AP Calculus Second Semester Exam

If f(x)=sin^-1(x), then f'(square root(3)/2)=

D. 2

Let g be a continuous function. Using the substitution u=2x-1, the integral from [2, 3] g(2x-1)dx is equal to which of the following?

B. (1/2) integral [2, 3] g(u)du

Let g(x)=x^4+4x^3. How many relative extrema does g have?

B. One

Let y=f(x) be a twice-differentiable function that such that f(1)=2 and dy/dx=y^3+3. What is the value of d^2y/dx^2 at x=1?

C. 132

The function f is defined above (with two different equations in a bracket after "f(x)"). The value of the integral from [-5, 3] f(x)dx is:

A. -2

The graph of the function f is shown above. For what values of a does lim from x->a f(x)=0?

B. 2 and 4

If y=cosx-ln(2x), then d^3y/dx^2=

A. sinx-2/x^3

The velocity of a particle moving along the x-axis is given by v(t)=sqrroot(t)-cos(e^t) for t>(or equal to)0. Which of the following statements describes the motion of the particle at t=1?

B. The particle is moving to the right with positive acceleration.

On a certain day, the rate at which material is deposited at a recycling center is modeled by the function R, where R(t) is measured in tons per hour and t is the number of hours since the center opened. Using a trapezoidal sum with the three subintervals indicated by the data in the table, what is the approximate number of tons of material deposited in the first 9 hours since the center opened? (Has table)

A. 68

Which of the following could be a slope field for the differential equation dy/dx=x^2+y?

B. (The lines are horizontal on the y-axis like this: -)

The function h is defined on the closed interval [-1, 3]. The graph of h', the derivative of h, is shown above. The graph consists of two semicircles with a common endpoint at x=1. Which of the following statements about h must be true? I. h(-1)=h(3) II. h is continuous at x=1 III. The graph of h has a vertical asymptote at x=1

B. II only

Let f be a function with first derivative defined by f'(x)=(3x^2-6)/(x^2) for x>0. It is known that f(1)=9 and f(3)=11. What value of x in the open interval (1, 3) satisfies the conclusion of the Mean Value Theorem for f on the closed interval [1, 3]?

B. square root of 3

Let y=f(x) be the particular solution to the differential equation dy/dx+(x+1)/y with the initial condition f(0)=-2. Which of the following is an expression for f(x)?

C. -(square root)x^2+2x+4

The first derivative of the function f is defined by f'(x)=(x+2e^-x)/(x^2+0.7). On what intervals is f increasing?

C. -infinity<x<infinity

The temperature in a room at midnight is 20 degrees Celsius. Over the next 24 hours, the temperature changes at a rate modeled by the differentiable function H, where H(t) is measured in degrees Celsius per hour and time t is measured in hours since midnight. Which of the following is the best interpretation of the integral [0, 6] H(t)dt?

C. Change in temp in Celcius between midnight and 6:00am.

The table above shows selected values of a continuous function f. For 0<(or equal to)x<(or equal to)13, what is the fewest possible number of times f(x)=4?

C. Three

A tire that is leaking air has an initial air pressure of 30 pounds per square inch (psi). The function t=f(p) models the amount of time t, in hours, it takes for the air pressure of the tire to reach p psi. What are the units for f'(p)?

D. Hours per psi

People are entering a building at a rate modeled by f(t) people per hour and exiting the building at a rate modeled by g(t) people per hour, where t is measured in hours. The functions f and g are non negative and differentiable for all time t. Which of the following inequalities indicates that the rate of change of the number of people in the building is increasing at time t?

D. f'(t)-g'(t)>0

Let f be the function with derivative given by f'(x)=sin(x^2-3). At what values of x in the interval -3<x<3 does f have a relation maximum?

A. -1.732 and 2.478 only

Integral [1, 2] (x^2-x-5)/(x+2)dx=

A. -3/2+ln4/3

A hemispherical water tank, shown above, has a radius of 6 meters and is losing water. The area of the surface of the water is A=12pih-pih^2 square meters, where h is the depth, in meters, of the water in the tank. When h=3 meters, the depth of the water is decreasing at a rate of 1/2 meter per minute. At that instant, what is the rate at which the area of the water's surface is decreasing with respect to time?

