Name AP Calculus AB Applications of the Derivative Test Review 9 92700 dr = 500 1. A ball is projected straight up from a 256-foot building with an initial velocity of 96ft/sec. Find the following without the use of a calculator: a. The position, velocity and acceleration functions b. When the ball reaches its maximum height. c. How high the ball goes. d. When the ball hits the ground. e. The impact velocity. f. Total distance traveled by the ball in the first 5 seconds. g. Displacement of the ball in the first 5 seconds. h. The velocity when the ball is 144 feet in the air. 3 7 1993 (30) . dr 500 1694 cuoio crimin mr 2 1931 f t = ou - 47712h Г- СО 14 h h=19 2. A spherical balloon is being filled with gas at a rate of 500 cubic cm / min. How fast is the radius of the balloon increasing when the radius is: A) 30 cm? B) 60 cm? dv=500 q TE at 3. A conical tank is being drained at a rate of 5 ft3 / sec. The tank has a base diameter of 20 feet and a depth of 14 feet. At what rate is the water level decreasing when the tank is only half full? -, 24 Str (500 rin ด d_= -5 Find the following for #'s 4 and 5: dt a. Coordinates of any relative extrema. b. Coordinates of the absolute extrema. c. Any intervals where f(x) is increasing, d. Any intervals where f(x) is decreasing. e. Any points of inflection. f. Any intervals where f(x) is concave up. g. Any intervals where f(x) is concave down. h. Sketch a possible graph for f(x). 144=10h -5 = 3 27- 20th 191 Toon 4. Given: f(x)= x? – 3.r? – 24x +2 on the closed interval (-4, 8). 5. Given: f(x) = (x+2)? on the closed interval [-3, 6). 6. Find the number guaranteed by the mean value theorem for the function f(x) = {2 on the closed interval (-1, 4]. I. II. f(x) is every f(-2)=3 and f(5)=1 antly four zeros in its don 7. The following graph is the graph of f'(x). The function f(x) has these properties: I. f(x) has a domain of (-2,5]. II. f(x) is everywhere differentiable on its domain. X=0 III. f(-2)= 3 and f(5) = 1 X=-1 X=1 IV. f(x) has exactly four zeros in its domain. X=4 N (a) For what x coordinates if any does f(x) have any relative extrema? Classify as a max or min and justify. X=3 (b) For what x-coordinates if any does f(x) have any points of inflection? Justify. (c) Sketch a possible graph of f(x) that fits the found information. 8. A plant storage area for a local nursery is to be constructed. The accountant for the nursery has stated that only $15,000 could be spent on the job. The storage area is to be rectangular in shape. One set of parallel sides must be constructed with a heavier grade of fencing in order to withstand the high traffic areas. The regular fence costs $4.00 per linear foot and the heavier grade fence costs $6.00 per linear foot. What are the dimensions of the storage area if the area is to be maximized? 9. A particle moves in a horizontal path such that its position is given by the function s(t)= t3 – 10t+27t-18 where S(t) is measured in meters and t is measured in seconds. Find the following: a) What are the velocity and acceleration functions in terms of t? b) When is the particle at rest? c) What is the position(s) of the particle when at rest? d) What is the position and velocity of the particle when the acceleration is -8 m/s? e) Sketch a motion schematic for the particle. Make sure to label position and velocity at each critical time. f) What was the total distance traveled in the first 5 seconds? g) Find the displacement in the first 5 seconds. h) What is the total distance traveled in the first 8 seconds? i) Find the displacement in the first 8 seconds. j) When is the particle moving to the right and left? Use interval notation for your answers. k) What is the average velocity of the particle in the first 9 seconds? 1) What is the average velocity of the particle between the 3rd and 7th second? 10. The following is a graph of f'(x). The function f has a domain of [-4, 3). a) Over what x values is fincreasing/decreasing? Justify your answer. b) For what x values does the graph of fhave any relative extrema? Classify as a max or c) d) e) min and justify your answer. Over what intervals is f concave up/down? Justify your answer. For what x values does f have points of inflection? Justify your answer. Sketch a possible graph of f(x) or 11. A particle moves in a horizontal path such that its position at any time t in seconds is given by the function s(t)=ť - 51-sin(2t)+2 where s(r) is measured in feet. Find the following: a) What are the velocity and acceleration functions? b) When is the particle at rest? c) What is the position of the particle when it is at rest? d) What is the position of the particle when the velocity is 3 ft/s? e) Over what intervals is the particle moving right or left? f) Find the displacement in the first 7 seconds. g) Find the total distance traveled in the first 7 seconds. h) What is the average velocity of the particle in the first 8 seconds?
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