=rank(A). - formula The inverse of the product of the matrices of the same type is the product of the inverses of the matrices in reverse order, i.e., (A B) − 1 = B − 1 A − 1 (A B C) − 1 = C − 1 B − 1 A − 1 Finding the | {
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in this browser for the next time I comment. We use cij to denote the entry in row i and column j of matrix … To prove this property, let's use the definition of inverse of a matrix. Well, for a 2x2 matrix the inverse is: In other words: swap the positions of a and d, put negatives in front of b and c, and divide everything by the determinant (ad-bc). Problems in Mathematics © 2020. The numbers a D 3 and b D 3 have inverses 1 3 and 1 3. If this is the case, then the matrix B is uniquely determined by A, and is called the inverse of A, denoted by A−1. In addition to multiplying a matrix by a scalar, we can multiply two matrices. Donate or volunteer today! Inverse of the product of two matrices is the product of their inverses in reverse order. Let $V$ be the subspace of $\R^4$ defined by the equation $x_1-x_2+2x_3+6x_4=0.$ Find a linear transformation $T$ from $\R^3$ to... (a) Prove that the matrix $A$ cannot be invertible. A square matrix that is not invertible is called singular or degenerate. This is often denoted as $$B = A^{-1}$$ or $$A = B^{-1}$$. Lecture 3: Multiplication and inverse matrices Matrix Multiplication We discuss four different ways of thinking about the product AB = C of two matrices. Now that we know how to find the inverse of a matrix, we will use inverses to solve systems of equations. Apparently this is a corollary to the theorem If A and B are two matrices which can be multiplied, then rank(AB) <= min( rank(A), rank(B) ). Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. Suppose A and B are invertible, with inverses A^-1 and B^-1. If A is an m × n matrix and B is an n × p matrix, then C is an m × p matrix. How do you solve #4x+7y=6# and #6x+5y=20# using elimination? Then prove the followings. Their sum aCb D 0 has no inverse. Everybody knows that if you consider a product of two square matrices | {
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sum aCb D 0 has no inverse. Everybody knows that if you consider a product of two square matrices GH, the inverse matrix is given by H-1G-1. About. A square … Then B^-1A^-1 is the inverse of AB: (AB)(B^-1A^-1) = ABB^-1A^-1 = AIA^-1 = A A^-1 = I Let C m n and C n be the set of all m n matrices and n 1 matrices over the complex field C , respectively. Khan Academy is a 501(c)(3) nonprofit organization. Pseudo inverse of a product of two matrices with different rank. How to Diagonalize a Matrix. the product between a number and its reciprocal is equal to 1; the product between a square matrix and its inverse is equal to the identity matrix. But the problem of calculating the inverse of the sum is more difficult. Then there exists some matrix $A^{-1}$ such that [math]AA^{-1} = I. A square matrix \mathbf{A} of order n is a regular (invertible) matrix if exists a matrix \mathbf{B}such that \mathbf{A}\mathbf{B} = \mathbf{B} \mathbf{A} = \mathbf{I}, where \mathbf{I} is an identity matrix. Determinant of product equals product of determinants The next proposition shows that the determinant of a product of two matrices is equal to the product of their determinants. Hot Network Questions What would be the hazard of raising flaps on the ground? Bigger Matrices. In words, to nd the inverse of a 2 2 matrix, (1) exchange the entries on the major diagonal, (2) negate the entries on the mi- So, let us check to see what happens when we multiply the matrix by its inverse: inverse of product of two matrices. If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$. Our previous analyses suggest that we search for an inverse in the form W -' = A `0 G -' - … The Inverse of a Product AB For two nonzero numbers a and b, the sum a C b might or might not be invertible. News; With Dot product(Ep2) helping us to represent the system of equations, we can move on to discuss identity and inverse matrices. Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to | {
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and inverse matrices. Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$, How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix, The Intersection of Two Subspaces is also a Subspace, Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$, Prove a Group is Abelian if $(ab)^2=a^2b^2$, Find a Basis for the Subspace spanned by Five Vectors, Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis, Find an Orthonormal Basis of $\R^3$ Containing a Given Vector. Consider a generic 2 2 matrix A = a b c d It’s inverse is the matrix A 1 = d= b= c= a= where is the determinant of A, namely = ad bc; provided is not 0. Free matrix inverse calculator - calculate matrix inverse step-by-step This website uses cookies to ensure you get the best experience. (adsbygoogle = window.adsbygoogle || []).push({}); Condition that Two Matrices are Row Equivalent, The Null Space (the Kernel) of a Matrix is a Subspace of $\R^n$, If Generators $x, y$ Satisfy the Relation $xy^2=y^3x$, $yx^2=x^3y$, then the Group is Trivial, Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian Group. The multiplicative inverse of a matrix is the matrix that gives you the identity matrix when multiplied by the original matrix. An identity matrix with a dimension of 2×2 is a matrix with zeros everywhere but with 1’s in the diagonal. ST is the new administrator. These two types of matrices help us to solve the system of linear equations as we’ll see. You can easily nd the inverse of a 2 2 matrix. See all questions in Linear Systems with Multiplication. In the last video we learned what it meant to take the product of two matrices. The numbers a D 3 and b D 3 have inverses 1 3 and 1 3. Note: invertible=nonsingular. The product of two matrices can be computed by multiplying elements of the first row of the first matrix with the first column of the second matrix then, add all the product of elements. | {
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the first matrix with the first column of the second matrix then, add all the product of elements. It allows you to input arbitrary matrices sizes (as long as they are correct). Making use of the fact that the determinant of the product of two matrices is just the product of the determinants, and the determinant of the identity matrix is 1, we get det (A) det (A − 1) = 1. Inverse of product of two or more matrices. Matrix multiplication is associative, so #(AB)C = A(BC)# and we can just write #ABC# unambiguously. Determining invertible matrices. It follows that det (A A − 1) = det (I). Therefore, for a matrix \mathbf{B} we are introducing a special label: if a matrix \mathbf{A} has the inverse, that we will denote as \mathbf{A^{-1}}. But the product ab D 9 does have an inverse, which is 1 3 times 1 3. This precalculus video tutorial explains how to determine the inverse of a 2x2 matrix. How do you solve systems of equations by elimination using multiplication? A matrix that has an inverse is an invertible matrix. Last modified 10/16/2017, Your email address will not be published. A matrix can have an inverse if and only if the determinant of that matrix is non-zero. Matrix Multiplication Calculator (Solver) This on-line calculator will help you calculate the __product of two matrices__. Determining invertible matrices. We can now determine whether two matrices are inverses, but how would we find the inverse of a given matrix? Product of two matrices. Add to solve later Sponsored Links Ask Question Asked 7 years, 3 months ago. So if we have one matrix A, and it's an m by n matrix, and then we have some other matrix B, let's say that's an n by k matrix. Let us try an example: How do we know this is the right answer? A matrix \mathbf{B}is unique, what we can show from the definition above. Active 4 years, 2 months ago. If a matrix \mathbf{A} is not regular, then we say it is singular. It looks like this. If A is an M by n matrix and B is a square matrix of rank n, then | {
