Calc Multiple Choice Answers

What is the area of the region in the first quadrant enclosed by the graphs of y=cosx, y=x, and the y-axis?

0.400

The base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line x+2y=8, as shown in the figure above. If cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

16.755

The function y is continuous on the closed interval [2,8] and has values that are given in the table above. Using the subintervals [2,5], [5,7], and [7,8], what is the trapezoidal approximation of the integral [2,8] f(x)dx?

160

26. What is the slope of the line tangent to the curve y=arctan(4x) at the point at which ?

2

Let F(x) be an antiderivative of [(lnx)^3]/x. If F(1)=0, then F(9)=

5.827

A particle moves along the x-axis with velocity given by v(t)=3t^2+6t for time t>=0. If the particle is at position x=2 at time t=0, what is the position of the particle at t=1?

6

If y=(x^3=1)62, then dy/dx=

6x^2(x^3=1)

The graph of f ', the derivative of f , is the line shown in the figure above. If f (0) = 5, then f (1) =

8

The graph of a function f is shown above. At which value of x is f continuous, but not differentiable?

A

If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h is decreasing at a rate of 3 inches per minute, which of the following must be true about the area A of the triangle?

A is decreasing

Let f be a function such that, as the limit of h approaches 0, [f(2+h)-f(2)]/h=5. Which of the following must be true?

I and II only

Let f be the function given above. Which of the following statements are true about f ?

I and II only

Let f be the function given by f(x)=lxl. Which of the following statements about f are true?

I and III only

Which of the following are antiderivatives of f(x)=sinxcosx?

I and III only

Let f be the function defined above. Which of the following statements about f are true?

I only

Let f be a function that is differentiable on the open interval (1,10). If f(2)=-5, f(5)=5, f(5)=5, and f(9)=-5, which of the following must be true?

I, II, and III

The first derivative of the function f is given by f'(x)=[(cos^2x)/x]-1/5. How many critical values does f on the open interval (0,10)?

Three

23. A rumor spreads among a population of N people at a rate proportional to the product of the number of people who have heard the rumor and the number of people who have not heard the rumor. If p denotes the number of people who have hear the rumor, which of the following differential equations could be used to model this situation with respect to time t, where k is a positive constant?

dp/dt=kp(N-p)

Shown above is a slope field for which of the following differential equations?

dy/dx=xy+y

If g is a differentiable function such that g(x)<0 for all real numbers x and if f'(x)=(x^2-4)g(x), which of the following is true?

f has a relative minimum at x=-2 and a relative maximum at x=2.

The graph of a function f is shown above. Which of the following statements about f is false?

f is continuous at x=a

The graph of f', the derivative of the function f, is shown above. Which of the following statements is true about f?

f is increasing for -2<=x<=0.

the graphs of the derivatives of the functions f, g, and h are shown above. Which of the functions f, g, or h have a relative maximum on the open interval a<x<b?

f only

The function f has the property that f(x), f '(x), and f "(x) are negative for all real values x. Which of the following could be the graph of f ?

f(x) must be decreasing and concave down

The graph of the piecewise linear function f is shown in the figure above. If g(x)= on [-2,x] the integral f(t)dt, which of the following values is greatest?

g(1)

For x>=0, the horizontal line y= 2 is an asymptote for the graph of the function f . Which of the following statements must be true?

limit, as x approaches infinity, of f(x)=2

A particle moves along the x-axis so that at time t>=0 its position is given by x(t)=2t^3-21t^2+72t-53. At what time t is the particle at rest?

t=3 and t=4

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

the graph of a parabola centered on graph

Let f be the function given by f(x)=2xe^x. The graph of f is concave down when

x< -1

24. Which of the following is the solution to the differential equation dy/dx+(x^2)/y with the initial condition y(3)=-2 ?

y=-sqrt[(2x^3)/3 - 14]

What are all the horizontal asymptotes of the graph of y=(4x^2+6x-1)/(5x^3+2x^2+1)

y=0 only

If f(x)=e^(2/x), then f'(x)=

(-2/x^2)*e^(2/x)

Let f be the function with derivative given by f'(x)=x^2-(2/x). On which of the following intervals is f decreasing?

(0, cubed root(2)]

If f is a continuous function and if F'(x)=f(x) for all real numbers x, then the integral [1,3] f(2x)dx=

(1/2)F(6)-(1/2)F(2)

The integral of x/(x^2-4)=

(1/2)ln(x^2-4) + C

The integral of (x^2)(cos(x^3))dx=

(1/3)sin(x^3) + C

If f(x)=x^2+2x, then d/dx(f(lnx))=

(2lnx+2)/x

The rate of change of the volume, V, of water in a tank with respect to time, t, is directly proportional to the square root of the volume. Which of the following is a differential equation that describes this relationship?

(dV/dt)=k*sqrt(V)

If f(x)=(x-1)(x^2+2)63, then f'(x)=

(x^2 +2)^2(7x^2-6x+2)

The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In terms of the circumference C, what is the rate of change of the area of the circle, in square centimeters per second?

-(0.1)C

Let f be the function given f(x)=3e^(2x) and let g be the function given by g(x)=6x^3. At what value of x do the graphs of f and g have parallel tangent lines?

-0.391

The graph of the function y=x^3+6x^2+7x-2cosx changes concavity at x=

-1.89

As x approaches 0, (5x^4+8x^2)/(3x^4-16x^2) is

-1/2

Let f be a differentiable function such that f(3) = 15, f(6) = 3, f ′(3) = -8, f ′(6) = -2. The function g is differentiable and g(x) = f -1(x) for all x. What is the value of g′(3)?

-1/2

The integral [sin(2x)+cos(2x)]dx=

-1/2cos(2x)+(1/2)sin(2x)+C

25. Let f be the function defined below, where c and d are constants. If f is differentiable at x = 2, what is the value of c + d?

