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8=23 16=24 32=25 64=26 Table 10.1: Cell growth data. 134 CHAPTER 10. EXPONENTIAL FUNCTIONS that the population of cells will double every hour, then reasoning as above will lead us to conclude that the formula N(t) = (3 106)2t × gives the population of cells after t hours. Now, as long as t represents a non-negative i... |
the Rules of Exponents To be completely honest, making sense of the expression y = bx for all numbers x requires the tools of Calculus, but it is possible to establish a reasonable comfort level 0 by handling the case when x is a rational number. If b and n is a positive integer (i.e. n = 1, 2, 3, 4,... ), then we can... |
, we have p q = b q√b p = q√bp. For any rational numbers r and s, and for all positive bases a and b: 136 CHAPTER 10. EXPONENTIAL FUNCTIONS y-axis n y=t n even = solution y-axis n y=t n odd y=b t-axis y=b* no solution or two solutions y=b t-axis exactly one solution Figure 10.4: How many solutions to tn = b?. 1. Produc... |
bx, for some b > 0, b = 1, and A0 6 = 0. We will refer to the formula in Definition 10.1.2 as the standard exponential form. Just as with standard forms for quadratic functions, we sometimes need to do a little calculation to put an equation in standard form. The constant A0 is called the initial value of the exponentia... |
is a horizontal asymptote for the leftmove left; i.e. hand portion of the graph. The graph becomes higher and higher above the horizontal axis as we move to the right; i.e., the graph is unbounded as we move to the right. The special case of y = 2x is representative of the function y = bx, but there are a few subtle p... |
flecting the graph of y = ( 1 b)x about the y-axis. Putting these remarks to- gether, if 0 < b < 1, we conclude that the graph of y = bx will look like Figure 10.6(b). Notice, the graphs in Figure 10.6(b) share qualitative features, mirroring the features outlined previously, with the “asymptote” and “unbounded” portion... |
, in turn, related to the loudness of the sound. A person can typically perceive sounds ranging from 20 Hz to 20,000 Hz. A# C# D# F# G# A# C# D# F# G#A# C#D# F# G# A# C# D# F# G#A# C# D# F# G# A# C# D# F# G# A# C# D# F# G#A# A B DC 220 Hz middle C Figure 10.7: A piano keyboard. A piano keyboard layout is shown in Figur... |
) 3π (b) 42+√5 (c) ππ (d) 5−√3 2 (e) 3π (f) √11π−7 Problem 10.2. Put each equation in standard exponential form: (a) y = 3(2−x) (b) y = 4−x/2 (c) y = ππx (d) y = 1 1 3 3+ x 2 (e) y = 5 0.3452x−7 (f) y = 4(0.0003467)−0.4x+2 (d) Anja, a third member of your lab working with the same yeast cells, took these 106 cells afte... |
dimes will you have stacked on the nth square? (c) Cherie, another member of your lab, looks at your notebook and says :...that formula is wrong, my calculations predict the formula for the number of yeast cells is given by the function (c) How many dimes will you have stacked on the 64th square? (d) Assuming a dime i... |
the oxygen transport efficiency at 20 torrs? At 40 torrs? At 60 torrs? Sketch the graph of M(p) − H(p); are there conditions under which transport efficiency is maximized (explain)? 144 CHAPTER 10. EXPONENTIAL FUNCTIONS Chapter 11 Exponential Modeling Example 11.0.1. A computer industry spokesperson has predicted that t... |
of the function N(x): P 0.2 0.4 0.6 0.8 1 P = (0.5, 100,000) Q = (1, 750,000). 800000 600000 400000 200000 Figure 11.1: Finding the equation for N(x) = N0bx. More importantly, this example illustrates a very important principal we can use when modeling with functions of exponential type. Important Fact 11.0.2. A funct... |
need to convert to a decimal by dividing by 100. 11.1. THE METHOD OF COMPOUND INTEREST 147 11.1.1 Two Examples Let’s consider an example: P0 = $1,000 invested at the annual interest percentage of 8% compounded yearly, so n = 1 and r = 0.08. To compute the value P(1) after one year, we will have P(1) = P0 + (periodic r... |
