AP Calculus AB Final Exam
If a trapezoidal sum overapproximates ∫0 to 4 f(x)dx, and a right Riemann sum underapproximates ∫0 to 4 f(x)dx, which of the following could be the graph of y=f(x)? *Remember decreases from 3 down
A
he graph of f′, the derivative of the function f, is shown above. Which of the following could be the graph of f ? *Remember it decreases then increases
C.
The advertising budget of a company is modeled by the differentiable function D, where D(t) is measured in dollars and t is the number of years since the company was founded. For 0≤t≤10, the company's advertising budget increased each year, but the rate of change of the advertising budget decreased each year. Which of the following could be a table of values of D ? * Remember by it goes up in by 8,00o0 first time
D.
The derivative of a function f is increasing for x<0 and decreasing for x>0. Which of the following could be the graph of f ?
E.
The size of a population of fish in a pond is modeled by the function P, where P(t) gives the number of fish and t gives the number of years after the first year of introduction of the fish to the pond for 0≤t≤10. The graph of the function P and the line tangent to P at t=4 are shown above. Which of the following gives the best estimate for the instantaneous rate of change of P at t=4 ? a. P (4) b. The slope of the line joining (0,P(0)) and (4,P(4)) c. The slope of the line joining (0,P(0)) and (8, P(0)) d. The slope of the line joining (3.9,P( 3.9)) and (4.1, P (4.1))
The slope of the line joining (3.9,P( 3.9)) and (4.1, P (4.1))
The function f given by f(x)=3x^5−4x^3−3x has a relative maximum at x= a. -1 b. −square root of5/5 c. 0 d. square roor of 5/5 e. 1
a. -1
If f(x)=ln(x+4+e−^3x), then f′(0) is a. -2/5 b. 1/5 c. 1/4 d. 2/5 e. nonexistent
a. -2/5
If y=cos2x, then dy/dx= a. -2sin 2x b. -sin 2x c. sin 2x d. 2 sin 2x e. 2 sin x
a. -2sin 2x
If f=1/2x^4/5−3/x^5, then dy/dx= a. 2/5x^1/5+ 15/x^6 b. 2/5x^1/5+15/x^4
a. 2/5x^1/5+ 15/x^6
An equation of the line tangent to the graph of f(x)=x(1−2x)3 at the point (1,−1) is a. y=-7x+6 b. y=−6x+5 c. y=-2x d. y-2x-3 e. y=7x-8
a. y=-7x+6
f f is the function given by f(x) = 3x2 − x3, then the average rate of change of f on the closed interval [1, 5 ] is a. -21 b. -13 c. -12 d.-9
b. -13
What is the total area of the regions between the curves y = 6x2 − 18x and y = −6x from x = 1 to x=3? a. 4 b. 12 c. 16 d. 20
b. 12
If y=−3cos(2x), then ⅆ^2y/dx^2 a. -12cos(2x) b. 12cos(2x) c.-3cos(2x) d. 3cos(2x)
b. 12cos(2x)
An equation of the line tangent to the graph of y=2x+3/3x−2 at the point (1, 5) is a. 13x-y=8 b. 13 x + y =8 c. x-13y=64 d. x+13y.=66 -e. 2x+3y=13
b. 13 x + y =8
The position of a particle moving along a straight line at any time t is given by s(t)=t^2+4t+4. What is the acceleration of the particle when t=4? Responses a. 0 b. 2 c. 4 d. 8 e. 12
b. 2
If ∫0 to 1 f(x)dx=2 and ∫0 to 4 f(x)dx=−3, then integral 1 to 4 (3(f(x) +2) dx= a. −13 b. −9 c. −7 d. 3 e. 21
b. −9
The function f is given by f(x)=x4+x2 -2. On which of the following intervals is f increasing? a. (-1/square root of 2, infinity) b. (-1/square root 2 of , 1 square root of 2) c. (0, infinity) d. (- infinity, 0) e. (- infinity, -1/ square root of 2)
c. (0, infinity)
The table above gives selected values for the differentiable function f. In which of the following intervals must there be a number c such that f′(c)=2 a. (0,4) b. (4,8) c. (8,12) d. (12,16)
c. (8,12)
