Test Your Strength AB Calculus: Section A 35 questions No calculator allowed Directions: Use the answer sheet to input your answers. Caps or lower case are OK. If you need to guess, put an X in the guess column. Take roughly an hour and a half on section A. ( ) x sin x cos x +1 1. Find lim x 0 e x x 1 A. 0 B. 1 C. D. nonexistent. The graph to the right is f x describes the graph? ( ) = e sin x+0.0001 0.01 x +1. Which of the following A. continuous only B. differentiable only C. both continuous and D. neither continuous differentiable nor differentiable 3. Find the equation of the line tangent to y = ln( x + x +1) at x = 1. A. y = x B. y = x + ln3 C. y = x 3 + ln3 1 3 D. y = x 1+ ln3 017 www.mastermathmentor.com - 1 - Test Your Strength AB Illegal to post on Internet
4. If f x x 3 ( ) = x + 4 A. 3 x + 4 x 3 C. 3x x + 4 x 3 3, find f ( x). ( ) ( x 3x 4) B. 6 x + 4 ( x 3) 3 ( ) x 4 D. 6 x + 4 ( )( x +1) ( x 3) 4 3x ln3 5. Find 3 dx ( 3x ln3) 3 A. + C B. 81 ( 3x ln3) 3 + C C. 7 ( 3x ln3) 3 + C D. 9 ( 3x ln3) 3 + C 3 6. Edna works in the deli at a supermarket. She is asked to slice cheese. The rate R that she slices cheese is given at various times by the table to the right. Which of the following statements must be correct? t ( sec) 0 10 0 30 ( ) 0 3 3 R slices/sec I. Between t = 0 and t = 30, the average rate of change of R is 1 15 II. The instantaneous rate of change of R at t = 15 is 0. slices sec. A. I only B. II only C. Both I and II D. Neither I nor II 017 www.mastermathmentor.com - - Test Your Strength AB Illegal to post on Internet
7. If x + x y 4y = 9, find the behavior of the curve at ( 3, ). A. Increasing, concave up B. Increasing, concave down C. Decreasing, concave up D. Decreasing, concave down 8. The expression 5 + 5 1 10 + 5 10 39 10 ( +... + 5 ) is a Riemann sum approximation for 4 A. 5 x dx B. 10 5 x dx C. 4 40 5 x 10 dx D. 1 10 0 40 0 5 x 10 dx 9. If f ( x) = tan 1 x, find lim h 0 1 f + h f 1 h. A. 0 B. 1 C. D. nonexistent 017 www.mastermathmentor.com - 3 - Test Your Strength AB Illegal to post on Internet
10. The functions f and g are differentiable and f ( g ( x) ) = tan x for all x. If f π f 3 = 1, what is the value of g π 4? A. 0 B. 1 C. D. 4 π 4 = 0, g π 4 = π 3 and 11. A melting spherical block of ice has its surface area changing at the rate of 4 inch /sec at the time the radius is 3 inches. How fast is the volume of the ice changing at that moment? S = 4πr, V = 4 3 πr3 A. 36 in3 sec B. 36π in3 sec C. 36π in 3 sec D. 36 π in 3 sec 1. A particle moving along the x-axis has velocity v t by the graph to the right and v t values of t is the particle changing directions? ( ) for 0 t 3 given ( ) = ( t 3) ( t 5) for 3 < t 6. For what A. t = and t = 4 only B. t = 3 only C. t = 1, 3, and 5 only D. never 017 www.mastermathmentor.com - 4 - Test Your Strength AB Illegal to post on Internet
13. The graph to the right is y = f could be the graph of f x ( )? ( x), defined on [, ]. Which of the following A. B. C. D. 14. If the graph of f ( x) is on the right and F ( x) = f t the following has the largest value? A. F C. F 4 ( ) B. F ( 0) ( ) D. F ( 5) x ( )dt, which of 017 www.mastermathmentor.com - 5 - Test Your Strength AB Illegal to post on Internet
15. e π ( ) sin ln x dx = x 1 A. 0 B. C. π D. π 1 16. The function g is differentiable for all real numbers. The table below gives values of the function and its first derivatives at selected values of x. If g 1 is the inverse function of g, what is the equation for the line tangent to the graph of y = g 1 ( x) at x = 5? x ( ) g ( x) g x 5 5 3 5 1 ( ) A. y = x B. y 5 = 3 x + 5 C. y + 5 = 1 ( 3 x 5 ) D. y 5 = 1 ( 3 x + 5 ) 17. The graph of x 4 = 4( x y ) has how many horizontal tangents? A. 1 B. C. 4 D. 5 017 www.mastermathmentor.com - 6 - Test Your Strength AB Illegal to post on Internet
18. If the position of a particle on a vertical line is determined by s( t) = 4t 3 3cost, 0 t π, what is the maximum acceleration of the particle? A. 36π 1 B. 6π +1 C. 4π +1 D. 1π 1 19. Consider the differential equation dy dx = x + 4y 1. If y = f x at what point does f have a relative maximum? ( ) is the solution to the differential equation, ( ) B. 0, 1 4 A. 3, C. (, 1) D. ( 5, 6) 0. The graph of f, the derivative of f, is shown to the right for 0 x 6. The areas of the regions between the graph of f and the axis are 6, 10 and 3 respectively. If f 0 ( ) = k, what is the difference between the maximum and minimum values of f on the interval 0 x 6? A. 10 B. k + 10 C. 1 D. k + 1 017 www.mastermathmentor.com - 7 - Test Your Strength AB Illegal to post on Internet
