Chapter 4
1.
Find the critical numbers of the function.
y  3x 2  12x
2.
The graph of the first derivative f (x) of a function f is shown below. At what values of x does f
have a local maximum or minimum?
3.
Find the absolute minimum value(s) of y  2 x 2  20x  9 on the interval [0, 6].
4.
Find the limit.
lim
x 
x7  3
x6  6
Select the correct answer.
a.
5.

b.

c.
0
7/6
d.
e.
1/ 2
Find the critical numbers of the function.
y
x
x 2  25
6.
Find an equation of the line through the point (9, 36) that cuts off the least area from the first quadrant.
7.
The graph of the derivative f (x) of a continuous function
have a local maximum or minimum?
f
is shown. At what values of x does f
8.
Evaluate the limit.
lim
1  cos x
x 0
x2  x
Select the correct answer.
a. -5
9.
b.
0
c. 
d. 1
e. 
Find the minimum points of the function.
F ( x)  x 4  500x
10. If 1,100 cm 2 of material is available to make a box with a square base and an open top, find the
largest possible volume of the box.
11. Sketch the curve.
y  x  3x 2 / 3
Select the correct answer.
a.
b.
c.
12. Find all the maximum and minimum values of the function.
F ( x) 
x5
x  42
13. Which of the following functions is graphed below?
Select the correct answer.
a.
y  3 x 2  3x  3
b.
y  3 x 3  3x  3
c.
y  x 2  3x  3
d.
y  x 2  3x  3
e.
y  x2  6
14. Find the point on the line y  10x  9 that is closest to the origin.
Select the correct answer.
a.
b.
c.
d.
e.
  90
,

 101
  90
,

 100
  90
,

 101
  92
,

 101
  89
,

 101
9 

101 
9 

101 
11 

101 
10 

101 
9 

101 
15. Find the smallest possible area of an isosceles triangle that is circumscribed about a circle of radius r.
16. Find f .
f ( x)  9 cos(3x)
Select the correct answer.
c.
f ( x)  y   cos(3x)  Cx 2  D
f ( x)  y  9 cos(x)  Cx  D
f ( x)  y   cos(3x)  Cx  D
d.
e.
f ( x)  y  cos(9x)  Cx 2  D
none of these
a.
b.
17. Find f .
f (t )  2t  3sin t ,
f (0)  5
Select the correct answer.
a.
f (t )  t 2  3cos t  2
b.
f (t )  t 2  3cos t  5
c.
f (t )  t 2  cos t  2
d.
e.
f (t )  2t 2  3cos t  5
none of these
18. Use Newton's method to find all the roots of the equation, correct to six decimal places.
2 x5  6 x 4  177 x3  11x 2  19 x  5  0
19. Find the maximum area of a rectangle that can be circumscribed about a given rectangle with length
L = 8 and width W = 3.
20. A particle is moving with the given data. Find the position of the particle.
v(t )  sin t  cos t , s(0)  0
1.
-2
2.
3, 5, 8, 10
3.
-41
4.
a
5.
5, -5
6.
y  4x  72
7.
4, 12, 17
8.
b
9.
5
10.
3511
11.
a
12.
1
4
13.
a
14.
a
15.
3 3r 2
16.
c
17.
a
18.
5, -1.707107, -0.292893
19.
60.5
20.
s(t )  1  cos t  sin t
1.
Find the absolute maximum value of y  81  x 2 on the interval [- 9, 9].
2.
Find the critical numbers of the function.
y  3x 2  30x
Select the correct answer.
a.
3.
-5
b.
c.
3
d.
0
e.
5
Find the limit.

lim y  y 2  2 y
y 
4.
30

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then
find all numbers c that satisfy the conclusion of Rolle's Theorem.
f ( x)  sin 5x,
 2 2
 5 , 5 