-2

As x approaches infinity, the limit [(2x-1)(3-x)]/[(x-1)(x+3)] is

-2

If f(x)=ln(x+4+e^(-3x)), then f '(0) =

-2/5

The derivative g' of a function g is continuous and has exactly two zeros. Selected values of g' are given in the table above. If the domain of g is the set of all real numbers, then g is decreasing on which of the following intervals?

-2<=x<+2 only

In the xy-plane the line x+y=k where k is a constant is tangent to the graph of y=x^2+3x+1 what is the value of k?

-3

If f(x)=cos(3x), then f'(pi/9)=

-3sqrt(3)/2

If y=(2x+3)/(3x+2), then dy/dx=

-5/(3x+2)^2

If y=2x-8, what is the minimum value of the product xy?

-8

On [0, pi/4], the integral of sinxdx=

-sqrt(2)/2 + 1

The integral of 1/(x^2)dx=

-x^(-1) + C

The second derivative of the function f is given by f''(x)=x(x-a)(x-b)^2. The graph of f'' is shown above. For what values of x does the graph of f have a point of inflection?

0 and a only

Population y grows according to the equation dy/dt=ky, where k is a constant and t is measure in years. If the population doubles every 10 years, then the value of k is

0.069

Let f be the function given by f(x)=2e^(4x^2). For what value of x is the slope of the line tangent to the graph of f at (x, f(x)) equal to 3?

0.168

The function f is twice differentiable with f(2) = 1, f ′(2)=4, and f ″(2) = 3. What is the value of the approximation of f (1.9) using the line tangent to the graph of f at x = 2?

0.6

If the derivative of f is given by f'(x)=(e^x)-(3x^2), at which of the following values of x does f have a relative maximum value?

0.91

A particle moves along a straight line. The graph of the particles' position x(t) at time t is shown above for 0 < t < 6. The graph has horizontal tangents at t = 1 and t = 5 and a point of inflection at t = 2. For what values of t is the velocity of the particle increasing?

0<t<2

The base of a solid S is the region enclosed by the graph of y=sqrt(lnx), the line x=e, and the x-axis. If the cross sections of S perpendicular to the x-axis are squares, then the volume S is

1

If 0<=k<(pi/2) and the areas under the curve y=cosx from x=k to x=(pi/2) is 0.1, then k=

1.120

At time t>=0, the acceleration of a particle moving on the x-axis is at a(t)=t+sint. At t=0, the velocity of the particle is -2. For what value t will the velocity of the particle be zero?

1.48

If a cannot = 0, then the limit, as x approaches a, [(x^2-a^2)/(x^4-a^4)] is

1/(2a^2)

Using the substitution u=2x+1, on [0,2] the integral of sqrt(2x+1)dx is equivalent to

1/2 of, on [1,5], the integral sqrt(u)du

As x approaches infinity, the limit of (x^3-2x^2+3x-4)/(4x^3-3x^2+2x-1)=

1/4

Let f be the function defined by f(x)=x^3+x. if g(x)=f^(-1)(x) and g(2)=1, what is the value of g'(2)?

1/4

Let f(x)=sqrt(x). If the rate of change of f at x=c is twice the rate of change at x=1, then c=

1/4

A table of values for a continuous function f is shown above. If four equal subintervals of [0,2] are used, which of the following is the trapezoidal approximation of the integral on [0,2] of f(x)dx?

12

The graph of the function f shown above has horizontal tangents at x= 2 and x = 5. Let g be the function defined by g(x)= on [0,x] the integral f(t)dt. For what values of x does the graph of g have a point of inflection?

2 and 5 only

Let g be a twice-differentiable function with g'(x)>0 and g''(x)>0 for all real numbers x, such that g(4)=12 and g(5)=18. Of the following, which is a possible value for g(6)?

27

If y=(x^2)sin(2x), then dy/dt=

2x(sin2x + xcos2x)

d/dx [on [0,x^2] the integral of sin(t^3)dt]=

2xsin(x^6)

If the line tangent to the graph of the function f at point (1,7) passes through the point (- 2,-2), then f'(1) is

3

Let f be a function with a second derivative given by f ″(x) = x2(x - 3)(x - 6). What are the x-coordinates of the points of inflection of the graph of f?

3 and 6 only

The graph of f is shown in the figure above. If the integral on [1,3] f(x)dx=2.3 and F'(x)=f(x), then F(3)-F(0)=

4.3

What is the slope of the tangent to the curve 3y^2 -2x^2=6-2xy at the point (3,2)?

4/9

A railroad track and a cross road at right angles. An observer stands on the road 70 m South of the crossing and watches the East bound train traveling at 60 meters per second. At how many meters per second is the train moving away from the observer at 4 seconds after it passes through the intersection?

57.60

The graph of the function f is shown above for 0 ≤ x ≤ 3. Of the following, which has the least value?

Right Riemann sum approximation of the integral on [1,3] of f(x)dx with 4 subintervals of equal length

The polynomial function f has selected values of its second derivative f" given in the table above. Which of the following statements is true?

The graph changes concavity in the integral [0,2]

If f(x)=(e^2x)/2x, then f'(x)=

[(e^2x)(2x-1)]/(2x^2)

Let f be the function defined by f(x)=4x^3-5x+3. Which of the following is an equation of the line tangent to the graph of f at the point where x = - 1?

y=7x+11

Which of the following is an equation of the line tangent to the graph of f(x)=x^4+2x^2 at the point where f'(x)=1?

y=x-0.122

A curve has slope 2x + 3at each point (x,y) on the curve. Which of the following is an equation for this curve if it passes through the point (1,2)?

y=x^2+3x-2