AB CALCULUS Page 1 of 6 NAME DATE 1. Evaluate each it: AB CALCULUS Show all work on separate paper. x 3 x 9 x 5x + 6 x 0 5x 3sin x x 7 x 3 x 3 5x (d) 5x 3 x +1 x x 4 (e) x x 9 3x 4 6x (f) h 0 sin( π 6 + h) sin π 6 h (g) x x + x 8 x (h) x a x a x a. Which of the following functions is/are continuous at x = 1? f (x) = ex 1 x 1 g(x) = ln(x 1) h(x) = sin x x 3. Find dy dx for each of the following functions: y = ln(5x) 9 x y = 3x 4 x + y = (3x + 5) cosx (d) y = x 3 + 7 (e) 4 y = sin 3 (6x) + x (f) y = f [g(x)] (g) y = t sint (h) y = sin 1 (3x) + tan x 1 x 3 (i) y = 5(x 3 +1) + 5Arc tan(x) 4. Let g(x) = f(x + 1). Find each of the following: g (x) g () g (x 3 d ) (d) dx g(x3 ) [ ] 1
AB CALCULUS Page of 6 5. Use the graph of f below to determine each of the following: 1 - -1 1 3 4 Where is f continuous? Where is f differentiable? 6. Find the value for a that makes the following function: continuous at x = differentiable at x = f (x) = x 3, x < ax, x 7. Suppose that f is continuous on [-5, 5] and has the following properties: f(0) =, f(3) = -, f(5) = 1, f > 0 on [-5, 0) and (1.5, 5], f is decreasing when x < 3, and f is increasing when x > 3. Sketch a possible graph of f. 8. Complete the following problem without the use of a calculator. Given f (x) = x 3 9x +1x, Find the critical value(s) of f Where is f increasing? decreasing? Where does f have a relative maximum? relative minimum? (d) What is the absolute maximum value of f on [0, 3]? (e) Where does f have an inflection point? (f) Where is f concave up? concave down? 9. Selected values of a function h where h (x) > 0 on [0, 4] are given in the table below. Use these values to estimate the value of h (3). x 0 1 3 4 h(x).8 1 1.4.8
AB CALCULUS Page 3 of 6 10. Given f (x) = ln(3x +1) Write the linearization of f at x = 1. Approximate f(1.) using your answer in. Is your approximation too large or too small? Explain. 11. Let x + xy + y = 7. Find an expression for dy dx. Find any points on the curve where the tangent is horizontal. Find any points on the curve where the tangent is vertical. 1. Given y = 4x 3 x + 5, find the: average rate of change on [-1, ] instantaneous rate of change at x = 1 13. Find d y for the following functions: dx y = x 3 3cos x + ln x y = 4x 3 x + y = 5 14. The radius of a sphere is increasing at a rate of ft/min. How fast is the volume changing when the radius is 3 feet? (V = 4 3 πr3 ) 15. What dimensions would maximize the area of a rectangle with a perimeter of 4 feet? 16. Let f (x) = 3x 3 5x + x +. Graph f on [-1, ]. Use the graph of f to estimate the value(s) of c guaranteed by the Mean Value Theorem on the interval [-1, ]. Find the value(s) of c. 3
AB CALCULUS Page 4 of 6 17. Given the graph of f, the derivative of f, on [-4, 4], find the following: 40 30 0 10 0-10 -0-30 -4-3 - -1 0 1 3 4 (d) (e) answer. On what interval(s) is f increasing? Decreasing? Why? Where does f have critical points? Where does f have relative maxima? Relative minima? Justify your Where does f have inflections points? On what interval(s) is f concave up? Concave down? 18. The Amazing Fortini is walking along a tightrope 70 feet in the air. His velocity, v(t), at time t, 0 t 1, is given in the graph below. 3 1-1 - -3-4 1 3 5 7 9 11 When does the tightrope walker change direction? When is his speed the greatest? What is his acceleration at 6 seconds? 4
AB CALCULUS Page 5 of 6 19. Chemicals from a storage tank are leaking into a pond. The rate of flow is measured at intervals and is recorded in the table below where t is measured in hours and R(t) in gallons per hour: T 0 4 6 8 10 1 14 16 R(t) 40 38 36 30 6 18 8 6 3 What is the average rate of flow during the first 8 seconds? Estimate the instantaneous flow at 4 seconds. 0. The figure below shows the graphs of f, f,and f. Identify each curve and justify your answer. a b c 1. Find the equation of the line tangent to the function y = 3x 10x + 5 at x = 3.. Find the equation of the line normal to the function y = cos 5x at x = π. 3. Given y = 4x 3, find the minimum value of xy. 5
AB CALCULUS Page 6 of 6 4. The function f is continuous and differentiable on [-5, 5] such that f( 5) = 10 and f(5) = 10. Decide which of the following statements are always true, sometimes true, or false. Justify your answer for each statement. f(0) = 0 f is an even function f = 0 for some c between 10 and 10 (d) f > 0 for all x between 5 and 5 (e) 10 f (x) 10 for all values of x between 5 and 5 (f) f = 0 for at least one c between 5 and 5 (g) f = for at least one c between 5 and 5 (h) f = 9 for at least one c between 5 and 5 5. Complete the following problem without the use of a calculator. The position of a particle on the x-axis is given by x(t) = t 3 6t + 9t +. Write an expression for the velocity of the particle. When is the particle at rest? Moving left? Moving right? What is the speed of the particle at t =? (d) Write an expression for the acceleration of the particle. (e) What is the minimum velocity of the particle on [0,3] 6. Find the antiderivative of: y = sec x tan x y = 7cos x 3 x 7. Evaluate each integral: 3sec x dx (x 4 4x + 7) dx (x 3 6 x ) dx 3 6