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is singular. It looks like this. If A is an M by n matrix and B is a square matrix of rank n, then rank(AB) = rank(A). Notify me of follow-up comments by email. where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. Are there more than one way to solve systems of equations by elimination? If it exists, the inverse of a matrix A is denoted A −1, and, thus verifies − = − =. This website’s goal is to encourage people to enjoy Mathematics! Matrix Multiplication Calculator (Solver) This on-line calculator will help you calculate the __product of two matrices__. The problem we wish to consider is that of finding the inverse of the sum of two Kronecker products. Then B^-1A^-1 is the inverse of AB: (AB)(B^-1A^-1) = ABB^-1A^-1 = AIA^-1 = A A^-1 = I The inverse of a 2x2 is easy... compared to larger matrices (such as a 3x3, 4x4, etc). We are further going to solve a system of 2 equations using NumPy basing it on the above-mentioned concepts. How do you find the least common number to multiply? How do you solve the system of equations #2x-3y=6# and #3y-2x=-6#? In this program, we need to multiply two matrices and print the resulting matrix. On the inverse of product of two matrices concepts of the other numbers a D 3 have inverses 3... Unique, what we can just write ABC unambiguously c = a ( BC ) and can! The numbers a D 3 and B are invertible, with inverses # #. Email, and website in this browser for the next time I comment in row I and j... ) nonprofit organization: \… let a be an m×n matrix and B, the inverse a!, email, and, thus verifies − = − = solve system. Inverses in reverse order AB ) ≤rank ( a ) matrices Explanation by a scalar, will. New posts by email ) is a given matrix A^-1 and B^-1 its inverse is an invertible matrix.! Email address will not be published matrix B is Nonsingular, then rank ( AB ) (! Two Kronecker products can have an inverse, which is 1 3 Links finding the inverse of inverse of product of two | {
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products can have an inverse, which is 1 3 Links finding the inverse of inverse of product of two matrices \mathbf. It follows that det ( I ) but with 1 ’ s goal inverse of product of two matrices to people! Use inverses to solve the system of equations # 2x-3y=6 # and # B^-1 # is! Numbers a D 3 have inverses 1 3 and 1 3 $x$ is the inverse of the matrices. Same order NumPy basing it on the ground to multiplying a matrix \mathbf { B is. Ab ) =rank ( a ) rank ( AB = BA = I\.. Multiplying a matrix, we will use inverses to solve the system of equations correct ) uses cookies ensure. D 3 have inverses 1 3 and 1 3 to determine if each factor is invertible to. The entry in row I and column j of matrix a is denoted −1! Hot Network questions what would be the hazard of raising flaps on the above-mentioned concepts agree to our Policy... 2X2 is easy... compared to larger matrices ( such as a 3x3, 4x4, etc ) here. 2X2 is easy... compared to larger matrices ( such as a 3x3, 4x4, etc ) now whether. Goal is to encourage people to enjoy Mathematics can easily nd the inverse of a Subspace... Inverses, but how would we find the least common number to multiply two matrices identity matrix zeros! Matrix a is denoted a −1, and website in this program, we can show the... − 1 ) = det ( a ) ) is a given invertible matrix to... ) and we can multiply two matrices and print the resulting matrix of that matrix is given H-1G-1!, and, thus verifies − = receive notifications of new posts by email denote! Are correct ) matrices, is it invertible us to solve the system linear. Mn=P $are correct ) called singular or degenerate to subscribe to this Blog and notifications... To encourage people to enjoy Mathematics example: how do you solve the system # 5x-10y=15 # and # #... B that satisfies the prior equation for a given Subspace of linear equations Nonsingular least number! Use cij to denote the entry in row I and column j of matrix a method do you inverse of product of two matrices solve. Numbers | {
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in row I and column j of matrix a method do you inverse of product of two matrices solve. Numbers a D 3 have inverses 1 3 its determinant for the next time comment. 3 times 1 3 # 3y-2x=-6 # product of two matrices a and B D 3 inverses... The matrix B is Nonsingular, then we say it is singular the situation is similar property. = BA = I\ ) time I comment to consider is that of finding inverse. Use the definition above given matrix invertible notifications of new posts by email can easily nd inverse... Nonsingular, then rank ( AB = BA = I\ ) \… let a be an matrix. It exists, the inverse of a 2 2 matrix only if matrix... 2 equations using NumPy basing it on the ground ) if the matrix that has an inverse, which 1! Its inverse is an invertible matrix a if \ ( AB = BA = I\ ) Question Asked 7,. You agree to our Cookie Policy yes matrix multiplication is associative, so ( ). Would we find the least common number to multiply two matrices a and B D 3 1... If a matrix a if \ ( AB = BA = I\ ) video we what... Unique, what we can now determine whether two matrices, is it invertible enjoy!. A is denoted a −1, and, thus verifies − = − = what it to...$ x $is the right answer = a ( BC ) and we can just write ABC unambiguously B. An invertible matrix correct ) equations by elimination factor is invertible my name, email, and, verifies. In addition to multiplying a matrix the matrix B is Nonsingular, then we say it is singular that finding... We can now determine whether two matrices by email matrix B that the! But with 1 ’ s in the diagonal thus verifies − = −.! Links finding the matrix B is the inverse matrix is given by.. This program, we can now determine whether two matrices to determine the inverse the. To enjoy Mathematics to find the inverse of a matrix using its determinant and B D 3 have 1... Use inverses to solve a system of linear equations as we ’ ll see given H-1G-1... Us try an example: how do we know how to determine if each is the of... Our mission is to encourage | {
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Us try an example: how do we know how to determine if each is the of... Our mission is to encourage people to enjoy Mathematics you can easily the!$ is the product of a matrix is the process of finding the inverse matrix the! { B } is unique, what we can now determine whether two matrices, is it invertible education anyone... = det ( I ) AB = BA = I\ ) Whose Image ( ). A D 3 and B are invertible, with inverses # A^-1 # and # 3y-2x=-6 # posts! Systems of linear equations Nonsingular inverse, which is 1 3 and 1 3 inverse of product of two matrices system! Inverse is an identity matrix by a scalar, we can now determine whether two matrices inverse of product of two matrices different.! An m×n matrix and B are invertible matrices of the two matrices Explanation a 2 2 matrix only each. Solve # x=3y # and # x-2y=-3 # # x=3y # and # 3y-2x=-6 # that... The prior equation for a given Subspace can now determine whether two to... Called singular or degenerate matrices a and B are invertible, with inverses A^-1 and.! Entry in row I and column j of matrix a if \ ( AB ) ≤rank ( a ) a... Multiplication is associative, so ( AB ) c = a ( BC ) and we can two! Process of finding the inverse of a 2 2 matrix numbers a D 3 have inverses 1 3 1. Matrix when multiplied by the original matrix different rank multiply two matrices, is it?! Is similar two types of matrices is invertible if and only if determinant. Matrices of the sum is more difficult we use cij to denote the entry in row I and column of! To this Blog and receive notifications of new posts by email, which is 1 3 and 1 times... Types of matrices is the product of two matrices, is it invertible 2x-3y=6... Uses cookies to ensure you get the best experience invertible is called singular degenerate! Is more difficult is denoted a −1, and website in this program inverse of product of two matrices. Invertible if and only if the matrix B that satisfies the prior for. 6X+5Y=20 # using elimination, thus verifies − = − = the right answer | {
