SECTION 6.5 1–8 Find the average value of the function on the given interval. 2 1. f sxd − 3x 1 8x, f21, 2g 2. f sxd − sx , f0, 4g 3. tsxd − 3 cos x, f2y2, y2g 4. tstd − t s3 1 t 2 5. f std − e sin t , cos t, Tstd − 50 1 14 sin f0, y2g 18. The velocity v of blood that flows in a blood vessel with radius R and length l at a distance r from the central axis is 9–12 (a) Find the average value of f on the given interval. (b) Find c such that fave − f scd. (c) Sketch the graph of f and a rectangle whose area is the same as the area under the graph of f . f2, 5g f1, 3g ; 11. f sxd − 2 sin x 2 sin 2x, 2 t 12 Find the average temperature during the period from 9 am to 9 pm. 8. hsud − sln udyu, f1, 5g 2x ; 12. f sxd − 2xe , (a) Use the Midpoint Rule to estimate the average velocity of the car during the first 12 seconds. (b) At what time was the instantaneous velocity equal to the average velocity? f1, 3g 7. hsxd − cos 4 x sin x, f0, g 10. f sxd − 1yx, 463 17. In a certain city the temperature (in °F ) t hours after 9 am was modeled by the function 6. f s xd − x 2ys x 3 1 3d2, f21, 1g 9. f sxd − sx 2 3d2, Average Value of a Function vsrd − P sR 2 2 r 2 d 4l where P is the pressure difference between the ends of the vessel and is the viscosity of the blood (see Example 3.7.7). Find the average velocity (with respect to r) over the interval 0 < r < R. Compare the average velocity with the maximum velocity. 19. The linear density in a rod 8 m long is 12ysx 1 1 kgym, where x is measured in meters from one end of the rod. Find the average density of the rod. f0, g f0, 2g 3 13. If f is continuous and y1 f sxd dx − 8, show that f takes on the value 4 at least once on the interval f1, 3g. 14. Find the numbers b such that the average value of f sxd − 2 1 6x 2 3x 2 on the interval f0, bg is equal to 3. 20. (a) A cup of coffee has temperature 95°C and takes 30 minutes to cool to 61°C in a room with temperature 20°C. Use Newton’s Law of Cooling (Section 3.8) to show that the temperature of the coffee after t minutes is Tstd − 20 1 75e2kt 15. Find the average value of f on f0, 8g. where k < 0.02. (b) What is the average temperature of the coffee during the first half hour? y 1 0 2 4 6 x 16. The velocity graph of an accelerating car is shown. √ (km/h) 60 40 21. In Example 3.8.1 we modeled the world population in the second half of the 20th century by the equation Pstd − 2560e 0.017185t. Use this equation to estimate the average world population during this time period. 22. If a freely falling body starts from rest, then its displacement is given by s − 12 tt 2. Let the velocity after a time T be v T . Show that if we compute the average of the velocities with respect to t we get vave − 12 v T , but if we compute the average of the velocities with respect to s we get vave − 23 v T . 20 0 4 8 12 t (seconds) 23. Use the result of Exercise 5.5.83 to compute the average volume of inhaled air in the lungs in one respiratory cycle. Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 464 CHAPTER 6 Applications of Integration 24. Use the diagram to show that if f is concave upward on fa, bg, then a1b fave . f 2 S D 25. Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.2) x to the function Fsxd − ya f std dt. 26. If fave fa, bg denotes the average value of f on the interval fa, bg and a , c , b, show that y f fave fa, bg − 0 a a+b 2 b APPLIED PROJECT c2a b2c fave fa, cg 1 fave fc, bg b2a b2a x CALCULUS AND BASEBALL In this project we explore three of the many applications of calculus to baseball. The physical interactions of the game, especially the collision of ball and bat, are quite complex and their models are discussed in detail in a book by Robert Adair, The Physics of Baseball, 3d ed. (New York, 2002). 1. It may surprise you to learn that the collision of baseball and bat lasts only about a thousandth of a second. Here we calculate the average force on the bat during this collision by first computing the change in the ball’s momentum. The momentum p of an object is the product of its mass m and its velocity v, that is, p − mv. Suppose an object, moving along a straight line, is acted on by a force F − Fstd that is a continuous function of time. (a) Show that the change in momentum over a time interval ft0 , t1 g is equal to the integral of F from t0 to t1; that is, show that t1 Batter’s box An overhead view of the position of a baseball bat, shown every fiftieth of a second during a typical swing. (Adapted from The Physics of Baseball) pst1 d 2 pst 0 d − y Fstd dt t0 This integral is called the impulse of the force over the time interval. (b) A pitcher throws a 90-miyh fastball to a batter, who hits a line drive directly back to the pitcher. The ball is in contact with the bat for 0.001 s and leaves the bat with velocity 110 miyh. A baseball weighs 5 oz and, in US Customary units, its mass is measured in slugs: m − wyt, where t − 32 ftys 2. (i) Find the change in the ball’s momentum. (ii) Find the average force on the bat. 2. In this problem we calculate the work required for a pitcher to throw a 90-miyh fastball by first considering kinetic energy. The kinetic energy K of an object of mass m and velocity v is given by K − 12 mv 2. Suppose an object of mass m, moving in a straight line, is acted on by a force F − Fssd that depends on its position s. According to Newton’s Second Law Fssd − ma − m dv dt where a and v denote the acceleration and velocity of the object. (a) Show that the work done in moving the object from a position s0 to a position s1 is equal to the change in the object’s kinetic energy; that is, show that s1 W − y Fssd ds − 12 mv122 12 mv 02 s0 Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.