Final Exam Review AP Calculus AB Find the slope of the curve at the given point P and an equation of the tangent line at P. 1) y = x2 + 11x - 15, P(1, -3) Use the graph to evaluate the limit. 2) lim x -1 f(x) Use the graph to estimate the specified limit. 3) Find lim f(x) and x (-1)- x (-1)+ lim f(x) Find numbers a and b so that f is continuous at every point. 4) f(x) = -12, ax + b, 6, x < -1-1 x 2 x > 2 Give an appropriate answer. 5) Let lim f(x) = 8 and x -8 lim g(x) = 3. Find x -8 lim [f(x) - g(x)]. x -8 1
Find the limit, if it exists. x 6) lim x -1 3x + 2 7) lim (6x2-9x - 10) x 10 8) lim x 15 1 x - 15 x3-6x + 8 9) lim x 0 x - 2 10) lim x 1 x4-1 x - 1 11) lim x 3-5 x2-9 12) lim x 7 x - 1-14x2-3x + 17 13) lim x - -6x2 + 8x + 13 Evaluate lim h 0 f(x0 + h) - f(x0) for the given h x0 and function f. 14) f(x) = 5x2-3 for x0 = 4 Use the table of values of f to estimate the limit. 15) Let f(x) = x - 4 x - 2, find x 4 lim f(x). x 3.9 3.99 3.999 4.001 4.01 4.1 f(x) Find an equation for the tangent to the curve at the given point. 16) y = x - x2, (3, -6) Find the slope of the curve at the indicated point. 17) y= x2 + 9x, x = -5 2
Calculate the derivative of the function. Then find the value of the derivative as specified. 18) f(x) = 8 x ; f (-1) Find the indicated derivative. 19) dp dq if p = 1 q + 2 Find the second derivative. 20) y = 4x2 + 7x - 7 Find the derivative of the function. 21) y = x 3 x - 1 Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 22) u(1) = 3, u (1) = -6, v(1) = 6, v (1) = -3. d (uv) at x = 1 dx 23) Find an equation for the tangent to the curve y = 27 x2 + 2 at the point (1, 9). The function s = f(t) gives the position of a body moving on a coordinate line, with s in meters and t in seconds. 24) s = 2t - t2, 0 t 2 Find the body's speed and acceleration at the end of the time interval. The figure shows the velocity v or position s of a body moving along a coordinate line as a function of time t. Use the figure to answer the question. 25) v (ft/sec) t (sec) When is the body moving backward? 3
26) At time t, the position of a body moving along the s-axis is s = t3-15t2 + 48t m. Find the total distance traveled by the body from t = 0 to t = 3. Find the derivative. 27) s = t7 cos t - 7t sin t - 7 cos t Find the indicated derivative. 28) Find y if y = 8x sin x. Given y = f(u) and u = g(x), find dy/dx. 29) y = u2, u = 6x - 1 Find the derivative of the function. 30) q = 15r - r7 Find dy/dt. 31) y = t5(t4 + 8) 5 32) The position of a particle moving along a coordinate line is s = 4 + 12t, with s in meters and t in seconds. Find the particle's velocity at t = 1 sec. Use implicit differentiation to find dy/dx. 33) 2xy - y2 = 1 34) x3 + 3x2y + y3 = 8 35) y x + 1 = 4 Use implicit differentiation to find dy/dx and d2y/dx2. 36) x2 + y2 = 3 37) Find the slope of the curve xy3 - x5y2 = -4 at (-1, 2). 38) Suppose that the radius r and volume V = 4 3 r 3 of a sphere are differentiable functions of t. Write an equation that relates dv/dt to dr/dt. 39) If xy2 = 4 and dx/dt = -5, then what is dy/dt when x = 4 and y = 1? 4