P(1/12), since a month is one-twelfth of a year. Arguing as before, paying special attention that the periodic rate is now r 12, we have n = 0.08 P(1/12) = P0 + (periodic rate)P0 = P0 1 +.08 12 = $1,000(1 + 0.006667) = $1,006.67. 148 CHAPTER 11. EXPONENTIAL MODELING After two compounding periods, the value is P(2/12),... |
. Savings Bond for $2,500. The conditions of the bond state that the U.S. Government will pay a minimum annual interest rate of r = 8.75%, compounded quarterly. Your Uncle has given you the bond as a gift, subject to the condition that you cash the bond at age 35 and buy a red Porsche. 11.2. THE NUMBER E AND THE EXPONE... |
1 + 0.08 12 12 1 + 0.08 52 52 1 + 0.08 365 365 = $1,083.00 = $1,083.22 = $1,083.28 8,760 hourly $1,000 1 + 0.08 8,760 8,760 =$1,083.29 150 CHAPTER 11. EXPONENTIAL MODELING We could continue on, considering “minute” and “second” compounding and what we will find is that the value will be at most $1,083.29. This illustra... |
horizontal asymptote, as z becomes BIG. We will let the letter “e” represent the spot where this horizontal line crosses the vertical axis and 2.7182818. This number is only an approximation, since e is known to e be an irrational number. What sets this irrational number apart from the ones you are familiar with (e.g.... |
possible scheme for computing future value. ⇒ Important Fact 11.2.2 (Continuous compounding). The future value of P0 dollars principal invested at an annual decimal interest rate of r under continuous compounding after t years is Q(t) = P0ert; this value is alnt, for any discrete compounding 1 + r ways greater than th... |
graphs of these two func- tions. (b) The graph of the equation x2 − y2 = 1 is shown below; this is called the unit hyperbola. For any value a, show that the point (x,y) = (cosh(a), sinh(a)) is on the unit hyperbola. (Hint: Verify that [cosh(x)]2 − [sinh(x)]2 = 1, for all x.) y (−1,0) (1,0) x (c) A hanging cable is mod... |
compute ln(5) = 1.60944. Conclude that the solution is t = 20.12 years. 12.1 The Inverse Function of y = ex If we sketch a picture of the exponential function on the domain of all real numbers and keep in mind the properties in Fact 10.2.1, then every horizontal line above the x-axis intersects the graph of y = ex exa... |
real numbers. natural logarithm function y = ln(x) can be obtained by flipping the graph of y = ex across the line y = x: Important Facts 12.1.1 (Graphical features of natural log). The function y = ln(x) has these features: • • • The largest domain is the set of positive numbers; e.g. ln(−1) makes no sense. The graph ... |
solve for x: − ln x2 + 1 = 5 ln(x) − ln x2 + 1. x2 − 1 = ln((x − 1)(x + 1)) = ln(x − 1) + 3x+1 = 12 3x+1 ln = ln(12) (x + 1) ln(3) = ln(12) ln(12) ln(3) x = − 1 = 1.2619. Example 12.1.5. If $2,000 is invested in a continuously compounding savings account and we want the value after 12 years to be $130,000, what is the... |
6 = 0 and a = 0. By studying the sign of the constant a, we can determine whether the function exhibits exponential growth or decay. For example, given the function A(t) = eat, if a > 0 (resp. a < 0), then the function exhibits exponential growth (resp. decay). Examples 12.2.2. (a) The function A(t) = 200 (2t) exhibit... |
function y = logb(x) has these features: • • • The largest domain is the set of positive numbers; e.g. logb(−1) is not defined. The graph has x-intercept 1 and is increasing if b > 1 (resp. decreasing if 0 < b < 1). The graph becomes closer and closer to the vertical axis as we approach x = 0; this says the y-axis is a... |
b) y =. We have just verified a useful conversion formula: Important Fact 12.3.3 (Log conversion formula). For x a positive number and b > 0, b = 1 a base, logb(x) = ln(x) ln(b). For example, log10(5) = log0.02(11) = log20 1 2 = ln(5) ln(10) ln(11) ln(0.02) ln( 1 2) ln(20) = 0.699 = −0.613 = −0.2314 6 6 12.4. MEASURING ... |
“loudness”. This loudness is directly related to the force being exerted on the eardrum, which we refer to as the intensity of the sound. We can try to measure the intensity using some sort of scale. This becomes challenging, since the human ear is an amazing instrument, capable of hearing a large range of sound inten... |