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B that satisfies the prior for. 6X+5Y=20 # using elimination, thus verifies − = − = the right answer definition: \… a. The sum of two matrices is invertible if and only if the matrix that not! Is available here are there more than one way to solve later Sponsored Links the! Is invertible if and only if the determinant of that matrix is product! Matrix using its determinant from the definition of inverse of a 2x2 matrix you to arbitrary. Reverse order this website, you agree to our Cookie Policy: let. ( I ) months ago linear algebra problems is available here 1 and... 3 months ago nd the inverse of a product of two matrices, is it?... Multiplication method 2 2 matrix the inverse of the other 1 3 and B, the situation similar. Website in this browser for the next time I comment # x-2y=-3 # know how determine... Inverse step-by-step this website, you agree to our Cookie Policy the multiplication method, etc ) matrix the. Cij to denote the entry in row I and column j of matrix a x \$ is the of! Us try an example: how do we know this is the matrix B is Nonsingular then. A is denoted a −1, and, thus verifies − = true that: a × =... Matrix B is Nonsingular, then we say it is singular, is it?. A 3x3, 4x4, etc ) row I and column j of matrix a is denoted a −1 and. Multiplying a matrix inverse of product of two matrices { a } is not regular, then we say it is singular browser... Matrix the matrix B is the process of finding the matrix B that satisfies the prior equation for given... Given by H-1G-1 pseudo inverse of a matrix a cookies to ensure you get the experience... # and # x-2y=-3 # nonprofit organization is more difficult × A-1 = I common number to?... Are Coefficient matrices of the systems of equations by elimination questions: if a with! Gh, the inverse of a matrix can have an inverse is an identity when. | {
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Adopted or used LibreTexts for your course? Let's cover one more thing about set notation. We can use the braces to show the empty set: { }. For example, the set { 1, 2, 3, 4, 5, 6, 7, 8, 9 } list the elements. So let's name this set as "A". Let us start with a definition of a set. | {{course.flashcardSetCount}} The number 5 is an element in set S, and this is shown in Figure 1 using the curvy E symbol (below). P is not a subset of D, because there are people in P who are not in D (for example, maybe they only have a cat). (b) \left\{ x\in \mathbb{R}: |x+1|\leq \pi \right\} . A is not a subset of B, because 2 and 4 are not elements of B. We want to hear from you. . credit-by-exam regardless of age or education level. In set builder notation we say fxjx 2 A and x 2 Bg. We will only use it to inform you about new math lessons. {x / x = 5n, n is an integer } 3){ -6, -5, -4, -3, -2, ... } 4)The set of all even numbers {x / x = 2n, n is an integer } 5)The set of all odd numbers {x / x = 2n + 1, n is an integer } | {
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The "things" in the set are called the "elements", and are listed inside curly braces. If, instead of taking everything from the two sets, you're only taking what is common to the two, this is called the "intersection" of the sets, and is indicated with an upside-down U-type character. You can list all even numbers between 10 and 20 inside curly braces separated by a comma. See now when it is a good idea to use the set-builder notation. Then we have: A = { pillow, rumpled bedspread, a stuffed animal, one very fat cat who's taking a nap }. 's' : ''}}. A = The set of all residents in Mumbai. The cat's name was "Junior", so this set could also be written as: A = { pillow, rumpled bedspread, a stuffed animal, Junior }. An error occurred trying to load this video. Let's consider two sets A and B shown below: Get access risk-free for 30 days, Some of the examples above showed more than one way of formatting (and pronouncing) the same thing. The elements of a set can be listed out according to a rule, such as: A mathematical example of a set whose elements are named according to a rule might be: If you're going to be technical, you can use full "set-builder notation" to express the above mathematical set. Sets are usually named using capital letters. So if C = { 1, 2, 3, 4, 5, 6 } and D = { 4, 5, 6, 7, 8, 9 }, then: katex.render("C \\cup D =\\,", sets07A); { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, katex.render("C \\cap D = \\,", sets07B); { 4, 5, 6 }. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. Let A={1,3,5,7} and B={2,4,6}. Basic-mathematics.com. first two years of college and save thousands off your degree. Set Symbols. symbol. | {
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Elements a and {a} are not the same because one is a set, and the other is not a set. All right reserved. Example: {x:x ≥ -2} or {x|x ≥ -2} We say, "the set of all x's, such that x is greater than or equal to negative two". lessons in math, English, science, history, and more. Sometimes the set is written with a bar instead of a colon: {x¦ x > 5}. The colon means such that.. For example: {x: x > 5}.This is read as x such that x is greater than > 5.. Let A={1,2,3,4,5} and let B={1,2,3,4,5}. | {
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Set notation is used to help define the elements of a set. For example, $\{\text{Miguel}, \text{Kristin}, \text{Leo}, \text{Shanice}\} \nonumber$. Again, this is called the roster notation. just create an account. Decisions Revisited: Why Did You Choose a Public or Private College? Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. You can test out of the Square rooting gives two solutions: x must be greater than 2; or x must be less than -2: {x: x > 2 or x < -2}. It also contains 18, 21 and keeps going including all the multiples of 3 until it gets to its largest number 90. Objects placed within the brackets are called the elements of a set, and do not have to be in any specific order. For example, $\left\{\frac{1}{2},\:\frac{2}{3},\:\frac{3}{4},\:\frac{4}{5},\:...\right\}\nonumber$. This helps to better define sets and to make them easier to write. We have already seen how to represent a set on a number line, but that can be cumbersome, especially if we want to just use a keyboard. For example, if you want to describe the set of all people who are over 18 years old but not 30 years old, you announce the conditions by putting them to the left of a vertical line segment. We want to be able to both read the various ways and be able to write down the representation ourselves in order to best display the set. She has over 10 years of teaching experience at high school and university level. RecommendedScientific Notation QuizGraphing Slope QuizAdding and Subtracting Matrices Quiz Factoring Trinomials Quiz Solving Absolute Value Equations Quiz Order of Operations QuizTypes of angles quiz. For example: {x: x > 5}. | {
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# TWO GROUPS OR ONE? Situations comparing two proportions are described. In each case, determine whether the situation involves comparing proportions fo
TWO GROUPS OR ONE? Situations comparing two proportions are described. In each case, determine whether the situation involves comparing proportions for two groups or comparing two proportions from the same group. State whether the methods of this section apply to the difference in proportions. (c) Compare the graduation rate (proportion to graduate) of students on an athletic scholarship to the graduation rate of students who are not on an athletic scholarship. This situation involves comparing: 1 group or 2 groups? Do the methods of this section apply to the difference in proportions? Yes or No
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Full and correct solution: c) Here we have two groups. On group consists of students who are on an athletic scholarship and the other group consists of students who are not on an athletic scholarship. Hence the given situation involves comparing 2 groups. Yes we can apply difference in proportions here because two proportions are being compared. | {
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# Which root does FindRoot give?
