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Name: Student ID: Section: Instructor:
Math 113 (Calculus II)
Final Exam Form A
Dec. 15, 2008 at 7:00 a.m.
Instructions:
- Work on scratch paper will not be graded.
- For questions 10 to 17, show all your work in the space provided. Full credit will be given only if the necessary work is shown justifying your answer. Please write neatly.
- Should you have need for more space than is alloted to answer a question, use the back of the page the problem is on and indicate this fact.
- Simplify your answers. Expressions such as ln(1), e^0 , sin(ฯ/2), etc. must be simplified for full credit.
- Calculators are not allowed.
For Instructor use only.
Possible Earned
MC 24
9 20
10 6
11 6
12 6
Sub 62
Possible Earned
13 6
14 6
15 6
16 6
17 6
Sub 30
Total 92
Multiple Choice. Fill in the answer to each problem on your scantron. Make sure your name, section and instructor is on your scantron.
- Here is a series
โ^ โ
k=
โ^ (โ1)n n (n^2 + 1)
. Which of the following is true?
a) The series does not converge absolutely by the root test.
b) The series diverges by the integral test.
c) The series converges conditionally by the ratio test.
d) The series converges conditionally.
e) The series converges absolutely by a limit comparison test.
f) The series converges absolutely by the ratio test.
g) The series neither diverges nor converges.
- Compute the sum โโ
k=
)k
a)
b)
c) 1
d)
e)
f) 2
g) The series diverges by the root test.
- Find the interval of convergence of the power series
โ^ โ
k=
3 k^ xk k^2
a)
b) (0,
) c) (โ
d)
[
]
e) [โ
) f) (โ 3 , 3)
g) The series converges for all values of x.
- Find the area between the graphs of y = sin (x) and y = cos (x) for x โ [0, ฯ/2].
a)
ฯ โ
b) โ2 + 2
2 c) 0
d) 2 e) None of the above.
Short Answer. Fill in the blank with the appropriate answer. 2 points each
- (20 points)
(a) What is the correct substitution to use in computing the integral,
0
1 + x^2 dx?
(b) Find lim nโโ n ln
1 + 2nโ^1
(c) Find the first 3 nonzero terms of the power series of sin
x^2
centered at 0.
(d) Let f (x) = cos (x). Find the first three terms of the power series of f centered at ฯ/4.
(e) What number equals
โ^ โ
k=
3 k^
(f) In the integral
0
1 + x^5
dx the substitution, x = sin (u) is used. Write the integral
which results. Do not try to work the integral.
(g) Find the antiderivative,
3 x^2 sec^2
x^3
dx.
(h) What is the formula for the arc length of the graph of the function y = f (x) for
x โ [a, b]?
(i) What is lim nโโ n^2 tan
4 n^2
(j) Identify
sec (2x) dx.
- (6 points) Find
(a)
4 x + 11 (x + 4) (x โ 1) dx
(b)
3 โ x^2 dx
- (6 points) The region between y = sin x which lies between x = 0, x = ฯ/ 2 , and the x axis is revolved about the line x = 0. Find the volume of the resulting solid of revolution.
- (6 points) The base of a solid is the inside of the circle, x^2 +y^2 โค 4. Cross sections perpendicular to the x axis are squares the length of a side corresponding to x being equal to the width of the base at that value of x. Find the volume of the resulting solid. A sketch of one such solid is shown.
x
y