(something like the ground-shaking rumble of a passing freight train), the sound pressure level needs to be relatively high to be heard; 100 db on average. As the frequency increases, the required sound pressure level for hearing tends to drop down to 0 db around 2000 Hz. An examination by a hearing specialist can det... |
, which is twice that of the first speaker. If I3 is the corresponding intensity of the sound, then as above, I3 = 10(β/10)I0. We are assuming that I3 = 2I1, so this gives us the equation I1 = 108.7I0 = 1 2 1 2 I3 10(β/10)I0 log10 108.7 = log10 8.7 = log10 1 2 1 2 10(β/10) + log10 8.7 = −0.30103 + 90 = β β 10 10(β/10) 1... |
If you invest Po dollars at 7% annual interest and the future value is computed by continuous compounding, how long will it take for your money to double? 163 (c) A rule of thumb used by many people to determine the length of time to double an investment is the rule of 70. The rule says it takes about t = 70 r years t... |
(a) Suppose the value of the house is $75,000 in 1962. Assume v(x) is a linear function. Find a formula for v(x). What is the value of the house in 1995? When will the house be valued at $200,000? (b) Suppose the value of the house is $75,000 in 1962 and $120,000 in 1967. Assume v(x) is a quadratic function. Find a fo... |
. At the start of the voyage, there were 500 ants in the cargo hold of the ship. One week into the voyage, there were 800 ants. Suppose the population of ants is an exponential function of time. (a) How long did it take the population to double? (b) How long did it take the population to triple? (c) When were there be ... |
to a new curve that (by the vertical line test) is the graph of a new function. The big caution in all this is that we are NOT ALLOWED to rotate or twist the curve; this kind of maneuver does lead to a new curve, but it may not be the graph of a function: See Figure 13.2. rotate NOT a function graph Figure 13.2: Rotat... |
line” in the same picture with the original p(x). Once we do this, it is easy to see how the graph of q(x) is really just the original line reflected across the x-axis. 6; so 2 ≥ ≤ ≥ ≤ y Next, take the original function equation y = p(x) = 2x + 2 and replace every occurrence of “x” by “−x.” This produces a new equation... |
the same as the domain for y = f(x). If the range for y = f(x) d. In other words, is c the reflection across the x-axis is the graph of y = −f(x). d, then the range of −y = f(x) is c −y ≤ ≤ ≤ ≤ y (ii) We can reflect the graph across the y-axis and the resulting curve is the graph of the new function obtained by replacin... |
x) = −2x − 2. Finally, by Fact 13.2.1 (i) applied to ℓ3, the line ℓ4 is the graph of the equation −y = −2x − 2, which we can write as the function y = f4(x) = 2x + 2. Figure 13.6 illustrates the fact that we need to be careful about the domain of the original function when using the reflection principle. For example, co... |
y = 3+√4 − x2. The upper right-hand dashed semicircle is of radius 2 and centered at (3, 3), so the corresponding equation must be y = 3 + 4 − (x − 3)2. p Keeping this same example, we can continue this kind p of shifting more generally by thinking about the effect of making the following three replacements in the equ... |
change. This example illustrates an important general principle referred to as the shifting principle. Important Facts 13.3.1 (Shifting). Let y = f(x) be a function equation. (i) If we replace “x” by “x − h” in the original function equation, then the graph of the resulting new function y = f(x − h) is obtained by hor... |
Grab the high point H on the curve and uniformly pull straight up, so that the high point now lies on the horizontal line y = 1 at (1, 1). Repeat this process by pulling L straight downward, so that the low point is now on the line y = −1 at (−1, −1). We end up with the ”stretched dashed curve” illustrated in Figure 1... |