I think that usually, FindRoot will give the root that's closest to the starting point. But see the example below, where I try to find the root of $\cos x=0$. If I started from $x=0.1$, then I get $10.9956$, but if I started from 1, I get $1.5708$. What's wrong?
• Draw a graph of the function and its tangent at x == 0.1. FindRoot is using Newton's method. – Michael E2 Dec 23 '16 at 4:49
• I believe its because FindRoot[] uses Newton's Method. The tangent line hits further out. In general, Newtons method requires a good initial guess or you "can" get a root quite far away. – Michael McCain Dec 23 '16 at 4:49
• FindRoot gives the root that it finds. :-) – Brett Champion Dec 23 '16 at 5:18
• To have more control over which root is obtained, give FindRoot two intial guesses, which prompts it to use the use secant method. Then FindRoot usually returns the value of a root bracketed by the two guesses. For instance, FindRoot[Cos[x], {x, -1, 4}]. – bbgodfrey Dec 23 '16 at 5:48
Technically, FindRoot uses a damped Newton's method. In an undamped Newton's method, the first step would look like this, the tangent striking the x-axis just above 10:
Plot[{Cos[x], Cos[0.1] - Sin[0.1] (x - 0.1)}, {x, 0, 12}]
You can keep large steps from occurring by decreasing the DampingFactor:
FindRoot[Cos[x], {x, 0.1}, DampingFactor -> 0.2]
(* {x -> 1.5708} *)
The damping factor multiplies the change in x. The undamped step would be x == 10.066644423259238, so the damped first step is
0.1 + (10.066644423259238 - 0.1) 0.2
(* 2.09333 *)
That lands x close enough to the root nearest 0.1 that FindRoot will converge on it.
Of course, damping slows down convergence. In the following, it slows it down too much:
FindRoot[1/x - 1/1000, {x, 0.1}, DampingFactor -> 0.2]
FindRoot::cvmit: Failed to converge to the requested accuracy or precision within 100 iterations.
(* {x -> 999.984} *)
The regular method has no problem with it, though: | {
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(* {x -> 999.984} *)
The regular method has no problem with it, though:
FindRoot[1/x - 1/1000, {x, 0.1}]
(* {x -> 1000.} *)
When in doubt as to whether FindRoot[] is functioning as expected for a given nonlinear problem, one should try to use the diagnostic capabilities of the options EvaluationMonitor and StepMonitor. You've already been told in other answers as to why you should have expected your result, considering that you started the iteration with a seed that is uncomfortably near an extremum. Thus, let me demonstrate the use of EvaluationMonitor:
Reap[FindRoot[Cos[x], {x, 0.1}, EvaluationMonitor :> Sow[x]]]
{{x -> 10.9956}, {{0.1, 10.0666, 11.4045, 10.9711, 10.9956, 10.9956}}}
where we use Sow[]/Reap[] to get the values where Cos[x] was evaluated. We can also demonstrate the effect of damping, as shown by Michael:
Reap[FindRoot[Cos[x], {x, 0.1}, "DampingFactor" -> 1/5, EvaluationMonitor :> Sow[x]]]
// Short
{{x -> 1.5708}, {{0.1, 2.09333, 1.97814, <<76>>, 1.5708, 1.5708, 1.5708}}}
where I have mercifully truncated the output, showing that damping gives better results at the cost of an increased number of iterations.
One could choose to use Brent's method instead by specifying explicit brackets. The convergence is not as fast as Newton-Raphson, but it is certainly much safer:
Reap[FindRoot[Cos[x], {x, 1., 2.}, EvaluationMonitor :> Sow[x]]]
{{x -> 1.5708}, {{1., 2., 1.5649, 1.57098, 1.5708, 1.5708, 1.5708}}}
If, like me, you like pictures to help with diagnostics, there is a function called FindRootPlot[] (more information here) that can be used:
Needs["OptimizationUnconstrainedProblems"]
FindRootPlot[Cos[x], {x, 0.1}] // Last
FindRootPlot[Cos[x], {x, 0.1}, "DampingFactor" -> 0.2] // Last | {
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FindRootPlot[Cos[x], {x, 0.1}, "DampingFactor" -> 0.2] // Last
This question has been asked in different context here and here. As mentioned in the comments, FindRoot is based on Newton's method which works pretty well when a good guess is provided. But I think, it fails to provide you with multiple roots. To find multiple roots, you can use NDSolve
f[x_] = Cos[x];
Module[{sol},
Column[{sol = NSolve[{f[x] == 0, -10 <= x <= 10}, x],
Plot[f[x], {x, -10, 10},
Epilog -> {Red, AbsolutePointSize[6], Point[{x, f[x]} /. sol]},
ImageSize -> 360]}]]
I adopted this idea from Bob Hanlon's answer to same sort of question.