40) Let f(x) = (x - 1)2/3 (a) Does f (1) exist? (b) Show that the only local extreme value of f occurs at x = 1. (c) Does the result of (b) contradict the Extreme Value Theorem? (d) Repeat parts (a) and (b) for f(x) = (x - c)2/3. Give reasons for your answers. Round your answer, if appropriate. 41) A man 6 ft tall walks at a rate of 7 ft/sec away from a lamppost that is 20 ft high. At what rate is the length of his shadow changing when he is 35 ft away from the lamppost? (Do not round your answer) 42) The volume of a sphere is increasing at a rate of 7 cm3/sec. Find the rate of change of its surface area when its volume is 32 cm3. (Do not round your answer.) 3 43) How close does the curve y = x come to the point can avoid square roots.) 4, 0? (Hint: If you minimize the square of the distance, you 7 Find the value or values of c that satisfy the equation the function and interval. 44) f(x) = x2 + 2x + 2, [-2, 1] f(b) - f(a) b - a = f (c) in the conclusion of the Mean Value Theorem for Use the graph of the function f(x) to locate the local extrema and identify the intervals where the function is concave up and concave down. 45) 5
Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. 46) y = x 2 x2 + 7 47) Find the inflection points (if any) on the graph of the function and the coordinates of the points on the graph where the function has a local maximum or local minimum value. Then graph the function in a region large enough to show all these points simultaneously. Add to your picture the graphs of the function's first and second derivatives. y = x5-4x4-200 48) The accompanying figure shows a portion of the graph of a function that is twice-differentiable at all x except at x = p. At each of the labeled points, classify y and y as positive, negative, or zero. 6
49) The graph below shows the position s = f(t) of a body moving back and forth on a coordinate line. (a) When is the body moving away from the origin? Toward the origin? At approximately what times is the (b) velocity equal to zero? (c) Acceleration equal to zero? (d) When is the acceleration positive? Negative? 50) If g(x) = 2f(x) + 3, find g (4) given that f (4) = 5. 51) Does the curve y = x ever have a negative slope? If so, where? Give reasons for your answer. Suppose u and v are differentiable functions of x. Use the given values of the functions and their derivatives to find the value of the indicated derivative. 52) u(2) = 6, u (2) = 4, v(2) = -3, v (2) = -5. d u dx v at x = 2 53) The curve y = ax2 + bx + c passes through the point (2, 10) and is tangent to the line y = 3x at the origin. Find a, b, and c. Find the linearization L(x) of f(x) at x = a. 54) f(x) = 3 x, a = 8 Write a differential formula that estimates the given change in surface area. 55) The change in the surface area S = 4 r2 of a sphere when the radius changes from r0 to r0 + dx 56) A = r2, where r is the radius, in centimeters. By approximately how much does the area of a circle decrease when the radius is decreased from 3.0 cm to 2.9 cm? (Use 3.14 for.) Find the derivative of y with respect to x. 57) y = ln 9x2 58) y = ln 1 + x x5 7
Find the derivative of y with respect to the independent variable. 59) y = (ln 3 ) 60) y =log4 x2 6 x+1 Find the derivative of y with respect to x, t, or, as appropriate. 61) y = e7-10x 62) y = (x2-2x + 4) ex Use logarithmic differentiation to find the derivative of y. x 63) y = x + 7 64) y = x(x + 8)(x + 3) Use l'hopital's Rule to evaluate the limit. 65) lim x -2 x2-4 x + 2 66) Which one is correct, and which one is wrong? Give reasons for your answers. x + 2 (a) lim x - 2 x2-4 = x -2 lim 1 2x = - 1 4 (b) lim x - 2 x + 2 x2-4 = 0-8 = 0 Find the derivative of y with respect to x. 67) y = 3 sin-1 (5x4) 68) y = tan-1 3x 69) y = -csc-1 6x + 1 9 Find the most general antiderivative. 70) ( t - 6 t) dt Evaluate the integral using the given substitution. 71) 28s3 ds 10 - s4, u = 10 - s 4 8
72) csc2 5 cot 5 d, u = cot 5 Evaluate the integral. x dx 73) (7x2 + 3)5 74) 10x 2 4 8 + 2x3 dx 75) sin t (8 + cos t)6 dt 9