, respectively. A number of possibilities are pictured in Figure 13.11(c). ± We refer to each new dashed curve as a verticaldilation of the original (solid) curve. This example illustrates an important principle. Important Facts 13.4.1 (Vertical dilation). Let c > 0 be a positive number and y = f(x) a function equation... |
)2. p step 1: start with upper semicircle x-axis step 2: reflect across step 3: stretch curve from step 2: to get y = −4f(x) Solution. The graph of y = f(x) is an upper semicircle of radius 1 centered at the point (−1,0). To obtain the picture of the graph of y = −4f(x), we first reflect y = f(x) across the x-axis; this g... |
location location. These two situations are indicated in Figure 13.13. We refer to each of the dashed curves as a horizontal dilation of the original (solid) curve. and the low point L = −1, − 1 2 2, − 1 1, 1 2 2, 1 − 1 2 1 2 13.5. VERTEX FORM AND ORDER OF OPERATIONS 173 The tricky point is to understand what happens ... |
y = x2 with four other functions, each of which corresponds to a horizontal shift, vertical shift, reflection or dilation. Once we have done this, we can read off the order of geometric operations using the order of composition. Along the way, pay special attention to the exact order in which we will be composing our f... |
axis. A reflection across the x-axis. Picture 2 2 2 −2 −2 −2 Table 13.1: Reflecting y = f(x). 176 CHAPTER 13. THREE CONSTRUCTION TOOLS Shifting (Assume c > 0 ) Symbolic Change New Equation Graphical Consequence Picture Replace x with (x − c). y = f(x − c) A shift to the right c units. 2 Replace x with (x + c). y = f(x + ... |
unit circle intersects each of the four lines exactly once. (b) Find the intersection points between the unit circle and each of the four lines. (c) Construct a diamond shaped region in which the circle of radius 1 centered at (−2, − 1) sits tangentially. Use the techniques of this section to help. y π π/2 f(x) −1 1 x... |
3) y = 3|2x − 1| + 5 10 ≥ 13.7. EXERCISES 179 (c) The graphs of y = 3|2x − 1| + 5 and y = −|x − 3| + 10 intersect to form a bounded region of the plane. Find the vertices of this region and sketch a picture. (c) The graph of z = a(x) from part (a) is given below. Sketch the graph and find the rule for the function z = 2... |
zero precisely when 0 < x < 1. Problem 13.6. An isosceles triangle has sides of length x, x and y. In addition, assume the triangle has perimeter 12. (a) Find the rule for a function that computes the area of the triangle as a function of x. Describe the largest possible domain of this function. (b) Assume that the max... |
(x), f(2x), f(2(x− 1)), 3f(2(x − 1)) and 3f(2(x − 1)) + 1 in the same coordinate system and explain which graphical operation(s) (vertical shifting, vertical dilation, horizontal shifting, horizontal dilation) have been carried out. (b) In general, explain what happens when you apply the four construction tools of Chap... |
(x) = 1 x whose graph is shown in Figure 14.1. This is an important example for the study of this class of functions, as we shall see. Let’s consider the graph of this function, f(x) = 1 x. We first begin by considering the domain of f. Since the numerator of 1 x is a constant, and the denominator is just x, the only wa... |
large in the negative direction, the curve y = 1/x gets closer and closer to the x-axis. We say that, in both the positive and negative directions, y = 1/x is asymptotic to the x-axis. A similar thing happens when we consider x near zero. If x is a small positive number (think 1/2, or 1/10, or 1/10000), then 1/x is a ... |
. 183 If we let g(x) = 1/x then we have shown that f(x) = A + Bg(x + C), and so the graph of the function f is just a horizontally shifted, vertically shifted and vertically dilated version of the graph of g. Also, if B turned out to be negative the graph would be vertically flipped, too. Why is that useful? It means th... |
x) is is close to a constant, and that constant is a c. Thus, the horizontal asymptote of f(x) = is the horizontal line y = a c. ax + b cx + d Example 14.0.1. Sketch the graph of the function f(x) = 3x−1 2x+7. Solution. We begin by finding the asymptotes of f. The denominator is equal to zero when 2x + 7 = 0, i.e., when... |