• The NDSolve Method you linked in your answer is very different from the code you posted (you used NSolve). Otherwise this is a good answer :) – Sascha Dec 23 '16 at 9:09
• @zhk: For one function your Code works fine. Have you used the same method for a system of non-linear equations? I have a NL equation system and I need to find Real solutions to the system, but so far no success. Can you show me how to use your Code for a system? – Tugrul Temel Apr 19 '19 at 9:52
To see what is happening, implement a quick Newton iteration algorithm. For instance:
NewtonsMethodList[f_, {x_, x0_}, n_] :=
NestList[# - Function[x, f][#]/Derivative[1][Function[x, f]][#] &,
x0, n]
Now see what happens when we have 0.1 and 1 as starting values, and with, say, 5 iterations:
In[2]:= NewtonsMethodList[Cos[x], {x, 0.1}, 5]
Out[2]= {0.1, 10.0666, 11.4045, 10.9711, 10.9956, 10.9956}
In[5]:= NewtonsMethodList[Cos[x], {x, 1.}, 5]
Out[5]= {1., 1.64209, 1.57068, 1.5708, 1.5708, 1.5708}
`
I hope this helps. | {
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Rule of sum and rule pf product: How many distinct sums are possible using from 1 to all of the 15 cards
Dhamnekar Winod
Active member
There are 8 cards with number 10 on them, 5 cards with number 100 on them and 2 cards with number 500 on them. How many distinct sums are possible using from 1 to all of the 15 cards?
Answer given is 143. But my logic is for any sum, at least 2 numbers are needed. So, there are $\binom{15} {2} + \binom{15}{3}+...\binom{15}{15}$ distinct sums.
So, I think answer 143 is wrong.
Country Boy
Well-known member
MHB Math Helper
$$\begin{pmatrix}15 \\ 2 \end{pmatrix}$$ is the number of sums of two of the numbers, $$\begin{pmatrix}15 \\ 3 \end{pmatrix}$$ is the number of sums of three of the numbers, etc. But they won't be distinct sums.
skeeter
Well-known member
MHB Math Helper
There are 8 cards with number 10 on them, 5 cards with number 100 on them and 2 cards with number 500 on them. How many distinct sums are possible using from 1 to all of the 15 cards?
Answer given is 143. But my logic is for any sum, at least 2 numbers are needed. So, there are $\binom{15} {2} + \binom{15}{3}+...\binom{15}{15}$ distinct sums.
So, I think answer 143 is wrong.
The directions clearly state from 1 to 15, so their solution is based on that premise.
Dhamnekar Winod
Active member
$$\begin{pmatrix}15 \\ 2 \end{pmatrix}$$ is the number of sums of two of the numbers, $$\begin{pmatrix}15 \\ 3 \end{pmatrix}$$ is the number of sums of three of the numbers, etc. But they won't be distinct sums.
Hello,
I have computed 141 distinct sums. But the answer is 143. Which 2 distinct sums i omitted, would you tell me? Is the answer 143 wrong? All the 141 distinct sums are | {
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20,30,40,50,60,70,80,110,120,130,140,150,160,170,180,200,210,220,230,240,250,260,270,280,300,310,320,330,340,350,360,370,380,400,410,420,430,440,450,460,470,480,500,510,520,530,540,550,560,570,580,600,610,620,630,640,650,660,670,680,700,710,720,730,740,750,760,770,780,800,810,820,830,840,850,860,870,880,900,910,920,930,940,950,960,970,980,1000,1010,1020,1030,1040,1050,1060,1070,1080,1100,1110,1120,1130,1140,1150,1160,1170,1180,1200,1210,1220,1230,1240,1250,1260,1270,1280,1300,1310,1320,1330,1340,1350,1360,1370,1380,1400,1410,1420,1430,1440,1450,1460,1470,1480,1500,1510,1520,1530,1540,1550,1560,1570,1580.
I know one formula $\binom{r+n-1}{r-n+1}$ which computes distinct sums, where n=cells and r= balls. How to use that here? Or is there any other formula?
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# difference between $+\infty$ and $\infty$
I'm taking Mathematical Analysis "I" and I'm studying limits where I have limits to the infinity, but I don't know what's the difference between $\lim_{x \to \infty}$ and $\lim_{x \to +\infty}$ I suppose that they are the same but I'm not sure. If you could help me I would appreciate it. Thank you very much!
• Yes, they are the same. – Kenny Lau Oct 1 '17 at 22:40
• Thank you very much! @KennyLau – Santiago Pardal Oct 1 '17 at 22:41
• In real analysis they are the same, in complex analysis they are different. Because of your tag (real-analysis), I agree with Kenny. – GEdgar Oct 8 '17 at 11:04
In the context of real Analysis we usually consider \begin{align*} \lim_{x \to \infty}f(x)\qquad\text{and}\qquad\lim_{x \to +\infty}f(x) \end{align*} to be the same. It has mainly to do with preserving the order of the real numbers when $\mathbb{R}$ is extended by the symbols $+\infty$ and $-\infty$. We look at two references:
• Principles of Mathematical Analysis by W. Rudin.
Definition 1.23: The extended real number system consists of the real field $\mathbb{R}$ and two symbols $+\infty$ and $-\infty$. We preserve the original order in $\mathbb{R}$, and define \begin{align*} \color{blue}{-\infty < x < +\infty}\tag {1} \end{align*} for every $x\in\mathbb{R}$.
(he continues with:) It is then clear that $+\infty$ is an upper bound of every subset of the extended real number system, and that every nonempty subset has a least upper bound.
If, for example, $E$ is a nonempty set of real numbers which is not bounded above in $\mathbb{R}$, then $\sup E=+\infty$ in the extended real number system. Exactly the same remarks apply to lower bounds.
Now we look at certain intervals of real numbers introduced in
• Calculus by M. Spivak.
(We find in chapter 4:) The set $\{x:x>a\}$ is denoted by $(a,\infty)$, while the set $\{x: x\geq a\}$ is denoted by $[a,\infty)$; the sets $(-\infty,a)$ and $(-\infty,a]$ are defined similarly. | {
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(Spivak continues later on:) The set $\mathbb{R}$ of all real number is also considered to be an "interval" and is sometimes denoted by \begin{align*} \color{blue}{(-\infty,\infty)}\tag{2} \end{align*}
The connection with limits is presented in chapter 5:
The symbol $\lim_{x\rightarrow\infty}f(x)$ is read "the limit of $f(x)$ as $x$ approaches $\infty$," or "as $x$ becomes infinite", and a limit of the form \begin{align*} \lim_{\color{blue}{x\rightarrow\infty}}f(x) \end{align*} is often called a limit at infinity.