A tank contains 50 liters of oil at time t=4 hours. Oil is being pumped into the tank at a rate R(t) where R(t) is measured in liters per hour, and t is measured in hours. Selected values of R(t) are given in the table above. Using a right Riemann sum with three subintervals and data from the table, what is the approximation of the number of liters of oil that are in the tank at time t=15 hours? a. 64.9 b. 68.2 c. 114.9 d. 116.6 e. 118.2
c. 114.9
Let g be the function with first derivative g′(x)=x^3+x for x>0. If g(2)=−7, what is the value of g5) ? a. 4.402 b. 11.402 c. 13.899 d. 20.899
c. 13.899
The graph of the function f is shown in the figure above. The value of limx→1+f(x) is a. -2 b. -1 c. 2 d. nonexistent
c. 2
Let f be a differentiable function with f(2)=3 and f′(2)=12. Using the line tangent to the graph of f at x=2 as a local linear approximation for f, what is the estimate for f(1.8) a. 2.5 b. 2.8 c. 2.9 d. 3.1
c. 2.9
Let f be the function with derivative defined by f'(x) = x3 - 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
lim x-> 3 x^2-9/x^2-2x-15 a. 0 b. 3/5 c. 3/4 d. nonexistent
c. 3/4
The graph of the function f is shown above. What is limx→2f(x) ? a. 1/2 b. 1 c. 4 d. The limit DNE
c. 4
The area of the region enclosed by the graphs of y=x and y=x^2−3x+3 is a. 2/3 b. 1 c. 4/3 d. 2 e. 14/3
c. 4/3
If f is the function defined by f(x)=x^2−4/x^2+x−6, then limx→2f(x) is a. 0 b. 2/3 c. 4/5 d. nonexistent
c. 4/5
A spherical snowball is melting in such a way that it maintains its shape. The volume of the snowball is decreasing at a constant rate of 5 cubic inches per minute. At the instant when the radius of the snowball is decreasing at a rate of 3/5 inch per minute, what is the radius of the snowball, in inches? (The volume V of a sphere with radius r is V=4/3pier^3.) b. 25/12 Pie c. 5/ Square root 12pie
c. 5/ Square root 12pie
A bug begins to crawl up a vertical wire at time t = 0. The velocity v of the bug at time t, 0 ≤ t ≤ 8, is given by the function whose graph is shown above. At what value of t does the bug change direction? a. 2 b. 4 c. 6 d. 7 e. 8
c. 6
The points (3, 0), (x,0) , (x, 1/x2) , and (3, 1/x2) are the vertices of a rectangle, where x≥3, as shown in the figure above. For what value of x does the rectangle have a maximum area? a. 3 b. 4 c. 6 d. 9 e. There is no such value of x.
c. 6
Let f be a function that is defined for all real numbers x. Of the following, which is the best interpretation of the statement limx→3f(x)=5 ? a. The value of the function f at x equals 3 is 5. b. The value of the function f at x equals 5 is 3. c. As x approaches 3, the values of f of x approach 5. d. As x approaches 5, the values of f(x) approach 3.
c. As x approaches 3, the values of f(x) approach 5.
The figure above shows the graph of the function f. Which of the following statements are true? I. limx→2−f(x)=f(2) II. limx→6−f(x)=limx→6+f(x) III. limx→6f(x)=f(6) a. II only b. III only c. I and II only d. II and III only e. I, II, and III
c. I and II only
On a certain day, the total number of pieces of candy produced by a factory since it opened is modeled by C, a differentiable function of the number of hours since the factory opened. Which of the following is the best interpretation of C'(3) = 500 ? a. The factory produces 500 pieces of candy during its 3rd hour of operation. b. The factory produces 500 pieces of candy in the first 3 hours after it opens. c. The factory is producing candy at a rate of 500 pieces per hour, 3 hours after it opens. d. The rate at which the factory is producing candy is increasing at a rate of 500 pieces per hour per hour, 3 hours after it opens.
c. The factory is producing candy at a rate of 500 pieces per hour, 3 hours after it opens.
