2015 AP calculus exam

If y= sin x cos x, then at x= pi/3, dy/dx= A. -1/2 B. -1/4 C. 1/4 D. 1/2 E. 1

A. -1/2

The top of a 15 foot long ladder rests against a vertical wall with the bottom of the ladder on level ground, as shown above. The ladder is sliding down the wall at a constant rate of 2 feet per second. At what rate, in radians per second, is the acute angle between the bottom of the ladder and the ground changing at the instant the bottom of the ladder is 9 feet from the base of the wall? A. -2/9 B. -1/6 C. -2/25 D. 2/25 E. 1/9

A. -2/9

What is the average rate of change of y=cos(2x) on the interval [0, pi/2]? A. -4/pi B. -1 C. 0 D. Square root of 2/2 E. 4/pi

A. -4/pi

If f(x)=square root of x +3/square root of x, then f'4=

A. 1/16

The first derivative of the function f is given by f'(x)= sin(x^2). At which of the following values of x does f have a local minimum? A. 2.507 B. 2.171 C. 1.772 D. 1.253 E. 0

A. 2.507

The values of a continuous function f for selected values of x are given in the table above. What is the value of the left Riemann sum approximation to |50 to 0 f(x) dx using the subintervals [0,25], [25,30], and [30,50]? A. 290 B. 360 C. 380 D. 380 E. 430

A. 290

The table above gives values of the differentiable functions f and g and their derivatives at x=0. If h(x)= f(xX)/g(x), what is the value of h'(0)? A. 8-3pi/4 B. 3pi-8/4 C. 4/pi D. 2-3pi/2 E. 8+3pi/4

A. 8-3pi/4

If lim f(x)= f(a), then which of the following statement about f must be true? A. f is continuous at x= a. B. f is differentiable at x= a C. For all values of x, f(x)= f(a) D. The line y= f(a) is tangent to the graph of f at x= a E. The line x= a is a vertical asymptote of the graph of f

A. f is continuous at x= a

Let f be the function given by f(x)= kx/x^2+1, where k is a constant. For what values of k, if any, is f strictly decreasing on the interval (-1, 1)? A. k<0 B. k= 0 C. k>0 D. k>1 only E. There are no such values of k

A. k<0

The vertical line x= 2 is an asymptote for the graph of the function f. Which of the following statements must be false? A. lim f(x) as x approaches 2= 0 B. lim f(x) as x approaches 2= -infinity C. lim f(x) as x approaches 2= infinity D. lim f(x) as x approaches infinity=2 E. lim f(x) as x approaches infinity= infinity

A. lim f(x) as x approaches 2= 0

The table above gives selected values for a differentiable and decreasing function f and its derivative. If f approaching 1 from the left of the inverse function of f, what is the value of the derivative of f approaching 1 from the left (2)? A. -80 B. -1/8 C. -1/80 D. 1/80 E. 1/8

B. -1/8

Let f be a differentiable function with selected values given in the table above. What is the average rate of change of f over the closes interval 0 less than or equal to x less than or equal to 10? A. -6 B. -5/2 C. -2 D. -2/5 E. 2/5

B. -5/2

A particle moves along the x-axis so that its position at time t>0 is given by x(t) and dx/dt= -10t^4 + 9t^2 + 8t. The acceleration of the particle is zero when t= A. 0.387 B. 0.831 C. 1.243 D. 1.647 E. 8.094

B. 0.831

lim sin(pi/3+h) - sin(pi/3)/ h as h approaches 0 is A. 0 B. 1/2 C. 1 D. Square root of 3/2 E. Nonexistent

B. 1/2

What is the area of the region enclosed by the graphs of y= square root of 4x-x^2 and y= x/2? A. 1.707 B. 2.829 C. 5.389 D. 8.886 E. 21.447

B. 2.829

If dy/dt= -10e^-t/2 and y(0)= 20, what is the value of y(6)? A. 20e^-6 B. 20e^-3 C. 20e^-2 D. 10e^-3 E. 5e^-3

