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Name
Student Number
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Math 113 – Fall 2006
Departmental Final Exam
Instructions:
- The time limit is 3 hours.
- Problem 1 consists of 9 short answer questions.
- Problems 2 through 8 are multiple choice questions.
- For problems 9 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 9.
- 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 – Fall 2006
Departmental Final Exam
Part I: Short Answer and Multiple Choice Questions Do not show your work for problem 1.
- Fill in the blanks with the correct answer.
(a) Does the improper integral
0
dx
ex^ + 1
converge (yes or no)
(b) The integral
cos x
sin 3 x
dx equals
(c) The integral
∫ (^) e 2
1
dx
2 x
equals
(d)
x^2
4
y^2
25
= 1 is the equation of a/an
(e) The radius of convergence of
n=
n x n is
(f) If n > 1, the integral
1
dx
xn^
equals
(g) The series x 2 −
x 4
x 6
x 8
+... is the MacLaurin series for the function
(h) The integral
x sin x dx equals
(i) The series 2 −
+... converges to
- Consider the region R that is the portion of the circle x^2 +y^2 = 1 that lies in the first quadrant.
What is the volume of the solid generated by revolving R about the line x + y = 2?
(a)
π
2
(d)
π^2
2
(g)
π^2
(b)
π
2
(e)
π^2
3
(h)
π^2
2
(c)
π
(f)
π^2
4
(i) None of the above
- The series
∑^ ∞
n=
3 n
n!
converges to
(a) ln 3 (d)
n+
n + 1
(g) cos 3
(b) ln 2 (e) ∞ (h) e 3 − 4
(c) ln(3) − 1 (f) e 3 (i) 3 e
- The interval of convergence of the power series
∑^ ∞
n=
n 2 (7x − 3) n is
(a)
(d) (0, 1) (g) (0, ∞)
(b)
(e)
(h) (−∞, ∞)
(c) (− 1 , 1) (f)
(i) None of these
- The integral
∫ (^) e+
2
(x − 1) ln(x − 1) dx is equal to
(a)
e^2 − 1
2
(d)
e^2 + 1
4
(b) e 2
e 2 − 1
4
(c)
e^2 + 1
2
(f) e^2 − 1
- The graph of the polar equation r = 2 cos(nθ) has how many petals?
(a) n petals if n is even, 2n petals if n is odd (e) n petals
(b) n/2 petals if n is odd, n petals if n is even (f) n/2 petals
(c) n petals if n is odd, 2n petals if n is even (g) None of these
(d) 2 n petals
The answers to the multiple choice MUST be entered on the grid on page 2. Otherwise, you will not receive credit.
Part II: Written Solutions
For problems 9 – 18, write your answers in the space provided. Neatly show your work for full credit.
- Evaluate each integral
(a)
dx
2 + x − x^2
(b)
sec 3 (2x) dx
- Find the centroid of the region that lies within the first quadrant and is bounded above by
y = 1 − x 2 .
- Find the area enclosed by the polar curves r = 2 − cos θ and r = 1.
- Use the first three non-zero terms of the MacLaurin series for e−x
2 to estimate the definite
integral
0
e −x^2 dx. Write your answer as a fraction, if possible.
- Find the mass of the circular region x^2 + y^2 ≤ 1, whose density at each point is twice the
distance from the point to the origin.
- Find the definite integral
0
x 3
1 − x^2 dx.
