Homework, Page 335
Homework Homework Assignment #4 Read Section 5.5 Page 335, Exercises: 1 – 49(EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 1. Write the area function of f (x) = 2x + 4 with lower limit a = –2 as an integral and find a formula for it. x 2 2 x 4 dx F x F 2 x 2 4 x 2 4 2 2 x2 4 x 4 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 4 5. Find G 1 , G 0 , and G , where G x 1 tan tdt x d x G 1 tan tdt 0 G x 1 tan tdt tan x dx 1 1 4 tan 4 1 G 1 0, G 0 0, G 1 4 G 0 tan 0 0, G Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 Find formulas for the functions represented by the integrals. x2 9. 1 tdt 2 tdt F x F 1 x2 1 x 1 2 2 2 2 x4 1 2 2 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 Find formulas for the functions represented by the integrals. x 13. sec2 d 4 x 4 4 tan x tan tan x 1 4 sec 2 d F x F x 4 sec 2 d tan x 1 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 Express the antiderivative F(x) of f (x) satisfying the given initial condition as an integral. 17. sec x, F 0 0 F x 0 sec tdt x Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 Calculate the derivative. d t 21. 100 cos 5 xdx dt d t 100 cos 5 xdx cos 5t dt Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 25. Make a rough sketch of the graph of the area function of g(x) shown in Figure 12. 1 2 3 4 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 Calculate the derivative. d x2 2 29. 0 sin tdt dx d x2 2 2 2 2 2 sin tdt sin x 2 x 2 x sin x 0 dx d x2 2 2 2 0 sin tdt 2 x sin x dx Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 Calculate the derivative. d 0 2 33. sin tdt 3 x dx d 0 d x3 2 2 2 3 2 2 2 3 x3 sin tdt 0 sin tdt sin x 3x 3x sin x dx dx d 0 2 2 2 3 sin tdt 3 x sin x 3 x dx Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 Let A x 0 f t dt and B x 2 f t dt with f x as in Figure 13. x 37. x Find the min and max of A(x) on [0, 6]. min 1.25, max 1.25 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 41. Area Functions and Concavity. Explain why the following statements are true. Assume f (x) is differentiable. (a) If c is an inflection point of A(x), then f ′(c) = 0. Since f(x) = A′(x), f ′(x) =A″(x) and A″(x) = 0 at inflections points of A(x). (b) A(x) is concave up if f (x) is increasing. If f(x) is increasing, then f ′(x) > 0, hence A″(x) > 0 and A(x) is concave up. (c) A(x) is concave down if f (x) is decreasing. If f(x) is decreasing, then f ′(x) < 0, hence A″(x) < 0 and A(x) is concave up. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 45. Sketch the graph of an increasing function f (x) such that both f ′(x) and A (x) are decreasing. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework, Page 335 49. Determine the function g (x) and all values of c such that 2 g t dt x x6 x c d 2 x x 6 2x 1 g x dx x 2 x 6 0 x 2 x 3 0 c 3, 2 2 g x x 1, c 3, 2 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 5: The Integral Section 5.5: Net or Total Change as the Integral of a Rate Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If water enters an empty bucket at the rate r(t), as shown in Figure 1, then the amount of water in the bucket at any time on the interval [0, 4] is equal to the area under the curve up to the vertical line at the time in question. Since the water enters the bucket at a varying rate, we can use the FTC to find the amount of water in the bucket at any time x, specifically: r t dt R x R 0 . Since the bucket is initially empty, the amount of water in the bucket is at time x is R( x) and the integral gives us the net change in the amount of water in the bucket. x 0 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company If s′(t) is both positive and negative on [t1, t2], the integral will total the signed areas and give us the net change in s. This is the basis of Theorem 1. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Example, Page 341 4. A survey shows that a mayoral candidate is gaining votes at the rate of 2000t + 1000 votes per day since she announced her candidacy. How many supporters will the candidate have after 60 days, assuming she had no supporters at t = 0. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The table shows the flow rate of cars passing an observation point in units of cars per hour. Estimate the number of cars using the highway in the two hour period by taking the average of the left and right Riemann sums. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Remembering that displacement is a vector quantity, the integral of velocity, another vector quantity, over an interval of time gives us the net displacement. To find distance traveled, we must take the integral of the absolute value of velocity, as illustrated in Figure 2. Figure 3 illustrates the path of the particle with velocity v(t) from Figure 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Theorem 2 formalizes what we observed in Figure 2. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Example, Page 341 8. A projectile is released with initial vertical velocity 100 m/s. Use the formula v(t) = 100 – 9.8t for velocity to determine distance traveled in the first 15 seconds. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Example, Page 341 18. Suppose that the marginal cost of producing x video recorders is 0.001x2 – 0.6x + 350 dollars. What is the cost of producing 300 units? If production is set at 300 units, what is the cost of producing 20 additional units? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Homework Homework Assignment #5 Read Section 5.6 Page 341, Exercises: 1 – 19(Odd) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company
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