��ed by the asymptotic nature of the linear-to-linear rational functions, and so this type of function provides a way to model such behavior. Given any linear-to-linear rational function, we can always divide the numerator and the denominator by the coefficient of x in the denominator. In this way, we can always assume ... |
-to-linear rational function, we know f(x) = ax + b x + c for constants a, b, and c. We need to find a, b and c. We know three things. First, f(10) = 20. So f(10) = 10a + b 10 + c = 20, which we can rewrite as 10a + b = 200 + 20c. Second, f(20) = 32. So f(20) = 20a + b 20 + c = 32, which we can rewrite as 20a + b = 640 ... |
However, the method is essentially identical. Let’s now apply these ideas to a real world problem. Example 14.1.3. Clyde makes extra money selling tickets in front of the Safeco Field. The amount he charges for a ticket depends on how many he has. If he only has one ticket, he charges $100 for it. If he has 10 tickets... |
the domain of each of the following functions. Find the x- and yintercepts of each function. Sketch a graph and indicate any vertical or horizontal asymptotes. Give equations for the asymptotes. (a) f(x) = 2x x−1 (c) h(x) = x+1 x−2 (e) k(x) = 8x+16 5x− 1 2 (b) g(x) = 3x+2 2x−5 (d) j(x) = 4x−12 x+8 (f) m(x) = 9x+24 35x... |
. 189 (e) In the long run, what will be the ratio of the prices of the ukuleles? Problem 14.4. Isobel is producing and selling casette tapes of her rock band. When she had sold 10 tapes, her net profit was $6. When she had sold 20 tapes, however, her net profit had shrunk to $4 due to increased production expenses. But w... |
FUNCTIONS (a) From 20 feet away, the street light has an intensity of 1 candle. What is k? (b) Find a function which gives the intensity of the light shining on Olav as a function of time, in seconds. (c) When will the light on Olav have maxi- mum intensity? (d) When will the intensity of the light be 2 candles? Probl... |
circular path. Natural Questions 15.0.1. How can we measure the angles ∠SPR, ∠QPR, and ∠QPS? How can we measure the arc lengths arc(RS), arc(SQ) and arc(RQ)? How can we measure the rate Cosmo is moving around the circle? If we know how to measure angles, can we compute the coordinates of R, S, and Q? Turning this arou... |
�AOB in Figure 15.2 constructed from two pieces of rigid wire, welded at the vertex. Sliding this model around inside the xy-plane will not distort its shape, only its position relative to the coordinate axis. So, we can slide the angle into position so that the initial side is coincident with the positive x-axis and t... |
at the origin. Divide this circle into 360 equal sized pie shaped wedges, beginning with the the point (r,0) on the circle; i.e. the place where the circle crosses the x-axis. We will refer to the arcs along the outside edges of these wedges as one-degree arcs. Why 360 equal sized arcs? The reason for doing so is hist... |
each called a one minute arc. Each one-minute arc can be further divided into 60 equal arcs, each called a one second arc. This then leads to angle measures of one minute, denoted 1 ′ and one second, denoted 1 ′′: 1◦ = 60 minutes = 3600 seconds. 15.3. DEGREE METHOD 195 90◦ 180◦ 270◦ 45◦ 315◦ −135◦ Figure 15.5: Example... |
�, 270◦, 45◦, −135◦, and 315◦. If the circle is of radius r and we want to compute the lengths of the arcs subtended by these six angles, then this can be done using the formula for the circumference of a circle (on the back of this text) and the following general principle: Important Fact 15.3.1. (length of a part) = ... |
have measure 1 radian Figure 15.7: Constructing an equilateral wedge. As before, begin with a circle Cr of radius r. Construct an equilateral wedge with all three sides of equal length r; see Figure 15.7. We define the measure of the interior angle of this wedge to be 1 radian. Once we have defined an angle of measure 1... |
units of θ are sometimes abbreviated as rad. It is important to appreciate the role of the radius of the circle Cr when using radian measure of an angle: An angle of radian measure θ will subtend an arc of length |θ| on the unit circle. In other words, radian measure of angles is exactly the same as arc length on the ... |
