ap calc 2012 mcc study guide
A particle moves along the x-axis. The velocity of the particle at time t is given by v(t), and the acceleration of the particle at time t is given by a(t). Which of the following gives the average velocity of the particle from time t = 0 to time t = 8 ? (A) [a(8)-a(0)]/8 (B) 1/8 ∫(0 to 8) v(t) dt (C) 1/8∫(0 to 8)|v(t)| dt (D)1/2 ∫(0 to 8) v(t) dt (E)[v(0)+v(8)]/2
(B) 1/8 ∫(0 to 8) v(t) dt
Let f be the function defined by f(x)= (ln x)/x. What is the absolute maximum value of f ? (A) 1 (B) 1/e (C) 0 (D) -e (E) f does not have an absolute maximum value.
(B) 1/e
lim ln(4+h)-ln4 is... h->0 h (A) 0 (B) 14 (C) 1 (D) e (E) nonexistent
(B) 14
A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 feet per second. If the height of the streetlight is 15 feet, what is the rate at which the person's shadow is lengthening? (A) 1.5 ft/sec (B) 2.667 ft/sec (C) 3.75 ft/sec (D) 6 ft/sec (E) 10 ft/sec
(B) 2.667 ft/sec
x: -2,0,3,5,6 f'(x): 3,1,4,7,5 Let f be a polynomial function with values of f'(x) at selected values of x given in the table above. Which of thefollowingmustbetruefor -2 < x < 6? (A) The graph of f is concave up. (B) The graph of f has at least two points of inflection. (C) f is increasing. (D) f has no critical points. (E) f has at least two relative extrema.
(B) The graph of f has at least two points of inflection.
A particle moves along the x-axis with its position at time t given by x(t) = (t - a)(t - b), where a and b are constants and a does not equal b. For which of the following values of t is the particle at rest? (A) t = ab (B) t = (a + b)/2 (C) t = a + b (D) t=2(a+b) (E) t=a and t=b
(B) t = (a + b)/2
Let f be a function such that ∫(6 to 12) f(2x) dx = 10. Which of the following must be true? (A) ∫(12 to 24) f(t)dt=5 (B) ∫(12 to 24)f(t)dt=20 (C) ∫(6 to 12)f(t)dt=5 (D) ∫(6 to 12)f(t)dt=20 (E) ∫(3 to 6)f(t)dt=5
(B) ∫(12 to 24)f(t)dt=20
The function f is defined by f (x) = x/(x+2) . What points (x,y) on the graph of f have the property that the line tangent to f at (x, y) has slope 1/2? (A) (0,0) only (B) (1/2,1/5)only (C) (0,0) and (-4,2) (D) (0,0)and 4,3 (2) (E) There are no such points.
(C) (0,0) and (-4,2)
If f'(x)= √(x^4+1)+(x^3-3x) ,then f has a local maximum at x= (A) -2.314 (B) -1.332 (C) 0.350 (D) 0.829 (E) 1.234
(C) 0.350
If y = (x^3 - cos x)^5 , then y' = (A) 5(x^3 - cos x)^4 (B) 5(3x^2 +sinx)^4 (C) 5(3x^2 +sinx) 4(D) 5(3x^2 +sinx)^4*(6x+cosx) 4(E) 5(x^3-cosx)^4*(3x^2+sinx)
(C) 5(3x^2 +sinx)
If f(x)=7x-3+lnx, then f'(1)= (A) 4 (B) 5 (C) 6 (D) 7 (E) 8
(C) 6
Let f be the function defined by f (x) = √|x-2| for all x. Which of the following statements is true? (A) f is continuous but not differentiable at x = 2. (B) f is differentiable at x = 2. (C) f is not continuous at x = 2. (D) limf(x) does not equal 0 x-->2 (E) x = 2 is a vertical asymptote of the graph of f.
(C) f is not continuous at x = 2.