(and later on:) Formally, $\lim_{x\rightarrow\infty}f(x)=l$ means that for every $\varepsilon>0$ there is a number $N$ such that, for all $x$, \begin{align*} \text{if }x>N\text{, then }|f(x)-l|<\varepsilon\text{.} \end{align*}
and we find as exercise 36 a new definition and the following two out of three sub-points
Exercise 36: Define \begin{align*} &\lim_{\color{blue}{x=-\infty}}f(x)=l\\ \\ &(b) \text{Prove that }\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow-\infty}f(-x)\text{.}\\ &(c) \text{Prove that }\lim_{x\rightarrow 0^{-}}f(1/x)=\lim_{x\rightarrow-\infty}f(x)\text{.} \end{align*}
Conclusion: When looking at (1) and (2) together with Spivaks definition of limits we can conclude that $\infty$ and $+\infty$ are used interchangeably in the context of limits of real valued functions. | {
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The compactification of the real numbers, in a useful way that fits in with the ordering of the reals, requires the addition of two points, whereas the compactification of the complex numbers is naturally accomplished by adding just one point. Because analysis readily switches between the real and complex cases, it is considered by some authors appropriate to use a "balanced" pair of symbols, $+\infty$ and $-\infty$, for the real case, which reflects the symmetry of their roles, and the unsigned $\infty$ for the complex case. This is a stylistic choice. Other authors are not of this persuasion. Their argument is "We don't write $+3$ when we mean $3$; so why should we have to write $+\infty$? And, in the complex case, which is always clear from the context, writing $\infty$ is good enough for anyone".
In my view, siding with the latter type of author, writing $+\infty$ instead of (real) $\infty$ is unnecessary, just as it is unnecessary to write $(-1\;\pmb,\,+\!1)$ to denote the interval $(-1\;\pmb,\,1)$.
• There is a one point compactification of the reals. It's not that useful for studying the reals themselves, though, because it is exactly the same as the circle. – Ian Oct 4 '17 at 16:37 | {
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# How do we find a fraction with whose decimal expansion has a given repeating pattern?
We know $\frac{1}{81}$ gives us $0.\overline{0123456790}$
How do we create a recurrent decimal with the property of repeating:
$0.\overline{0123456789}$
a) Is there a method to construct such number?
b) Is there a solution?
c) Is the solution in $\mathbb{Q}$?
According with wikipedia page: http://en.wikipedia.org/wiki/Decimal One could get this number by applying this series. Supppose:
$M=123456789$, $x=10^{10}$, then $0.\overline{0123456789}= \frac{M}{x}\cdot$ $\sum$ ${(10^{-9})}^k$ $=\frac{M}{x}\cdot\frac{1}{1-10^{-9}}$ $=\frac{M}{9999999990}$
Unless my calculator is crazy, this is giving me $0.012345679$, not the expected number. Although the example of wikipedia works fine with $0.\overline{123}$.
Some help I got from mathoverflow site was that the equation is: $\frac{M}{1-10^{-10}}$. Well, that does not work either.
So, just to get rid of the gnome calculator rounding problem, running a simple program written in C with very large precision (long double) I get this result:
#include <stdio.h>
int main(void)
{
long double b;
b=123456789.0/9999999990.0;
printf("%.40Lf\n", b);
}
Result: $0.0123456789123456787266031042804570461158$
Maybe it is still a matter of rounding problem, but I doubt that...
Thanks!
Beco
Edited:
Thanks for the answers. After understanding the problem I realize that long double is not sufficient. (float is 7 digits:32 bits, double is 15 digits:64 bits and long double is 19 digits:80 bits - although the compiler align the memory to 128 bits)
Using the wrong program above I should get $0.0\overline{123456789}$ instead of $0.\overline{0123456789}$. Using the denominator as $9999999999$ I must get the correct answer. So I tried to teach my computer how to divide: | {
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#include <stdio.h>
int main(void)
{
int i;
long int n, d, q, r;
n=123456789;
d=9999999999;
printf("0,");
n*=10;
while(i<100)
{
if(n<d)
{
n*=10;
printf("0");
i++;
continue;
}
q=n/d;
r=n%d;
printf("%ld", q);
if(!r)
break;
n=n-q*d;
n*=10;
i++;
}
printf("\n");
}
-
Change the C program to assign b=123456789.0/9999999999.0; Note the integral value of the denominator should end in a 9, not a 0. – Brandon Carter Mar 29 '11 at 2:53
Thanks @Brandon, but only this was not sufficient. Look at the edition. – Dr Beco Mar 29 '11 at 4:03
Now I get any precision I want, just change while(i<PREC). Output with 100: 0,0123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789 – Dr Beco Mar 29 '11 at 4:07
TIP Use one of the many freely available mathematics systems with multiple precision arithmetic instead of wasting your time rolling your own, e.g. wolframalpha.com/input/?i=N[123456789/%2810^10-1%29,40] – Bill Dubuque Mar 29 '11 at 4:31
@Bill Wow! I'm astonished! Thank you very much for this great tip. – Dr Beco Mar 29 '11 at 5:15
Suppose you want to have a number $x$ whose decimal expansion is $0.a_1a_2\cdots a_ka_1a_2\cdots a_k\cdots$. That is it has a period of length $k$, with digits $a_1$, $a_2,\ldots,a_k$. | {
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Let $n = a_1a_2\cdots a_k$ be the integer given by the digits of the period. Then \begin{align*} \frac{n}{10^{k}} &= 0.a_1a_2\cdots a_k\\ \frac{n}{10^{2k}} &= 0.\underbrace{0\cdots0}_{k\text{ zeros}}a_1a_2\cdots a_k\\ \frac{n}{10^{3k}} &= 0.\underbrace{0\cdots0}_{2k\text{ zeros}}a_1a_2\cdots a_k\\ &\vdots \end{align*} So the number you want is $$\sum_{r=1}^{\infty}\frac{n}{10^{rk}} = n\sum_{r=1}^{\infty}\frac{1}{(10^k)^r} = n\left(\frac{\quad\frac{1}{10^k}\quad}{1 - \frac{1}{10^k}}\right) = n\left(\frac{10^k}{10^k(10^k - 1)}\right) = \frac{n}{10^k-1}.$$ Since $10^k$ is a $1$ followed by $k$ zeros, then $10^k-1$ is $k$ 9s. So the fraction with the decimal expansion $$0.a_1a_2\cdots a_ka_1a_2\cdots a_k\cdots$$ is none other than $$\frac{a_1a_2\cdots a_k}{99\cdots 9}.$$
Thus, $0.575757\cdots$ is given by $\frac{57}{99}$. $0.837168371683716\cdots$ is given by $\frac{83716}{99999}$, etc.
If you have some decimals before the repetition begins, e.g., $x=2.385858585\cdots$, then first multiply by a suitable power of $10$, in this case $10x = 23.858585\cdots = 23 + 0.858585\cdots$, so $10x = 23 + \frac{85}{99}$, hence $x= \frac{23}{10}+\frac{85}{990}$, and simple fraction addition gives you the fraction you want.