A. 3pi square meters per minute

The function y=e^3x-5x+7 is a solution to which of the following differential equations?

A. y"-3y'-15=0

Integral [-1, 3] (x^2-2x)dx?

B. 4/3

Let R be the shaded region bounded by the graph of y=(square root)x, the graph of y=x-2, and the x-axis , as shown in the figure above. Which of the following gives the volume of the solid generated when R is revolved about the x-axis?

C. pi integral [0, 2] xdx + pi integral [2, 4] x-(x-2)^2dx

The table above gives the values of f, f', g, and g' for selected values of x. If h(x)=f(g(x)), what is the value of h'(x)?

B. -14

To help restore a beach, sand is being added to the beach at a rate of s(t)=65+24sin(0.3t) tons per hour, where t is measured in hours since 5:00am. How many tons of sand are added to the beach over the 3-hour period from 7:00am to 10:00am?

A. 255.368

Over the time interval 0<(or equal to)t<(or equal to)5, a particle moves along the x-axis. The graph of the particle's velocity, v, is shown above. Over the time interval, the particle's displacements is 3 and the particle travels a total distance of 13. What is the value of the integral [2, 4] v(t)dt?

B. -5

lim from x->e ((x^20-3x)-(e^20-3e))/(x-e)=

B. 20e^19-3

Let f be the function defined above, where c is a constant. If f is continuous at x=1, what is the value of c?

B. 3

Let f be a differentiable function such that f(2)=4 and f'(2)=1/2. What is the approximation for f(2.1) found by using the line tangent to the graph of f at x=2?

B. 3.95

The graph of f', the derivative of the function f, is shown above. Which of the following could be the graph of f?

A. Line on left, slanted parabola on right

What is the total area of the regions between the curves y=6x^2-18x and y=-6x from x=1 to x=3?

B. 12

The graph of the function f, shown above (the V shaped graph), consists of three line segments. If the function g is an antiderivative of f such that g(2)=5, for how many values of c, where 0<(or equal to)c<(or equal to)6, does g(c)=3?

B. One

The graph of y=f(x) consists of a semicircle with endpoints at (2, -6) and (12, -6), as shown in the figure above. What is the value of the integral [2, 12] f(x) dx?

C. -60+25pi/2

The velocity of a particle moving along the x-axis is given by v(t)=2-t^2 for time t>0. What is the average velocity of the particle from time t=1 to time t=3?

C. -7/3

lim from x->3 (tan(x-3))/(3e^x-3)

C. 1/2

The function f is defined by f(x)=x^3+4x+2. If g is the inverse function of f and g(2)=0, what is the value of g'(2)?

C. 1/4

Consider a triangle in the xy-plane. Two vertices of the triangle are ont he x-axis at (1, 0) and (5, 0), and a third vertex is on the graph of y=ln(2x)-1/2x+5 for 1/2<(or equal to)x<(or equal to)8. What ist he maximum area of such a triangle?

C. 2ln4+8

The graph of the function f is shown above. At what value of x does f have a jump discontinuity?

C. 7

If f(x)=e^2x(x^3+1), then f'(2)=

D. 30e^4

The second derivative of a function f is given by f"(x)=sin(3x)-cos(x^2). How many points of inflection does the graph of f have on the interval 0<x<3?

D. Five

Which of the following limits is equal to integral [2, 5] x^2dx?

D. Look for 3k/n and 3/n

Let f be a differentiable function such that f(1)=pi and f'(x)=sqrroot(x^3+6). What is the value of f(5)?

D. 27.814

An object moves along a straight line so that at any time t its acceleration is given by a(t)=6t. At time t=0, the objects velocity is 10 and the position is 7. What is the object's position at t=2?

D. 35

The function g is defined by g(x)=x^2+bx, where b is constant. If the line tangent to the graph of g at x=-1 is parallel to the line that contains the points (0, -2) and (3, 4), what is the value of b?

D. 4