Select the correct answer.
a.
b.
c.
d.
e.
1
3
, c2 
10
10
1
1
c1   , c2  
10
10
1
3
,
c1 
c2  
10
10
1
3
c1   , c2  
10
10
none
c1  
5.
Show that the equation x5  3x  1  0 has exactly one real root.
6.
The graph of the second derivative f (x) of a function f is shown. State the x-coordinates of the
inflection points of f.
7.
The graph of the derivative f (x) of a continuous function f is shown. On what intervals is f
decreasing?
8.
For what values of c does the curve have maximum and minimum points?
F ( x)  4 x3  cx2  4 x
9.
Suppose the line y  5x  1 is tangent to the curve y  f (x) when x  8 . If Newton's method is
used to locate a root of the equation f ( x)  0 and the initial approximation is x1  8 , find the
second approximation x2 .
10. Find the limit.
lim
x 
x7  3
x6  6
11. Consider the following problem: A farmer with 890 ft of fencing wants to enclose a rectangular area
and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest
possible total area of the four pens?
Select the correct answer.
a. 19,825.5 ft 2
e. 19,791.5 ft 2
b.
19,802.5 ft 2
c.
19,801.5 ft 2
d.
19,902.5 ft 2
12. Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of
side L = 9 cm if one side of the rectangle lies on the base of the triangle. Round the result to the nearest
tenth.
Select the correct answer.
a. 5.5 cm, 4.4 cm
e. 4.5 cm, 4 cm
b.
4 cm, 3.91 cm
c.
7.5 cm, 2.9 cm
d.
4.5 cm, 3.9 cm
13. Find the most general antiderivative of the function.
f ( x)  9 x 2  10x  3
14. Consider the figure below, where a = 7, b = 1 and l = 6. How far from the point A should the point P
be chosen on the line segment AB so as to maximize the angle  ?
Round the result to the nearest hundredth.
15. Sketch the curve.
y  2x 3  3x
Select the correct answer.
a.
b.
c.
16. Sketch the curve.
y
x2
3x  6
Select the correct answer.
a.
b.
c.
17. Which of the following functions is graphed below?
Select the correct answer.
a.
y  3 x 2  3x  3
b.
y  3 x 3  3x  3
c.
y  x 2  3x  3
d.
y  x 2  3x  3
e.
y  x2  6
18. Find two positive numbers whose product is 196 and whose sum is a minimum.
Select the correct answer.
a.
4, 49
these
b.
2, 98
c.
14, 14
d.
6, 24
e.
none of
19. Use Newton's method with the specified initial approximation x1 to find x3 , the third approximation
to the root of the given equation. (Give your answer to four decimal places.)
x 4  13  0, x1  2
20. An aircraft manufacturer wants to determine the best selling price for a new airplane. The company
estimates that the initial cost of designing the airplane and setting up the factories in which to build it
will be 900 million dollars. The additional cost of manufacturing each plane can be modeled by the
function m( x)  1,600x  10x 4 / 5  0.17x 2 where x is the number of aircraft produced and m is the
manufacturing cost, in millions of dollars. The company estimates that if it charges a price p (in
millions of dollars) for each plane, it will be able to sell x( p)  390  5.8 p .
Find the cost function.
1.
9
2.
a
3.
-1
4.
d
f ( x ) = x 5 + 3x + 1. Since f is continuous and f (- 1) = - 3 and f (0) = 1, the equation f (x) = 0 has at
5.
least one root at (- 1, 0) by the Intermediate Value Theorem. Suppose that the equation has more
than one root; say a and b are both roots with a < b. Then f (a) = 0 = f (b) so by Rolle's Theorem f '
(x) = 5x 4 + 3 = 0 has a root in ( a, b ). But this is impossible since clearly f ( x)  3  0 for all real x.
6.
2, 4, 9
7.
(3, 9)  (12, 13)
8.
| c |  48
9.
1/5
10. 
11. b
12. d
13. 3x 3  5x 2  3x  C
14. 3.26
15. b
16. a
17. a
18. c
19. 1.8989
20. C( x)  900  1600x  10x 4 / 5  0.17x 2