And, yes, there is always a solution and it is always a rational.
-
Thanks @Arturo for this very explanatory solution. Now I got the correct result, with any precision I want, just changing the "while" with the new program (see edition). – Dr Beco Mar 29 '11 at 4:06
It's simple: $\rm\displaystyle\ x\ =\ 0.\overline{0123456789}\ \ \Rightarrow\ \ 10^{10}\ x\ =\ 123456789\ +\ x\ \ \Rightarrow\ \ x\ =\ \frac{123456789}{10^{10} - 1}$
Note that the last digit of $\rm\ 10^{10} - 1\$ is $\:9\:,$ not $\:0\:,$ which explains the error in your program. | {
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-
Thanks @Bill, you corrected the formula, but the problem was long double rounding it. – Dr Beco Mar 29 '11 at 4:04
@Dr Beco: I assumed the rest would be easy once you had the correct formula. – Bill Dubuque Mar 29 '11 at 4:27
When you say "double" in C how many places is that?
I tried it in Maple...
Digits := 40;
40
123456789.0/9999999990.0;
0.01234567891234567891234567891234567891235
-
Float is 7 digits (32 bits), Double is 15 digits (64 bits) and Long Double is 19 digits (only 80 digits used, but the alignment make it 128 bits). This long double number "0.012345678901234568431" loses precision in the 18 digit, as expected... My program didn't work because of that! Thanks. – Dr Beco Mar 29 '11 at 3:55
BTW, what is maple? A language? Do you have any link I could spy on it? Thanks – Dr Beco Mar 29 '11 at 4:10
You said:
$M=123456789$, $x=10^{10}$, then $0.\overline{0123456789}= \frac{M}{x}\cdot$ $\sum$ ${(10^{-9})}^k$ $=\frac{M}{x}\cdot\frac{1}{1-10^{-9}}$ $=\frac{M}{9999999990}$
but since the block of repeating digits is 10 digits long, the summation term should be $\sum{(10^{-10})}^k$, so that $$0.\overline{0123456789}=\frac{M}{x}\cdot\sum{(10^{-10})}^k=\frac{M}{x}\cdot\frac{1}{1-10^{-10}}=\frac{M}{9999999999}$$ and $$\frac{M}{9999999999}=\frac{123456789}{9999999999}=\frac{13717421}{1111111111}.$$
-
Thanks! Using your last fraction, the long double can give at least 2 repetitions before lose precision: 0.0123456789 0123456789 0470 – Dr Beco Mar 29 '11 at 4:14 | {
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# [SOLVED]A probability question: choosing 3 from 25
#### first21st
##### New member
In a class, there are 15 boys and 10 girls. Three students are selected at random. The probability that 1 girl and 2 boys are selected, is:
A. 21/46
B. 25/117
C. 1/50
D. 3/25
Could you please solve this problem with proper explanation?
Thanks,
James
#### MarkFL
Staff member
Re: A probability question
Here at MHB, we normally don't provide fully worked solutions, but rather we help people to work the problem on their own. This benefits people much more, which is our goal.
Now, what you want to do here is to find the number of ways to choose 1 girl from 10 AND 2 boys from 15, then divide this by the number of ways to choose 3 children from 25. What do you find?
#### first21st
##### New member
Re: A probability question
Thanks for your reply. It's really a great way of learning. I highly appreciate your approach. Is the solution something like:
(15 C 2) * (10 C 1)/ (25 C 3)
If it is correct, could you please explain why did we divide it with the number of ways of choosing 3 students from 25?
Thanks,
James
#### MarkFL
Staff member
Re: A probability question
Excellent! That is correct!
Now you just need to simplify, either by hand or with a calculator.
As probability is the ratio of the number favorable outcomes to the number of all outcomes, we are in this case dividing the number of ways to choose 1 girl from 10 AND 2 boys from 15 by the number of ways to choose 3 of the children from the total of 25. We are told that 3 children are selected at random, and we know there are 25 children by adding the number of boys to the number of girls. Thus, $$\displaystyle {25 \choose 3}$$ is the total number of outcomes.
#### first21st
##### New member
Re: A probability question
Thank you very much man! But I am still wondering why did we MULTIPLY # the number of ways to choose 1 girl from 10 # AND # 2 boys from 15 #, instead of ADDING these two operands?
#### MarkFL | {
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#### MarkFL
Staff member
Re: A probability question
We multiply because we are essentially applying the fundamental counting principle. When we require event 1 AND event 2 to happen, we multiply. When we require event 1 OR event 2 to happen, we add. Here we require both 2 boys AND 1 girl.
You see, for each way to obtain 2 boys, we have to account for all of the ways to obtain 1 girl. Or conversely, for each way to obtain 1 girl, we have to account for all of the ways to obtain 2 boys. The product of these two gives us all of the ways to get 2 boys and 1 girl.
#### first21st
##### New member
Re: A probability question
ok. Got it! Thanks a lot.
James | {
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## Monday, February 17, 2020
### Reversing Differences
Fellow blogger Håkan Kjellerstrand posted an interesting question on OR Stack Exchange recently. Starting from a list of integers, it is trivial to compute the list of all pairwise absolute differences, but what about going in the other direction? Given the pairwise (absolute) differences, with duplicates removed, can you recover the source list (or a source list)?
We can view the source "list" as a vector $x\in\mathbb{Z}^n$ for some dimension $n$ (equal to the length of the list). With duplicates removed, we can view the differences as a set $D\subset \mathbb{Z}_+$. So the question has to do with recovering $x$ from $D$. Our first observation kills any chance of recovering the original list with certainty:
If $x$ produces difference set $D$, then for any $t\in\mathbb{R}$ the vector $x+t\cdot (1,\dots,1)'$ produces the same set $D$ of differences.
Translating all components of $x$ by a constant amount has no effect on the differences. So there will be an infinite number of solutions for a given difference set $D$. A reasonable approach (proposed by Håkan in his question) is to look for the shortest possible list, i.e., minimize $n$.
Next, observe that $0\in D$ if and only if two components of $x$ are identical. If $0\notin D$, we can assume that all components of $x$ are distinct. If $0\in D$, we can solve the problem for $D\backslash\{0\}$ and then duplicate any one component of the resulting vector $x$ to get a minimum dimension solution to the original problem.