If the velocity of a particle moving along the x-axis is v(t)=2t−4 and if at t=0 its position is 4, then at any time t its position x(t) is a. t^2-4t b. t^2-4t-4 c. t^2-4t+4 d. 2t^2-4t e. 2t^2-4t+4
c. t^2-4t+4
Let f be the function given by f(x)=�x^3−6x^2+8x−2. What is the instantaneous rate of change of f at a. -5 b. -15/4 c. -1 d. 5 e. 17
c.-1
What is the area of the region between the graphs of y=x2 and y=-x from x=0 to x=2? a. 2/3 b. 8/3 c. 4 d. 14/3 e. 16/3
d. 14/3
A particle travels along the x-axis so that at time t≥0 its velocity is given by v(t)=t^2−6t+8. What is the total distance the particle travels from t = 0 to t = 3? a. 1 b. 6 c. 20/3 d. 22/3 e. 8
d. 22/3
If ∫0 to 3f(x)dx=6 and ∫3 to 5f(x)dx=4, then ∫0 to 5(3+2f(x))dx= a. 10 b. 20 c. 23 d. 35 e. 50
d. 35
Water is pumped out of a lake at the rate R(t)=12 square root t over/t+1 cubic meters per minute, where t is measured in minutes. How much water is pumped from time t = 0 to t = 5? a. 9.439 cubic meters b. 10.954 cubic meters c. 43.816 cubic meters d. 47.193 cubic meters e. 54.772 cubic meters
d. 47.193 cubic meters
Let R be the region bounded by the graph of y=lnx, the vertical line x=e^2, and the x-axis, as shown in the figure. Which of the following integrals are equal to the area of R? a. I only b. III only c. I and II only d. I and III only
d. I and III only
he graph of f′, the derivative of the function f, is shown in the figure above. On what open intervals is the graph of f concave up? a. b<x<d b. <x<e c. d < x< e only d. a<x<b and d<x<e
d. a<x<b and d<x<e
The area of the shaded region in the figure above is represented by which of the following integrals? a. integral a to c (lf(x)l-lg(x)l)dx b. integral a to c f(x) dx - integral a to c g(x) dx c. integral a to c (g(x)-f(x))dx d. integral a to c (f(x)-g(x))dx
d. integral a to c (f(x)-g(x)) dx
he table above gives values of a function f at selected values of x. Which of the following conclusions is supported by the data in the table? a. limx→3f(x)=0 b. limx→3f(x)=3 c. limx-- f(x)= 10 d. limx--f(x) does not exist
d. limf--f(x) does not exist
If y3 + y = x2, 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
If f(x)=(2x+1)^4, then the 4th derivative of f(x) at x = 0 is a. 0 b. 24 c. 48 d. 240 e. 384
e. 384
Three graphs labeled I, II, and III are shown above. One is the graph of f, one is the graph of f', and one is the graph of f''. Which of the following correctly identifies each of the three graphs? a. f-I, f'-II, f""-III b. f-I, f'-III, f''-II c. f-II, f'-1, f''-III d. f-II, f'-III, f''-I e. f-III, f'-II, f''-I
e. f-III, f'-II, f''-I
f f is the function defined above, then f′(−1) is a. -2 b. 2 c. 3 d. 5 e. nonexistent
e. nonexistent
If 3x2 + 2xy + y2 = 2, then the value of dy/dx at x = 1 is a. -2 b. 0 c. 2 d. 4 e. not defined
e. not defined
Integral (x3 − 3x) dx = a. 3x2 - 3 + C b. 4 x 4 - 6 x 2 + C c. x4/3 − 3x2+C d. x 4 /4 − 3x + C e. x4/4 − 3x2/2 + C
e. x4/4 − 3x2/2 + C
A pump allows water to flow into a pool at the rate of r(t)=2t liters per minute, where t is the time in minutes since the pump was turned on. Which of the following defines a function that measures the accumulation of water in the pool during the time period from t=1 to t=x as the variable x moves along the t-axis as shown in the figure above? f(x)=2x f(x)=1+2x f(x)= integral 0 to 1 2tdt f(x)=integral 1 to x 2tdt
f(x)=integral 1 to x 2tdt