The graph of f', the derivative of f, is shown in the figure above. The function f has a local maximum at x = (A) -3 (B) -1 (C)1 (D)3 (E)4
(C)1
For -1.5 < x < 1.5, let f be a function with first derivative given by f'(x) = e^(x4 -2x2 +1) - 2. Which of the following are all intervals on which the graph of f is concave down? (A) (-0.418, 0.418) only (B) (-1,1) (C) (-1.354, -0.409) and (0.409, 1.354) (D) (-1.5, -1) and (0, 1) (E) (-1.5, -1.354), (-0.409, 0), and (1.354, 1.5)
(D) (-1.5, -1) and (0, 1)
Using the substitution u=√x , ∫(1 to 4) (e^√x)/(√x) dx is equal to which of the following? (A) 2∫(1 to 16)e^u du (B) 2∫(1 to 4)e^u du (C) 2∫(1 to 2)e^u du (D) 1/2∫(1 to 2)e^u du (E) ∫(1 to 4)e^u du
(D) 1/2∫(1 to 2)e^u du
Water is pumped into a tank at a rate of r(t) = 30(1-e^-0.16t) gallons per minute, where t is the number of minutes since the pump was turned on. If the tank contained 800 gallons of water when the pump was turned on, how much water, to the nearest gallon, is in the tank after 20 minutes? (A) 380 gallons (B) 420 gallons (C) 829 gallons (D) 1220 gallons (E) 1376gallons
(D) 1220 gallons
The graph of y = (e^tan x) - 2 crosses the x-axis at one point in the interval [0, 1]. What is the slope of the graph at this point? (A) 0.606 (B) 2 (C) 2.242 (D) 2.961 (E) 3.747
(D) 2.961
The graph above gives the velocity, v, in ft/sec, of a car for 0<t<8, where t is the time in seconds. Of the following, which is the best estimate of the distance traveled by the car from t = 0 until the car comes to a complete stop? (A) 21ft (B) 26ft (C) 180ft (D) 210ft (E) 260ft
(D) 210ft
The graph of the function f is shown above. Which of the following statements is false? (A) lim f(x) exists. x-->2 (B) lim f(x) exists. x-->3 (C) lim f(x) exists. x--> 4 (D) lim f(x) exists. x-->5 (E) The function f is continuous at x = 3.
(D) lim f(x) exists. x-->5
.Let f(x)=(2x+1)^3 and let g be the inverse function of f. Given that f(0)=1, what is the value of g'(1)? (A)-2/27 (B) 1/54 (C) 1/27 (D)1/6 (E)6
(D)1/6
∫(secxtanxdx) = (A) secx+C (B) tanx+C (C) (sec^2x/2)+C (D) (tan^2x/2)+C (E) (sec^2xtan^2x/2)+C
(E) (sec^2xtan^2x/2)+C
The graph of f'', the second derivative of f, is shown above for -2<x<4. What are all intervals on which the graph of the function f is concave down? (A) -1 < x < 1 (B) 0<x<2 (C) 1 < x < 3 only (D) -2<x<-1 only (E) -2<x<-1 and 1<x<3
(E) -2<x<-1 and 1<x<3
A particle moves along a line so that its acceleration for t ≥ 0 is given by a(t) = (t+3)/(√t^3+1). If the particle's velocity at t=0 is 5, what is the velocity of the particle at t=3? (A) 0.713 (B) 1.134 (C) 6.134 (D) 6.710 (E) 11.710
(E) 11.710
t (hours): 4,7,.12,15 R(t) (liters/hour): 6.5,6.2,5.9,5.6 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
(E) 118.2
A particle moves along the x-axis. The velocity of the particle at time t is 6t - t2. What is the total distance traveled by the particle from time t = 0 to t = 3? (A) 3 (B) 6 (C) 9 (D) 18 (E) 27
(E) 27
The graph of the function f is shown in the figure above. For which of the following values of x is f'(x) positive and increasing? (A) a (B) b (C) c (D) d (E) e
(E) e
The line y = 5 is a horizontal asymptote to the graph of which of the following functions? (A) y=[sin(5x)]/x (B) y=5x (C) y= 1/(x-5) (D) y= 5x/(1-x) (E) y=(20x^2-x)/(1+4x^2)
(E) y=(20x^2-x)/(1+4x^2)
Which of the following is the solution to the differential equation dy/dx = 2sin x with the initial condition y(pi)=1? (A) y=2cosx+3 (B) y=2cosx-1 (C) y=-2cosx+3 (D) y=-2cosx+1 (E) y=-2cosx-1
(E) y=-2cosx-1
if y=xsinx, then dy/dx= a)sinx+cosx b)sinx+xcosx c)sin-xcosx d)x(sinx+cosx) e)x(sinx-cosx)
b)sinx+xcosx
Let f be the function given by f (x) = 300x - x3. On which of the following intervals is the function f increasing? (A) (-∞, -10] and [10, ∞) (B) [-10,10] (C) [0,10]only (D) [0,10 root3]only (E) [0, ∞)
(A) (-∞, -10] and [10, ∞)
If(x+2y)*dy/dx=2x-y, what is the value of d^2y/d^2x at the point(3,0)? (A) - 10/3 (B) 0 (C) 2 (D) 10/3 (E) Undefined
(A) - 10/3
Let g be the function given by g(x) = (x^2)(e^kx), where k is a constant. For what value of k does g have a critical point at x = 2/3? (A) -3 (B) -3/2 (C) -1/3 (D) 0 (E) There is no such k.