it is rigged so that we can easily compute lengths of arcs and areas of circular sectors (i.e. “pie-shaped regions”). This is a key reason why we will almost always prefer to work with radian measure. Example 15.5.1. If a 16 inch pizza is cut into 12 equal slices, what is the area of a single slice? This can be solved... |
ians. Consequently, by Important Fact 15.5.2 the area A(t) of the irrigated region after t hours is A(t) = 1 2 (1200)2θ(t) = 1 2 (1200)2 π 6 t = 120,000πt square feet. After 1 hour, the irrigated area is A(1) = 120,000π = 376,991 sq. ft. Likewise, after 37 minutes, which is 37 60 hours, the area of the irrigated region... |
call a line segment connecting two points on a circle a chord of the circle. The above example illustrates a general principal for approximating the length of any chord. A smaller angle will improve the accuracy of the arc length approximation. Important Fact 15.5.5 (Chord Approximation). In Figure 15.14, if the centr... |
). When b = 0, we are at the North or South poles on the earth. In a similar spirit, we could imagine slicing the earth with a plane Q which is perpendicular to the equatorial plane and passes through the center of the earth. The resulting intersection will trace out a circle of radius 3,960 miles on the surface of the... |
to study the distance between two locations. A great circle of a sphere is defined to be a circle lying on the sphere with the same center as the sphere. For example, the equator and any line of longitude are great circles. However, lines of latitude are not great circles (except the special case of the equator). Great... |
degrees/ minutes/ seconds. Problem 15.2. A nautical mile is a unit of distance frequently used in ocean navigation. It is defined as the length of an arc s along a great circle on the earth when the subtending angle has measure 1 ′ = “one minute” = 1/60 of one degree. Assume the radius of the earth is 3,960 miles. (a) ... |
angle of 110◦, how much area is swept clean? (b) Through how much of an angle would the wiper sweep if the area cleaned was 10 square inches? Problem 15.5. Astronomical measurements are often made by computing the small angle formed by the extremities of a distant object and using the estimating technique in 15.5.1. I... |
swept out per unit time” by the moving object, starting from some initial position. We need to somehow indicate the direction in which the angle is being swept out. This can be done by indicating “clockwise” our “counterclockwise”. Alternatively, we can adopt the convention that the positive rotational direction is co... |
20π 7 ft sec. 16.2. DIFFERENT WAYS TO MEASURE CIRCULAR MOTION 209 Important Fact 16.1.1. This discussion is an example of what is usually called “units analysis”. The key idea we have illustrated is how to convert between two different types of units: rev min converts to − ft min 16.2 Different Ways to Measure → Circu... |
now know two general relationships for circular motion: (i) s = rθ, where s=arclength (a linear distance), r=radius of the circular path and θ=angle swept in RADIAN measure; this was the content of Fact 15.4.1 on page 197. (ii) θ = ωt, where θ is the measure of an angle swept, ω= angular speed and t represents time el... |
wise direction and the radius r = 1 228 = 14 inches. The other given quantity, “40 MPH”, involves miles, so we need to decide which common units to work with. Either will work, but since the problem is focused on the wheel, we will utilize inches. If the speedometer reads 40 MPH, this is the linear speed of a specified ... |
“time” unit comes from v and the “angular” unit will always be radians. As a comparison with the solution above, we can convert ω into RPM units: rad sec 1 rev 2π rad ω = 50.28 = 8 rev sec. All of the problems in this section can be worked using either the “unit conversion method” or the “v = ωr method”. 16.3 Music Li... |
