Student (Print)
Section
Last, First Middle
Student (Sign)
Student ID
Instructor
MATH 152
Exam 3
Fall 2000
Test Form A
1-9
/45
10
/10
11
/15
Part II is work out. Show all your work. Partial credit will be given.
12
/10
You may not use a calculator.
13
/10
14
/10
Part I is multiple choice. There is no partial credit.
TOTAL
1
Part I: Multiple Choice (5 points each)
There is no partial credit. You may not use a calculator.
!
.
1. Compute
a.
b.
c.
d.
25
3
125
3
125
2
75
2
3 n 5 2"n
n0
e. .
!Ÿ
.
2. The series
n1
a.
b.
c.
d.
"1
n
n
is
absolutely convergent
convergent but not absolutely convergent
divergent
none of the above
!Ÿ
.
3. Determine the interval of convergence of the power series
a.
b.
c.
d.
e.
Ÿ2, 4 ¢
¡2, 4 Ÿ"4, "2 ¢
¡"4, "2 Ÿ"., . n0
x3 n
.
n1
2
Ÿ C
C
4. Find the Maclaurin series for f x sin 2x.
a.
b.
c.
d.
e.
3 3
5 5
2x " 2 x 2 x
3!
5!
3
5
2x
2x
2x "
"
3!
5!
3
5
7
x" x x " x
3!
5!
7!
3
5
7
x
x
x
x
3!
5!
7!
3
5
2x 2x 2x 3!
5!
7 7
" 2 x 7!
2x 7 7!
C
C
2x 7 7!
C
Ÿ 5. Find the 3 rd degree Taylor polynomial for f x e "2x about x 1.
a.
e "2
2e "2
b. e "2 " 2e "2
c. e "2 2e "2
d. e "2 " 2e "2
e. e "2 " 2e "2
Ÿx " 1 Ÿx " 1 2e
Ÿx " 1 4e
Ÿx " 1 4e
Ÿx " 1 " 4e
!
.
6. Compute
a. "4
b. "2
n0
2e "2
n 1
2
n"1
"2
"2
"2
"2
Ÿx " 1 Ÿx " 1 Ÿx " 1 Ÿx " 1 Ÿx " 1 2
2
2
2
2
Ÿ
Ÿ
4 e "2 x " 1 3
3
" 4 e "2 x " 1 3
3
8e "2 x " 1 3
" 8e "2 x " 1 3
" 8e "2 x " 1 3
Ÿ
Ÿ
Ÿ
!
.
.
HINT: Differentiate
xn.
n0
c. 4
9
d. 2
e. 4
3
Ÿ
Ÿ
7. Find a unit vector which is orthogonal to the plane containing the points P 1, 0, 0 , Q 0, 2, 0 ,
Ÿ
and R 0, 0, 3 .
6 , "3 , 2
a.
11 11 11
b. 6, "3, 2
c. 6, 3, 2
6 , "3 , 2
d.
7 7 7
6, 3, 2
e.
7 7 7
Ÿ
Ÿ
Ÿ 8. Find the equation of the circle for which one diameter has endpoints P 1, 1
a.
b.
c.
d.
e.
Ÿx 3 Ÿx 3 Ÿx " 4 Ÿx " 3 Ÿx 4 2
2
2
2
2
Ÿy 4 Ÿy 4 Ÿy " 5 Ÿy " 4 Ÿy 5 2
2
2
2
2
25
5
25
5
25
Ÿ
Ÿ
9. Consider the triangle with vertices A 0, 0, 0 , B 1, 1, 0
a.
b.
c.
d.
e.
Ÿ and Q 7, 9 .
Ÿ
and C 0, 2, 2 . Find the angle at A.
0°
30°
45°
60°
90°
4
Part II: Work Out
Show all your work. Partial credit will be given.
You may not use a calculator.
10. (10 points)
Ÿ a. Find the power series representation for f x convergence.
;
1
1 x2
centered at x 0 and its radius of
Ÿ b. Find the power series representation for f x x tan "1 x centered at x 0 and its radius of
convergence.
HINT:
tan "1 x
x
0
1
1 t2
dt
5
11. (15 points) Determine if each of the following series converges or diverges. Say why.
Be sure to name or quote the test(s) you use and check out all requirements of the test.
! nŸŸ"ln1n .
a.
n
n2
2
Circle one: Converges
Diverges
Circle one: Converges
Diverges
Circle one: Converges
Diverges
Explain:
! Ÿ"n!3 .
b.
n
n0
Explain:
! Ÿ"1n 4 5
.
c.
n1
n1 n"1
2 n
Explain:
6
12. (10 points) Determine if each of the following series converges or diverges. Say why.
Be sure to name or quote the test(s) you use and check out all requirements of the test.
If it converges, find the sum. If it diverges, does it diverge to ., ". or neither?
!
.
a.
n1
1
1 3n
Circle one: Converges
Diverges
Circle one: Converges
Diverges
Explain:
!
.
b.
n1
n"1 " n
n
n1
Explain:
7
13. (10 points) Find the volume of the parallelepiped with adjacent edges PQ, PR and PS, where
Ÿ
Ÿ
Ÿ
P 1, 1, 1 , Q 2, 3, 1 , R 4, 1, 5
Ÿ
and S 1, 3, 4 .
!Ÿ
.
14. (10 points) Determine the interval of convergence of the power series
n1
x"2 n
. Be sure to
n 1/3 3 n
show how you checked the convergence at the endpoints.
8