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Math 113 – Winter 2007
Departmental Final Exam
Instructions:
- The time limit is 3 hours.
- Problem 1 consists of 5 short answer questions.
- Problem 2 consists of 5 T/F questions.
- Problems 3 through 7 are multiple choice questions.
- For problems 8 through 18 give the best answer and justify it with suitable reasons and/or relevant work.
- Work on scratch paper will not be graded. Do not show your work for problem 1 through 7.
- Please write neatly.
- Notes, books, and calculators are not allowed.
- Expressions such as ln(1), e^0 , sin(π/2), etc. must be simplified for full credit.
For administrative use only:
M.C. /
Total /
Math 113 – Winter 2007
Departmental Final Exam
Part I: Short Answer and Multiple Choice Questions Do not show your work for problems in this part.
- Fill in the blanks with the correct answer.
(a) 1 + x + x^2 + · · · is the Maclaurin series for
(b)
ln (x) dx =
(c) What technique must be used to integrate
x + 1 x^3 + 4x dx?
(d)
∑^ ∞
n=
(−1)n−^1 x^2 n−^1 (2n − 1)! is the Maclaurin series for
(e) What substitution should be used to find
4 − x^2 dx x =
- True/False: Write T if statement always holds, F otherwise.
Let
an =
n=1 an^ be an arbitrary series.
(a) 1 − 12 + 13 − 14 + · · · converges absolutely.
(b)
sec (x) dx = ln |sec (x) + tan (x)| + C
(c) The improper integral
1
dx x^3 +1 converges.
(d) If
n=1 an^ converges absolutely, then it converges.
(e) If
n=1 a (^2) n converges, then ∑∞ n=1 an^ also converges.
- Find the area enclosed by the spiral r = f (θ) = eθ^ where θ ∈ [0, π].
(a)
e^2 π^ −
(b)
e^2 π^ −
(c) eπ^ − 1
(d)
eπ^ −
(e)
e^2 π^ − 1 (f) None of the above.
The answers to the multiple choice MUST be entered on the grid on the previous page. Oth- erwise, you will not receive credit.
Part II: Written Solutions
For problems 8 – 18, write your answers in the space provided. Neatly show your work for full credit.
- Determine whether each series converges absolutely, conditionally or fails to converge. State your conclusion next to the series.
(a)
∑^ ∞
n=
(n − 10) ln n
(b)
∑^ ∞
n=
2 sin (n) n^2 + 1
(c)
∑^ ∞
n=
n cos (nπ) n^2 + 1
- Find
x^2 e−xdx.
- Evaluate the improper integral
1
dx x^2 (x + 1)
or show that it does not converge.
- Use partial fraction decomposition to evaluate the following integral: ∫ 6 x^2 − 3 x − 18 (x − 5)(x^2 + 2x + 4) dx
- Find the power series of ln (1 − x) centered at 0 and give its radius of convergence.
- Consider the ellipse, x^2 − 6 x + 2y^2 + 16y + 37 = 0. Find the vertices, express in standard form, and sketch the graph.
y
x
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