1. If dy dx = ( x + x +1) ( x x +1) and y = 1 when x = 0. Find y when x = 1. A. 1 B. 8 15 C. D. 3 15. On Saturday morning, a post office is open for only hours. Prior to its opening, people line up outside waiting for the doors to open. The number of people in line at any time t, 0 t, is modeled by a differentiable function P. Values of P t number of times at which P ( t) = 0? ( ) are shown in the table below. For 0 t, what is the fewest Time 9:00 9:15 9:30 9:45 10:00 10:15 10:30 10:45 11:00 P t ( ) people 4 18 10 10 14 19 6 0 4 A. 1 B. C. 3 D. 4 3. Find 1 0 e x e x +1 dx A. 1 ln e +1 B. ln e +1 ( ) ln C. 1 ( 1+ e) D. 1 ln ( e +1) 017 www.mastermathmentor.com - 8 - Test Your Strength AB Illegal to post on Internet
4. The area between y = x and y = x on [0, ] is rotated about the x-axis. Find the volume of the solid. A. 64π 15 B. 16π 15 C. 8π D. 4π 3 e, find the difference between f 1 x and the linear approximation to f 1 line to f at x = 1. 5. If f ( x) = x using the tangent A. 3 e 3 e B. C. 1 e e e D. 3 e 6. Let P be the population of some geographical area. If the population of the area changes over time based on the following criteria, which of them signals exponential growth of the population? I. The change of population is directly proportional to itself. II. The change of population is inversely proportional to itself. III The change of population is directly proportional to e Population. A. I only B. I and II only C. II and III only D. I and III only 017 www.mastermathmentor.com - 9 - Test Your Strength AB Illegal to post on Internet
7. A particle moves along a horizontal line on the interval t [0, 5] with acceleration as defined in the graph to the right, made up of straight lines. If v 0 ( ) = 0, for what values of t does the particle change direction? A. t = 1 and 3 only B. t = and 4 only C. t = 1,, 3, and 4 D. never 8. A rectangle in inscribed within the area enclosed by the curve y = x, x = 4, and the x-axis as shown in the figure to the right. What is the maximum possible ratio of the area of this rectangle and the shaded area? 1 A. 3 C. 1 B. 1 3 D. 3 017 www.mastermathmentor.com - 10 - Test Your Strength AB Illegal to post on Internet
9. The graph of the function f, shown to the right has a horizontal tangent at x = 1, 3, 5, and 7. Let g be the function defined by ( ) = f t g x have? x 3 ( )dt. How many inflection points does the graph of g A. 1 B. C. 3 D.4 30. A piecewise function is defined as f x interval [0, 5].. Find the average value of f on the x 4 + 4 x 1, < x 5 ( ) = x,0 x A. 95 9 B. 91 15 C. 67 15 D. 79 18 017 www.mastermathmentor.com - 11 - Test Your Strength AB Illegal to post on Internet
31. The graph of f ( x) is on the graph to the right. Which of the following statements are true? I. lim x 0 f x II. lim x f x III. lim x ( ) exists ( ) = f ( x) exists A. I only B. II only C. III only D. I and II only 3. The function f is continuous on the closed interval [0, 10] and has the values given in the table to the right. The function is constructed such that the trapezoidal approximation for 10 f x dx is between 10 and 0 ( ) 0 inclusive. What is the difference between the maximum and minimum values of k? A. B. 4 C. 6 D. 8 x 0 5 7 10 f ( x) 4 6 k 8 4 017 www.mastermathmentor.com - 1 - Test Your Strength AB Illegal to post on Internet
33. ln( tan x) sin xcosx dx = A. 1 4 ln ( sec x) + C B. 1 ln ( sec x) + C C. ( ) ln tan x 4 ( ) + C D. ln tan x + C 34. The equation ( x + y ) = ( x 3 + y ) 3 graphs the bean curve. Approximate the value of y at x = 1 the linear approximation to the curve at the point (1, 1). by using A. 1 B. 3 C. 1 D. 3 017 www.mastermathmentor.com - 13 - Test Your Strength AB Illegal to post on Internet
35. A slope field is shown in the figure to the right. What could be the differential equation that generated it? A. dy dx = x x + y C. dy dx = x + x + y ( ) B. dy dx = y + x y D. dy dx = y + x + y 017 www.mastermathmentor.com - 14 - Test Your Strength AB Illegal to post on Internet