Review for Chapter 8
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1 Review for Chapter 8 The function v(t) is the velocity of an object moving on the x-axis at any time t 0 where the velocity is measured in ft/sec. a) Determine when the object is moving right, left and stopped b) Determine the acceleration of the object at t =. c) Determine the average acceleration of the object over the given time interval (set-up only) d) Determine the displacement of the object over the given time interval e) Determine the average velocity of the object over the given time interval (set-up only) f) Determine the total distance traveled by the object over the given time interval *do star problems without a calculator *1) ( t) e v t,4 0 * ) v( t) 3sin t 0,
2 v,5 c 3) ( t) t 3t 4 (for extra practice) 0 c 4) v( t) sin t cos t 0, (if there is time) 5) A particle moves along a y-axis. Its velocity, measured in cm/hr, is given in the graph below and initial position at t = 0 is y(0) = 4. The values in each region represent the area of the region. a) Find the position at each time: t = 3 t = 6 t = 8 b) What is the displacement of the particle for the first 8 hours? c) What is the displacement by the object for 3 t 8 c) What is the total distance traveled on 0,8? d) At what time t is the speed the greatest? e) When does acceleration equal 0?
3 6) An object is moving along an x-axis and its velocity v(t) is given in the graph below. The velocity is measured in feet/sec. The object s position is x(0) = 3 v(t) ft/sec (, ) 3 9 (13, 4) t (sec) a) Find the position at each time. t = 0 (4, -) (7, -) t = 9 t = 13 b) Find the displacement on 0,13. c) Find the total distance traveled on 0,13. d) Find the average velocity of the object on 0,13. e) Find when the object is at 3, other than at t = 0. Justify. f) At the end of the trip (t = 13), is the object left, right, or at the same place from where it started? g) Find a(3)
4 7) A snail is moving along an x-axis and its velocity v(t) is given in the graph below. The velocity is measured in cm/day. v(t) cm/day (0, ) 9 (13, 4) t (day) Show all analyses that leads to your solutions. a) Find the snail s acceleration at t =. b) Find the displacement of the snail on 0,4 (4, -) (7, -) c) Find the displacement of the snail on 4,11 d) Find the total distance the snail traveled on 0,13 e) Given that at t = 4 the snail was at position x = 10 on an x-axis. Find the position of the snail at each time t = 4 t = 9 t = 13 t = t = 0 f) Write a function P(t) to determine the position of the snail on the x-axis at any given time t. g) At time t = 9, is the snail left, right, or at the same place from the snail position at t = 4? Justify. h) Find when the object is at 10, other than at t = 4. Justify.
5 8) The rate of milk consumption in the United States is given by C(t)=35.08e t/30 where t is the number of years after Jan 1, 1980 and measured in kiloliters per year. a) Find the rate of milk consumption in b) Find the total amount of milk consumed from c) Find the total amount of milk consumed from d) Find the average rate of milk consumption from 1980 to ) Below is a chart of velocities of a ball rolling down an incline in meters per minute. 0,1 min. a) Find the average acceleration on mins m/min b) When is acceleration negative? c) Estimate the acceleration of 3 minutes 1 d) Use a right rectangular approximation to estimate v ( t) dt 0 e) Using correct units, what is d) telling you? f) Find the average velocity of the ball on 0,1
6 For problems 10, 11, 1. (velocities measured in cm/hr.) a) Determine when the object is moving right, left and stopped b) Determine the acceleration of the object at t = 1. c) Determine the average acceleration of the object over the given time interval (set-up only) d) Determine the displacement of the object over the given time interval e) Determine the average velocity of the object over the given time interval (set-up only) f) Determine the total distance traveled by the object over the given time interval 10) (calc. ok) v( t) 6sin 3t 0 t t 11) (calc. ok) v( t) 0 t 3 1 t 1) (no calc.) v ( t) 6t 18t 1 0 t 3
7 13) Water is poured into a container and drained out. The velocity v(t) of the water is given in the graph below. v(t) liters/sec Show all analyses that leads to your solutions. a) What is the velocity of the water at t = 3? b) Is the speed of the water going into the container greatest at t = 3 or t = 9? Justify. c) Find the displacement of the water on 1,8 d) Find the displacement of the water on 5,10 e) Find the total amount of water poured into and drained from the container on 0,10 f) Given that container held 3 liters of water at t = 5. Find the amount of water in the container at each time. t = 8 t = 10 t = 1 t = 0 g) Write a function L(t) to determine the amount of water in the container at any given time t. L(t) = h) At what time does the container have the most liters of water on the 10 minute interval? Justify. i) Within the 10 minute time interval, did the container hold 14 liters of water? Justify.
8 14) The rate of consumption of fuel in Arkansas after 1990 (in billions of barrels per year) is modeled by the function C 5.01e 15. t a) Find the rate of fuel consumption in 199. b) Find the total amount consumed from January 1, 1990 to January 1, c) Find the total amount consumed from January 1, 1993 to January 1, d) Find the average rate of consumption of fuel in Arkansas from January 1, 1993 to January 1, ) y 1 x 0 y cos x x a) Find the enclosed area. b) Find the volume generated by revolving the region about the x-axis. c) Find the volume generated by revolving the region about the y-axis. d) Find the volume generated by revolving the region about x 3. (set-up only, do not evaluate) e) Find the volume generated by revolving the region about y 1. (set-up only, do not evaluate) f) Find the volume generated by semi-circle cross sections perpendicular to the x-axis. (set-up only, do not evaluate)
9 16) y x y 0 x y a) Find the enclosed area. b) Find the volume generated by revolving the region about the x-axis. c) Find the volume generated by revolving the region about the y-axis. d) Find the volume generated by revolving the region about x 1. (set-up only, do not evaluate) e) Find the volume generated by revolving the region about y. (set-up only, do not evaluate) f) Find the volume generated by rectangle cross sections with the height twice the base perpendicular to the y-axis. (set-up only, do not evaluate) g) Refer to part a. Write an equation to find the value of k that divides the area into equal parts. (set-up only, do not evaluate)
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