Unit 5 MCQ AP Calc AB

Let f be a function with first derivative given by f′(x)=x(x−5)2(x+1). At what values of x does f have a relative maximum?

-1 only

The graph of f′, the derivative of the function f, is shown above for 0<x<9. Which of the following statements is true for 0<x<9 ?

f has one relative minimum and two relative maxima.

Let f be the function defined by f(x)=x5−10x3. The graph of f′, the derivative of f, is shown above. On which of the following intervals is the graph of f concave up?

-√3<x<0 and x>√3

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?

f III f' II f'' I

The graph of f′, the derivative of the continuous function f, is shown above on the interval −2<x<16. Which of the following statements is true about f on the interval −2<x<16 ?

f has three relative extrema, and the graph of f has four points of inflection.

The graph of f′, the derivative of the function f, is shown above. On which of the following open intervals is the graph of f concave down?

(−5,−3) and (1,6)

Let f be the function defined by f(x)=sinx+cosx. What is the absolute minimum value of f on the interval [0,2π] ?

-√2

The figure above shows the graph of f on the interval [a,b]. Which of the following could be the graph of f′, the derivative of f, on the interval [a,b] ?

B. Rises then decreases at origin then rises around 5 and goes back down then rises around 10

Selected values of a continuous function f are given in the table above. Which of the following statements could be false?

By the Mean Value Theorem applied to f on the interval [2,5], there is a value c such that f′(c)=10

The graph of y=f(x) is shown above. Which of the following could be the graph of y=f′′(x) ?

C. Concave down to 1 then rises

In the xy-plane, how many horizontal or vertical tangent lines does the curve xy2=2+xy have?

One vertical but no horizontal

Let C be the curve defined by x2+y2=36. Consider all points (x,y) on curve C where y>0. Which of the following statements provides a justification for the concavity of the curve?

The curve is concave down because y′′=−36/y^3<0

Let f be the function defined by f(x)=x3−6x2+9x+4 for 0<x<3. Which of the following statements is true?

f is decreasing on the interval (1,3) because f′(x)<0 on the interval (1,3).

Let f be the function defined by f(x)=xlnx for x>0. On what open interval is f decreasing?

0<x<1/e only

Consider the curve defined by x^2=e^x−y for x>0. At what value of x does the curve have a horizontal tangent?

2

Let f be the function given by f(x)=2x3+3x2+1. What is the absolute maximum value of f on the closed interval [−3,1] ?

6

An electrical power station is located on the edge of a lake, as shown in the figure above. An electrical line needs to be connected from the station to an island in the lake that is located 4 miles due south and 1 mile due east of the station. It costs $5,000 per mile to install an electrical line on land and $10,000 per mile to install an electrical line underwater. If C represents a cost function, which of the following methods best explains how to determine the minimum cost, in dollars, for connecting the electrical line from the station to the island?

Let C(x)=10,000(4−x)^2+1+5,000x. Solve C′(x)=0 and find the values of x where C′(x) changes sign from negative to positive. Evaluate C for those values of x to determine the minimum cost.

Let f be the function defined by f(x)=xsinx with domain [0,∞). The function f has no absolute minimum and no absolute maximum on its domain. Why does this not contradict the Extreme Value Theorem?

The domain of f is not a closed and bounded interval.

The derivative of the function f is given by f′(x)=x2−2−3xcosx. On which of the following intervals in [−4,3] is f decreasing?

[−3.444, −1.806] and [−0.660, 1.509]

The point (−3,4) is on the curve defined by x2y3=576. Which of the following statements is true about the curve at the point (−3,4) ?

dy/dx>0 and d2y/dx2>0

The second derivative of the function f is given by f′′(x)=sin(x28)−2cosx. The function f has many critical points, two of which are at x=0 and x=6.949. Which of the following statements is true?

f has a local maximum at x=0 and at x=6.949.

The second derivative of the function f is given by f′′(x)=x2cos(x2+2x6). At what values of x in the interval (−4,3) does the graph of f have a point of inflection?

2.229 only

Let f be the function given by f(x)=5cos2(x2)+ln(x+1)−3. The derivative of f is given by f′(x)=−5cos(x2)sin(x2)+1x+1. What value of c satisfies the conclusion of the Mean Value Theorem applied to f on the interval [1,4] ?

2.749 because f′(2.749)=f(4)−f(1)/3

The temperature inside a vehicle is modeled by the function f, where f(t) is measured in degrees Fahrenheit and t is measured in minutes. The first derivative of f is given by f′(t)=t2−3t+cost. At what times t, for 0<t<4, does the temperature attain a local minimum?

3.299 only

Let f be the function given by f(x)=x+4(x−1)(x+3) on the closed interval [−5,5]. On which of the following closed intervals is the function f guaranteed by the Extreme Value Theorem to have an absolute maximum and an absolute minimum?

[−2,0]

Let g be the function defined by g(x)=(x2−x+1)ex. What is the absolute maximum value of g on the interval [−4,1] ?

e

The total cost, in dollars, to order x units of a certain product is modeled by C(x)=5x2+320. According to the model, for what size order is the cost per unit a minimum?

An order of 8 units has a minimum cost per unit.

In the xy-plane, the point (0,−2) is on the curve C. If dydx=4x9y for the curve, which of the following statements is true?

At the point (0,−2), the curve C has a relative maximum because dy/dx=0 and d2y/dx2<0

Let f be the function given by f(x)=x(x−4)(x+2) on the closed interval [−7,7]. Of the following intervals, on which can the Mean Value Theorem be applied to f ? I.[−1,3] because f is continuous on [−1,3] and differentiable on (−1,3). II.[5,7] because f is continuous on [5,7] and differentiable on (5,7). III.[1,5] because f is continuous on [1,5] and differentiable on (1,5).

I and II only

A curve in the xy-plane is defined by the equation x3+y2−12x+16y=28. Which of the following statements are true? I At points where x=−2, the lines tangent to the curve are horizontal. II At points where y=−8, the lines tangent to the curve are vertical. III The line tangent to the curve at the point (−1,1) has slope 12.

I, II, and III

The graph of f′′, the second derivative of the continuous function f, is shown above on the interval [0,9]. On this interval f has only one critical point, which occurs at x=6. Which of the following statements is true about the function f on the interval [0,9] ?

The absolute minimum of f is at x=6.

The function f is continuous on the interval (0,9) and is twice differentiable except at x=6, where the derivatives do not exist (DNE). Information about the first and second derivatives of f for some values of x in the interval (0,9) is given in the table above. Which of the following statements could be false?

The graph of f has a point of inflection at x=8

The acceleration, in meters per second per second, of a race car is modeled by A(t)=t3−152t2+12t+10, where t is measured in seconds. What is the car's maximum acceleration on the time interval 0≤t≤6 ?

The maximum acceleration of the race car is 28 meters per second per second and occurs at t=6 seconds.

The Second Derivative Test cannot be used to conclude that x=2 is the location of a relative minimum or relative maximum for which of the following functions?

f(x)=x3−6x2+12x+1, where f′(x)=3x2−12x+12

The graph of f′, the derivative of the continuous function f, is shown above on the interval −8<x<7. The graph of f′ has horizontal tangent lines at x=−6, x=−3, x=2, and x=6.3, and a vertical tangent line at x=−4. On which of the following intervals is the graph of f both decreasing and concave up ?

(−8,−6), (−3,0), and (6.3,7) only

Let f be a differentiable function with f(0)=−4 and f(10)=11. Which of the following must be true for some c in the interval (0,10) ?

f′(c)=11−(−4)/10−0 since the Mean Value Theorem applies.