B. 20e^-3

Which of the following is the anti derivative of 3sec^2x+2? A. 3tan x B. 3tan x+2x C. 3sec x+2x D. sec^3x+2x E. 6sec^2xtan x

B. 3tan x+2x

The base of a solid is the region bounded by the x-axis and the graph of y= square root of 1-x^2. For the solid, each cross section perpendicular to the x-axis is a square. What is the volume of the solid? A. 2/3 B. 4/3 C. 2 D. 2 pi/3 E. 4 pi/3

B. 4/3

Derive (5e^2x+1/x)dx

B. 5/2e^2x+2/x^2+c

f(x)= {x^2sin(pi x) for x<2 and x^2+cx-18 for x greater than or equal to 2. Let f be the function defined above, where c is a constant. For what value of c, if any, is f continuous at x=2? A. 2 B. 7 C. 9 D. 4 pi - 4 E. There is no such value of c

B. 7

The graph of the function h is shown in the figure above. Of the following, which has the greatest value? A. Average value of h over (-3,2) B. Average rate of change of h over (-3, 2) C. {2 to -3 h(x) dx D. {0 to -3 h(x) dx E. h'(0)

B. Average rate of change of h over (-3, 2)

The function P(t) models the population of the world, in billions of people, where t is the number of years since January 1, 2010. Which of the following is the best interpretation of the statement P'(1)= 0.076? A. On February 1, 2010, the population of the world was increasing at a rate of 0.076 billion people per year B. On January 1, 2011, the population of the world was increasing at a rate of 0.076 billion people per year C. On January 1, 2011, the population of the world was 0.076 billion people

B. On January 1, 2011, the population of the world was increasing at a rate of 0.076 billion people per year

If f"(x)= x(x+2)^2, then the graph of f is concave up for A. x<0 B. x>0 C. -2<x<0 D. x<-2 and x>0 E. All real numbers

B. x>0

Which of the following is an equation for the line tangent to the graph of y= 3- | x to -1 (e^-t^3) dt at the point where x= -1? A. y-3= -3e(x+1) B. y-3= -e(x+1) C. y-3= 0 D. y-3= 1/e(x+1) E. y-3= 3e(x+1)

B. y-3= -e(x+1)

Shown above is a slope field for the differential equation dy/dx=y^2(4-y^2). If y= g(x) is the solution to the differential equation with the initial condition g(-2)= -1, then lim g(x) is A. - infinity B. -2 C. 0 D. 2 E. 3

C. 0

The function f is defined by f(x)= 2x^3-4x^2+1. The application of the Mean Value Theorem to f on the interval 1 less than or equal to x less than or equal to 3 guarantees the existence of a value c, where 1<c< 3, such that f'(c)= A. 0 B. 9 C. 10 D. 14 E. 16

C. 10

Let f be the function given by f(x)= (x-4)(2x-3)/(x-1)^2. If the line y= b is a horizontal asymptote to the graph of f, then b= A. 0 B. 1 C. 2 D. 3 E. 4

C. 2

An object moves along a straight line so that at any time t greater than or equal to 0 it's velocity is given by v(t)= 2cos(3t). What is the distance traveled by the object from t=0 to the first time that it stops? A. 0 B. Pi/6 C. 2/3 D. Pi/3 E. 4/3

C. 2/3

Let f be the function with derivative defined by f'(x)= x^3-4x. At which of the following values of x does the graph of f have a point of inflection? A. 0 B. 2/3 C. 2/square root of 3 D. 4/3 E. 2

C. 2/square root of 3

The velocity v, in meters per second, of a certain type of wave is given by v(h)= 3 square root of h, where h is the depth, in meters, of the water through which the wave moves. What is the rate of change, in meters per second per meter, of the velocity of the wave with respect to the depth of the water, when the depth is 2 meters? A. -3/4 square root of 2 B. -3/8 square root of 2 C. 3/2 square root of 2 D. 3/ square root of 2 E. 4/ square root of 2