(A) -3
Let f be the function defined above. For what value of k is f continuous at x = 2 ? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5
(A) 0
Let R be the region in the first quadrant bounded below by the graph of y = x^2 and above by the graph of y =√x. R is the base of a solid whose cross sections perpendicular to the x-axis are squares. What is the volume of the solid? (A) 0.129 (B) 0.300 (C) 0.333 (D) 0.700 (E) 1.271
(A) 0.129
The function f is defined by f(x)= {2 x<3 and {x-1 for x>3 for What is the value of ∫(1 to5)f(x)dx? (A) 2 (B) 6 (C) 8 (D) 10 (E) 12
(A) 2
What is the area of the region in the first quadrant bounded by the graph of y = e^(x/2) and the line x = 2? (A) 2e-2 (B) 2e (C) (e/2) -1 (D) (e-1)/2 (E) e-1
(A) 2e-2
If f(x)=√x^2-4 and g(x)=3x-2, then the derivative of f(g(x))at x=3 is... (A) 7/√5 (B)14/√5 (C)18/√5 (D)15/√21 (E)30/√21
(A) 7/√5
If P(t) is the size of a population at time t, which of the following differential equations describes linear growth in the size of the population? (A) dP/dt = 200 (B) dP/dt = 200t (C) dP/dt=100t^2 (D) dP/dt = 200P (E) dP/dt=100P^2
(A) dP/dt = 200
Let f be a function that is continuous on the closed interval [2, 4] with f (2) = 10 and f (4) = 20. Which of the following is guaranteed by the Intermediate Value Theorem? (A) f (x) = 13 has at least one solution in the open interval (2, 4). (B) f(3)=15 (C) f attains a maximum on the open interval (2, 4). (D) f'(x) = 5 has at least one solution in the open interval (2, 4). (E) f'(x) > 0 for all x in the open interval (2, 4).
(A) f (x) = 13 has at least one solution in the open interval (2, 4).
Let g be a function with first derivative given by g'(x) = ∫(0 to x) e^-t^2 dt. Which of the following must be true on the interval 0 < x < 2 ? (A) g is increasing, and the graph of g is concave up. (B) g is increasing, and the graph of g is concave down. (C) g is decreasing, and the graph of g is concave up. (D) g is decreasing, and the graph of g is concave down. (E) g is decreasing, and the graph of g has a point of inflection on 0 < x < 2.
(A) g is increasing, and the graph of g is concave up.
The graph of a differentiable function f is shown above. If h(x) = ∫(0 to x) f(t) dt, which of the following is true? (A) h(6) < h'(6) < h''(6) (B) h(6) < h''(6) < h'(6) (C) h'(6) < h(6) < h''(6) (D) h''(6) < h(6) < h'(6) (E) h''(6) < h'(6) < h(6)
(A) h(6) < h'(6) < h''(6)
For t ≥ 0, the position of a particle moving along the x-axis is given by x(t) = sin t - cos t. What is the acceleration of the particle at the point where the velocity is first equal to 0 ? (A)-√2 (B)-1 (C)0 (D)1 (E)√2
(A)-√2