lead-in groove, the needle gradually works its way to the exit groove. However, whereas the angular speed of the LP is a constant 33 1 3 RPM, the linear speed at the needle can vary quite a bit, depending on the needle location. 214 CHAPTER 16. MEASURING CIRCULAR MOTION 6 1 Figure 16.6: Lead-in and exit grooves. Examp... |
disc. The pits in the silver coating will cause the reflected laser light to vary in intensity. A sensor detects this variation, converting it to a digital signal (the analogue to digital or AD conversion). This is fed into a digital to analogue or DA conversion device, which sends a signal to your stereo, again produc... |
350 RPM. 16.4 Belt and Wheel Problems The industrial revolution spawned a number of elaborate machines involving systems of belts and wheels. Computing the speed of various belts and wheels in such a system may seem complicated at first glance. The situation can range from a simple system of two wheels with a belt conn... |
sprocket are both rigidly mounted on a common axis of rotation. Step 3: Given ωB, find vB. Use the fact vB = rBωB = rBωA = vA. rB rA Step 4: Observe vB = vchain = vC; this is because the chain is directly connecting the two sprockets and assumed not to slip. 16.4. BELT AND WHEEL PROBLEMS 217 • Step 5: Given vC, find ωC.... |
CIRCULAR MOTION 16.5 Exercises Problem 16.1. The restaurant in the Space Needle in Seattle rotates at the rate of one revolution per hour. (a) Through how many radians does it turn in 100 minutes? (b) How long does it take the restaurant to rotate through 4 radians? (c) How far does a person sitting by the window move... |
should rotate counterclockwise with an angular speed of 12 RPM; (ii) the linear speed of a rider should be 200 mph; (iii) the lowest point on the ride should be 4 feet above the level ground. 12 RPM θ P 4 feet (a) Find the radius of the ferris wheel. (b) Once the wheel is built, John suggests that Tiff should take the... |
sprocket has radius 3 inches and the front sprocket has radius r inches. Suppose you are pedaling the front sprocket at the rate of 1.5 rev sec and your forward speed is 11 mph on the bike. What is the radius of the front sprocket? Problem 16.8. You are designing a system of wheels and belts as pictured below. You wan... |
are equal: for x, we obtain x = 15 4 221 The billiard table layout. 4 ft 5 ft find this location this pocket for the big money 6 ft 12 ft 4 θ θ 5 − x x Mathmatically modeling the bank shot. Figure 17.2: A pocket billiard banking problem. 222 CHAPTER 17. THE CIRCULAR FUNCTIONS This discussion is enough to win the tourne... |
In the middle triangle, sin(θ) = 1, tan(θ) = 1. In the right-hand triangle, √2 sin(θ) = 1. The symbols “sin”, “cos”, and “tan” are abbreviations for the words sine, cosine and tangent, respectively. As we have defined them, the trigonometric ratios depend on the dimensions of the triangle. However, the same ratios are ... |
right triangle. Angle θ Trigonometric Ratio Deg Rad sin(θ) cos(θ) tan(θ) 0◦ 30◦ 45◦ 60◦ 902 2 √3 2 1 1 √3 2 √2 2 1 2 0 0 1 √3 1 √3 Undefined Table 17.1: Exact Trigonometric Ratios 224 CHAPTER 17. THE CIRCULAR FUNCTIONS!!! CAUTION!!! Some people make a big deal of “approximate” vs. “exact” answers; we won’t worry about ... |
of the angle ∠ABC is only accurate within 2 ′, find the possible error in |AC|. ± C d A B 100 310 18 ′ Solution. The trigonometric ratio relating these two sides would be the tangent and we can convert θ into decimal form, arriving at: tan(31◦18 ′) = tan(31.3◦) = |AC| |BA| = d 100 Figure 17.7: The distance spanning a r... |
|TE| = |TL| − |EL| = 2.850 − 2.544 = 0.306 miles = 1,616 feet. The elevation of the peak above sea level is given by: Peak elevation = plane altitude + |EL| = |SP| + |EL| = 2,000 + (2.544)(5,280) = 15,432 feet. Example 17.3.4. A Forest Service helicopter needs to determine the width of a deep canyon. While hovering, t... |
23 feet. As noted above, |ED| = |CD| − |CE| = 2,724 − 1,723 = 1,001 feet is the width of the canyon. 17.4 Circular Functions S = (x,y) y 20 θ x P R If Cosmo is located somewhere in the first quadrant of Figure 17.1, represented by the location S, we can use the trigonometric ratios to describe his coordinates. Impose th... |