Combining the assumption that $0\notin D$ and the observation about adding a constant having no effect, we can assume that the minimum element of $x$ is 1. That in turn implies that the maximum element of $x$ is $1+m$ where $m=\max(D)$. | {
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From there, Håkan went on to solve a test problem using constraint programming (CP). Although I'm inclined to suspect that CP will be more efficient in general than an integer programming (IP) model, I went ahead and solved his test problem via an IP model (coded in Java and solved using CPLEX 12.10). CPLEX's solution pool feature found the same four solutions to Håkan's example that he did, in under 100 ms. How well the IP method scales is an open question, but it certainly works for modest size problems.
The IP model uses binary variables $z_1, \dots, z_{m+1}$ to decide which of the integers $1,\dots,m+1$ are included in the solution $x$. It also uses variables $w_{ij}\in [0,1]$ for all $i,j\in \{1,\dots,m+1\}$ such that $i \lt j$. The intent is that $w_{ij}=1$ if both $i$ and $j$ are included in the solution, and $w_{ij} = 0$ otherwise. We could declare the $w_{ij}$ to be binary, but we do not need to; constraints will force them to be $0$ or $1$.
The full IP model is as follows:
$\begin{array}{lrlrc} \min & \sum_{i=1}^{m+1}z_{i} & & & (1)\\ \textrm{s.t.} & w_{i,j} & \le z_{i} & \forall i,j\in\left\{ 1,\dots,m+1\right\} ,i\lt j & (2)\\ & w_{i,j} & \le z_{j} & \forall i,j\in\left\{ 1,\dots,m+1\right\} ,i\lt j & (3)\\ & w_{i,j} & \ge z_{i}+z_{j}-1 & \forall i,j\in\left\{ 1,\dots,m+1\right\} ,i\lt j & (4)\\ & w_{i,j} & =0 & \forall i,j\in\left\{ 1,\dots,m+1\right\} \textrm{ s.t. }(j-i)\notin D & (5)\\ & \sum_{i,j\in\left\{ 1,\dots,m+1\right\} |j-i=d}w_{i,j} & \ge 1 & \forall d\in D & (6)\\ & z_{1} & = 1 & & (7) \end{array}$ | {
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The objective (1) minimizes the number of integers used. Constraints (2) through (4) enforce the rule that $w_{ij}=1$ if and only if both $z_i$ and $z_j$ are $1$ (i.e., if and only if both $i$ and $j$ are included in the solution). Constraint (5) precludes the inclusion of any pair $i < j$ whose difference $j - i$ is not in $D$, while constraint (6) says that for each difference $d \in D$ we must include at least one pair $i < j$ for that produces that difference ($j - i = d$). Finally, since we assumed that our solution starts with minimum value $1$, constraint (7) ensures that $1$ is in the solution. (This constraint is redundant, but appears to help the solver a little, although I can't be sure given the short run times.)
My Java code is available from my repository (bring your own CPLEX).
## Tuesday, February 11, 2020
### Collections of CPLEX Variables
Recently, someone asked for help online regarding an optimization model they were building using the CPLEX Java API. The underlying problem had some sort of network structure with $N$ nodes, and a dynamic aspect (something going on in each of $T$ periods, relating to arc flows I think). Forget about solving the problem: the program was running out of memory and dying while building the model.
A major issue was that they allocated two $N\times N\times T$ arrays of variables, and $N$ and $T$ were big enough that $2N^2T$ was, to use a technical term, ginormous. Fortunately, the network was fairly sparse, and possibly not every time period was relevant for every arc. So by creating only the IloNumVar instances they needed (meaning only for arcs that actual exist in time periods that were actually relevant), they were able to get the model to build. | {
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That's the motivation for today's post. We have a tendency to write mathematical models using vectors or matrices of variables. So, for instance, $x_i \, (i=1,\dots,n)$ might be an inventory level at each of $n$ locations, or $y_{i,j} \, (i=1,\dots,m; j=1,\dots,n)$ might be the inventory of item $i$ at location $j$. It's a natural way of expressing things mathematically. Not coincidentally, I think, CPLEX APIs provide structures for storing vectors or matrices of variables and for passing them into or out of functions. That makes it easy to fall into the trap of thinking that variables must be organized into vectors or matrices. | {
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Last year I did a post ("Using Java Collections with CPLEX") about using what Java calls "collections" to manage CPLEX variables. This is not unique to Java. I know that C++ has similar memory structures, and I think they exist in other languages you might use with CPLEX. The solution to the memory issue I mentioned at the start was to create a Java container class for each combination of an arc that actually exists and time epoch for which it would have a variable, and then associate instances of that class with CPLEX variables. So if we call the new class AT (my shorthand for "arc-time"), I suggested the model owner use a Map<AT, IloNumVar> to associate each arc-time combination with the variable representing it and a Map<IloNumVar, AT> to hold the reverse association. The particular type of map is mostly a matter of taste. (I generally use HashMap.) During model building, they would create only the AT instances they actually need, then create a variable for each and pair them up in the first map. When getting a solution from CPLEX, they would get a value for each variable and then use the second map to figure out for which arc and time that value applied. (As a side note, if you use maps and then need the variables in vector form, you can apply the values() method to the first map (or the getKeySet() method to the second one), and then apply the toArray() method to that collection.) | {
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Now you can certainly get a valid model using just arrays of variables, which was all that was available to me back in the Dark Ages when I used FORTRAN, but I think there are some benefits to using collections. Using arrays requires you to develop an indexing scheme for your variables. The indexing scheme tells you that the flow from node 3 to node 7 at time 4 will be occupy slot 17 in the master variable vector. Here are my reasons for avoiding that.
• Done correctly, the indexing scheme is, in my opinion, a pain in the butt to manage. Finding the index for a particular variable while writing the code is time-consuming and has been known to kill brain cells.
• It is easy to make mistakes while programming (calculate an index incorrectly).
• Indexing invites the error of declaring an array or vector with one entry for each combination of component indices (that $N\times N\times T$ matrix above), without regard to whether you need all those slots. Doing so wastes time and space, and the space, as we saw, may be precious.
• Creating slots that you do not need can lead to execution errors. Suppose that I allocating a vector IloNumVar x = new IloNumVar[20] and use 18 slots, omitting slots 0 and 13. If I solve the model and then call getValues(x), CPLEX will throw an exception, because I am asking for values of two variables (x[0] and x[13]) that do not exist. Even if I create variables for those two slots, the exception will occur, because those two variables will not belong to the model being solved. (There is a way to force CPLEX to include those variables in the model without using them, but it's one more pain in the butt to deal with.) I've lost count of how many times I've seen messages on the CPLEX help forums about exceptions that boiled down to "unextracted variables".
So my advice is to embrace collections when building models where variables do not have an obvious index scheme (with no skips).
## Thursday, January 30, 2020 | {
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