x

y

y f x

Mathematics 31 – Curve Sketching "A"

Name: _________________________ Score: ________ 30

Multiple Choice - Show work for 6 *questions on loose leaf

*1. The graph of y=5 x4−x5has a point of inflection at

A. (0,0 ) B. (3 ,162 ) C. (4 ,256 ) D. (0,0 )& (3 ,162 )

2. Let f be a function whose domain is the closed interval [−3,4 ] and let the derivative of f have

the graph shown in the figure below. The function f is increasing on which of the following intervals?

A. (−1 ,−3 )∪(2,4 ) B. (−2,0 )∪(3,4 ) C. (−3 ,−2 )∪(0,3 ) D. (−1,2 )

3. Suppose that f is an odd function; that is, f (−x )=−f ( x ), for all x . Suppose that f'( x0 )

exists. Which of the following must necessarily be equal tof'(−x0 )?

A. f'( x0 ) B. −f

'( x0 ) C.

1

f ' ( x0 ) D.

−1

f ' ( x0 )

4.limn→∞

3n3−5nn3−2n2+1

A. -5 B. -2 C. 1 D. 3

*5. For what value of x does the function f ( x )=( x−2 ) ( x−3 )2 have a relative maximum?

A.

52 B.

−73 C.

−52 D.

73

6. If the graph of y=ax+b

x+c has a horizontal asymptote, y=2, and a vertical asymptote, x=−3 , then a+c=

A. -5 B. -1 C. 1 D. 5

*7. The graph of y=3 x4−16 x3+24 x2+48 is concave down for

A. x<0 C.x<−2∧x>−2

3

B.x< 2

3∧x>2

D.

23<x<2

*8. The function f is given by f ( x )=x4+x2−2. On which of the following intervals is f

increasing?

A.(−1

√2,∞)

B. (−1

√2,

1

√2 )C. (0 ,∞ ) D. (−∞ ,0 )

9. The graphs of the derivatives of the functions f ,g& h are shown below. Which of the

functions have a relative maximum on the open interval, a< x<b?

A. f B. g C. h D. All of these

10. Given f ( x )= x−1

x+1,state the equations of all vertical and horizontal asymptotes of f ( x ).

A.

x=1y=−1 B.

x=−1y=1 C.

x=1 ,−1y=0 D. x=−1

*11. Given f ( x )= 2x2

x+1, state the equation of the oblique asymptote of f ( x ).

A. y=2x+2 B. y=2x C. y=2x−2 D. y=x−1

12. A polynomial p( x )has a relative maximum at (−2,4 ) ,a relative minimum at (1,1 ) , a relative

maximum at (5,7 )and no other critical points. How many real zeroes does p( x )have?

A. 1 B. 2 C. 3 D. 4

*13. Give the value of xwhere the function f ( x )=x3−21

2x2+30 x−3

has a local minimum.

A. -5 B. 5 C. 2 D. -2

14. Which of the following pairs of graphs could represent the graph of a function and the graph of its derivative?

A. I B. II C. III D. I & III

*15. Find the locations of the absolute extreme values of f ( x )=3 x4−4 x3−8on the interval [−2,2 ] .

A. Maximum at x=2 ,minimum at x=1

B. Maximum at x=−2 ,minimum at x=0

C. Maximum at x=2 ,minimum at x=−2

D. Maximum at x=−2 ,minimum at x=1

16. Use the following properties of a function f ( x ) to determine the location of any local extrema. (+ means strictly positive and - means strictly negative.)

Note: DNE means "Does Not Exist"

A. No local extrema

B. No local maximum and local minimum at x=4

C. Local maximum at x=−3 and local minimum at x=1

D. Local maximum at x=1 and local minimum at x=−3

17. The graph of y=f ( x ) is shown in the figure below. On which of the following intervals are dydx

>0and

d2 ydx2

>0?

I. a< x<b II. b< x<c III. c<x<d

A. I only B. II only C. I and II D. II and III

18. The graph of the derivative function f'( x )is shown below. At what value of x does function

f ( x )have a relative maximum?

A. -2 B. -1 C. 1 D. 3

19. Below is a graph of the derivative y=f '( x ) . Use the graph to identify the location of the local

minimum(s) of the function f ( x ).

A. x=0 B. x=±2 C. x=−3∧1 D. x=1

20. The graph shown below shows the derivative f ' of the function f . At what values of x

does function f have a point of inflection.

A. c∧e B. a C. b∧d D. a∧c

21. Find the width of a rectangle that has a perimeter of 64 feet and a maximum area. Let A

represent the area of the rectangle, w the width of the rectangle and l the length of the rectangle. Which equation should you optimize to solve the problem efficiently?

A. A=64w−w2C.

A=64w

+w

B. A=32w−w2D.

A=128w

+2w

*22. The top of a 25-foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. When the top of the ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

A.

−78 ft /min C.

78 ft /min

B.

−724 ft /min C.

724 ft /min

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

A. 57.6 B. 57.88 C. 59.20 D. 67.40

24. Determine from the graph whether the function has any absolute extrema on the interval, [ 0,3 .5 ] .

A. Absolute minimum onlyB. Absolute maximum onlyC. Both absolute minimum and absolute maximumD. No absolute extrema