103
Final Math Exam
Fall 2009
3
6
0
1. If
6
0
3
f ( x ) dx = 12 and f ( x ) dx = 42 , find the value of ( 2 f ( x ) 3) dx.
a) 5 0
b) 51
c) 5 2
d) 56
e) 53
f) 54
g) 55
h) None of these
8
2. Find the value of the integral
3
0
1
x
+ 1 dx.
32
x
a) 14
b) 6
c) 12
e) 18
f) 72
g)
e2
3. Find the value of
( ln x )
e
d) 20
5
2
h) 168
2
x
dx.
a) ln2
b)
1
ln 2
2
c)
1
2
d)
3
2
e) 1
f)
1
ln 2
g) 0
h)
7
3
ln 9
4. Find the value of
e
x /2
dx.
0
a) 4
b) 2
c) 6
d) 3
e) 16
f) 9
g) 8
h) 1
5. Find all critical numbers for the function f ( x ) = 3 9 x 2 .
a) 0
b) -3
c) 0, -3
d) No critical numbers
e) 3
f) 3, -3
g) 0, 3, -3
h) None of these
x2 + 4 x + 5
if x < 2
1
6. At what value(s) of x is the function f ( x ) =
if 2 x 2 discontinuous?
2x
if x > 2
1+ x 2
a) -2
b) 0
c) -2,0, and 2
d) -2 and 0
e) 2
f) -2 and 2
g) 0 and 2
h) f is continuous everywhere
(
)
7. Find the interval on which the graph of f ( x ) = ln x 2 + 1 is concave upward.
a) (-1,1)
b) (-1,2)
c) (-2,1)
d) (-2,2)
e) (-1,3)
f) (-3,2)
g) (-3,3)
h)
( , )
8. The curve y = x 3 + x 2 x has two horizontal tangents. Find the distance between these two horizontal lines.
a)
11
9
b)
22
27
c)
32
27
d)
5
3
e)
14
9
f)
4
3
g)
13
9
h)
7
3
9. If f ( x ) =
a)
x
, find f .
tan x
4
2
2
b)
e) 1 2
1
2
x
ln ( 3 + 2e5 x )
6x
a) 5
b)
1
5
c)
1
6
f)
1
10
g) 2 10
e)
5
3
d)
g) 2
f) 2
10. Evaluate the limit lim
c) 1
h)
.
d)
3
5
h)
5
6
2
11. Let V be the volume of a cylinder having height h and radius r, and assume that h and r vary with time. When
the height is 5 in. and is increasing at 0.2 in./s. , the radius is 3 in. and is decreasing at 0.1 in./s. How fast is the
volume changing at that instant?
12. A rectangle with base on the x -axis has its upper vertices on the curve y = 12 x 2 . Find the maximum area of
such a rectangle. Be sure to prove that you have found the maximum area.
x
13. The graph of f below consists of line segments and semicircles. Let g ( x ) =
f ( t ) dt .
0
Answer the following questions.
(a) g (14 )
(b) g (10 )
(c) g ( 6 )
(d) What is the absolute minimum value of g on the interval [0,14]?
14. Given the graph of y = f ( x ) , answer the questions that follow.
(a) Find all values of x at which (Explain your answers for full credit)
(i) f is increasing.
(ii) f is decreasing.
(iii) f ( x ) > 0.
(iv) f has an inflection point.
(v) f has a local maximum
(b) Sketch a graph which could represent y = f ( x ) .
Math 103 Fall 2009
Final Exam
Answers
1. B
2. E
3. H
4. A
5. G
6. A
7. A
8. C
9. A
10. H
11.
6
5
12. 32 un.2
13. a) 4 +
5
2
b) 12 +
9
2
c) 2
d) 0
14. a)
i)
( x1 , x3 )
ii)
( x0 , x1 ) ( x3 , x4 )
iii)
( x0 , x2 )
iv) x2 v) x3
b)
x0
x1
x2
x3
x4