C. 3/2 square root of 2

Lim x^2-9/x^2-2x-15 as x approaches -3 is A. 0 B. 3/5 C. 3/4 D. 1 E. Nonexistent

C. 3/4

f(x)= {3x-2 if x<1 and {ln (3x-2) if x greater than or equal to 1. Let f be the function defined above. Which of the following statements about f are true? I. lim f(x) as x approaches 1 from the left= lim f(x) as x approaches 1 coming from the right II. lim f'(x) as x approaches 1 from the left= lim f'(x) as x approaches 1 coming from the right III. f is differentiable at x=1 A. None B. I only C. II only D. II and III only E. I, II, and III

C. II only

The graph of f', the derivative of f, is shown above. The line tangent to the graph of f' at x=0 is vertical, and f' is not differentiable at x=2. Which of the following statements is true? A. f' does not exist at x=2 B. f is decreasing on the interval (2,4) C. The graph of f has a point of inflection at x=2 D. The graph of f has a point of inflection at x= 0 E. f has a local maximum at x=0

C. The graph of f has a point of inflection at x=2

The graph of y= f(x) on the closed interval [0,4] is shown above. Which of the following could be the graph of y=f'(x)?

D

The rate at which motor oil is leaking from an automobile is modeled by the function L defined by L(t)= 1+ sin(t^2) for time greater than or equal to 0. L(t) is measured in liters per hour, and t us measures in hours. How much oil leaks out of the automobile during the first half hour? A. 1.998 liters B. 1.247 liters C. 0.969 liters D. 0.541 liters E. 0.531 liters

D. 0.541 liters

The temperature F, in degrees Fahrenheit, of a cup of coffee t minutes after it is poured is given by F(t)= 72 + 118e^-0.093t. To the nearest degree, what is the average temperature of the coffee between t= 0 and t= 10 minutes? A. 93 degrees B. 119 degrees C. 146 degrees D. 149 degrees E. 154 degrees

D. 149 degrees

The function f is continuous on the closed interval (1,7). If 7 to 1 f(x) dx= 42 and 3 to 7 f(x) dx= -32, then 3 to 1 2f(x) dx= A. -148 B. 10 C. 12 D. 20 E. 148

D. 20

If f(x)=x^2-4 and g is a differentiable function of x, what is the derivative of f(g(x))? A. 2g(x) B. 2g'(x) C. 2xg'(x) D. 2g(x)g'(x) E. 2g(x)-4

D. 2g(x)g'(x)

If f'(x)= cos(x^2) and f(3)= 7, then f(2)= A. 0.241 B. 5.831 C. 6.416 D. 6.759 E. 7. 241

D. 6.759

Which of he following is the solution to the differential equation dy/dx= 5y^2 with the initial condition y(0)= 3? A. y= square root of 9e^5x B. y= square root of 1/9e^5x C. y= square root of e^5x+9 D. y= 3/1-15x E. 3/1+15x

D. y=3/1-15x

Derive (x^2+5)^6 dx=

E. 1/21 (x^3+5)^7+c

The figure above shows the graph of f', the derivative of a function f, for 0 less than or equal to x less than or equal to 2. What is the value of x at which the absolute minimum of f occurs? A. 0 B. 1/2 C. 1 D. 3/2 E. 2

E. 2

If y^3+y=x^2, then dy/dx= A. 0 B. x/2 C. 2x/3y^2 D. 2x-3y^2 E. 2x/1+3y^2

E. 2x/1+3y^2

The graph of the piecewise linear function f is shown above. Let h be the function given by h(x)= {x to -1 f(t) dt. On which of the following intervals is h increasing? A. 1,3 B. 0,5 C. 2,5 only D. 2, 9 E. 3, 9 only

E. 3, 9 only

Let y= f(x) define a twice-differentiable function and let y= t(x) be the line tangent to the graph of f at x= 2. If t(x) greater than or equal to f(x) for all real x, which of the following must be true? A. f(2) greater than or equal to 0 B. f'(2) greater than or equal to 0 C. f'(2) less than or equal to 0 D. f"(2) greater than or equal to 0 E. f"(2) less than or equal to 0

E. f"(2) less than or equal to 0