on the unit circle. Definition 17.4.1. Let θ be an angle in standard central position inside the unit circle, as in Figure 17.11. This angle determines a point P on the unit circle. Define two new functions, cos(θ) and sin(θ), on the domain of all θ values as follows: cos(θ) sin(θ) def= horizontal x-coordinate of P on u... |
θ), sin(θ)) in Figure 17.11 only works if the angle θ is viewed in central standard position. You must do some additional work if the angle is placed in a different position; see the next Example. y-axis r = 1 kilometer 0.025 rad sec x-axis Michael starts here Angela starts here 0.03 rad sec (a) Angela and Michael on t... |
So, after 18 seconds Angela’s location will be A(18) = 17.4. CIRCULAR FUNCTIONS 229 17.4.2 Relating circular functions and right triangles If the point P on the unit circle is located in the first quadrant, then we can compute cos(θ) and sin(θ) using trigonometric ratios. In general, it’s useful to relate right triangl... |
) and want to compute the coordinates of points on this circle? The circular functions can be used to answer this more general question. Picture our circle Cr centered at the origin in the same picture with unit circle C1 and the angle θ in standard central position for each circle. As pictured, we can view θ = ∠ROP = ... |
(2.9416), sin(2.9416)) = (−0.9801, 0.1987) T = (2 cos(2.9416), 2 sin(2.9416)) = (−1.9602, 0.3973) U = (3 cos(2.9416), 3 sin(2.9416)) = (−2.9403, 0.5961). 17.6. OTHER BASIC CIRCULAR FUNCTION 231 Example 17.5.3. Suppose Cosmo begins at the position R in the figure, walking around the circle of radius 20 feet with an angul... |
= 0. This calculation motivates a tan(θ) = sin(θ) cos(θ), provided cos(θ) = 0. 6 6 232 CHAPTER 17. THE CIRCULAR FUNCTIONS The only time cos(θ) = 0 is when the corresponding point P on the unit circle has x-coordinate 0. But, this only happens at the positions (0, 1) and (0, −1) on the unit circle, corresponding to ang... |
the Northwest flight when it is 20 miles North of SeaTac. Find the location of the Alaska flight when it is 50 miles West of SeaTac. Find the location of the Delta flight when it is 30 miles East of SeaTac. impose a coordinate Solution. We system in Figure 17.21(a), where “East” (resp. “North”) points along the positive ... |
FUNCTIONS 17.7 Exercises Problem 17.1. John has been hired to design an exciting carnival ride. Tiff, the carnival owner, has decided to create the world’s greatest ferris wheel. Tiff isn’t into math; she simply has a vision and has told John these constraints on her dream: (i) the wheel should rotate counterclockwise... |
seconds to reach the northermost point of the track. Impose a coordinate system with the center of the track at the origin, and the northernmost point on the positive y-axis. (a) Give Marla’s coordinates at her starting point. (b) Give Marla’s coordinates when she has been running for 10 seconds. (c) Give Marla’s coor... |
the head of the Queen Anne Community Group and one of your members asks you to make sure that the radio station does not exceed the limits of the permit. After finding a relatively flat area nearby the tower (not necessarily the same altitude as the bottom of the tower), and standing some unknown distance away from the ... |
or string over the power lines? Problem 17.12. In the pictures below, a bug has landed on the rim of a jelly jar and is moving around the rim. The location where the bug initially lands is described and its angular speed is given. Impose a coordinate system with the origin at the center of the circle of motion. In eac... |
moves from 0 π 2 radians around unit circle to x-axis (a) What do you see on the y-axis? (b) What do you see on the x-axis? Figure 18.2: Projecting the coordinates of points onto the y-axis and the x-axis. By studying the coordinates of the ball as it moves in the first quadπ/2 radians. rant, we will be studying cos(θ)... |
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