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Name Student Number Section Number Instructor
Math 113 – Fall 2005 Departmental Final Exam
Instructions:
- The time limit is 3 hours.
- Problem 1 consists of 13 short answer questions.
- Problem 2 consists of 3 T/F questions.
- Problems 3 through 9 are multiple choice questions.
- For problems 10 through 18 give the best answer andrelevant work. justify it with suitable reasons and/or
- 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 2005 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) The integral
cos(x + 2) dx equals (b) The integral
sec x tan x dx equals (c) The integral
0
dx 1 + x^2 equals (d) The integral
0 √^ dx 1 − x^2 equals (e) The integral
tan^2 x dx equals (f) The integral
0 √^ dx x equals (g) The integral
0
dx x^3 equals (h) The integral
∫ (^) x √1 + x 2 dx equals (i) Give the limit of the sequence
1 − (^) n^1
)n} as n → ∞ if it is convergent, otherwise write DIVERGENT.
(j) State the integration by parts formula:
(k) Give a limit definition of the improper integral
0
sin √ x x dx
(l) Let State the (2n)-th term of the MacLaurin series for sinx^ x
- The series x^2 + x^4 + x 2 6 + x 6 8 + · · · =
∑^ ∞
n=
x(2n+2) n! converges to the function (a) (^) 1+x^2 x 2 (e) x^2 (sin x^2 + cos x^2 ) (b) x^2 tan−^1 x (f) sin x^2 + cos x^2 (c) ex^2 +2^ (g) None of these (d) x^2 ex^2
- The improper integral
∫^ ∞
0
xe−xdx converges to (a) 0 (e) 2 (b) 1 /e (f) e (c) 1 / 2 (g) None of these (d) 1 (h) It doesn’t converge
- The length of the curve y = cosh x from x = 0 to x = 1 is (a) sinh 1 (e) ∞ (b) cosh 1 (f) a real number in (0,1) (c) cosh^2 1 − cosh^2 0 (g) Imaginary (d) 1 (h) None of these
- The area enclosed by the polar curve r = 3 + sin θ is (a) 5 π (e) 4. 5 π (b) 4 π (f) 19 π (c) 9 π (g) 9 π^2 (d) π/ 4 (h) None of these
- The interval of convergence of the power series
∑^ ∞
n=1^ n
(^2) (5x − 3)n (^) is (a) (− 3 / 5 , 3 /5) (e) (2/ 5 , 4 /5) (i) None of the above (b) (− 5 / 3 , 5 /3) (f) (1/ 5 , 1) (c) (0, 1) (g) (0, ∞) (d) (− 1 , 1) (h) (−∞, ∞)
- The coefficient of x^3 in the series expansion of (1 + x)^1 /^4 is (a) 413 = 641 (e) (^4203) 3! = 965 (i) None of the above (b) (^431) 3! = 3841 (f) (^4213) 3! = 1287 (c) (^436) 3! = 641 (g) (^4253) 3! = 38425 (d) (^4153) 3! = 1285 (h) (^4353) 3! = 38435
The answers to the multiple choice MUST be entered on the grid on the previous page. Oth-erwise, you will not receive credit.
- The region bounded by y = x and y = 2x^2 is revolved about the y-axis ; find the volume of the solid generated.
- Find the area of the surface of revolution generated by revolving the curveabout the x-axis. y = √x, 0 ≤ x ≤ 4,
- Find the centroid of the region bounded by the curves y = √1 + x^2 , x = 1 and y = 1 + x. Express you answer in terms of unevaluated integrals. (Note: You should simplify the inte- grands as much as possible.)
- If a region in the first quadrant, with area 10around the line x = −5, find the resulting volume of revolution.π and centroid at the point (1, 12), is revolved
- (a) Determine the power series expansion of ∫^ tan−^1 x dx.
(b) Find first two nonzero terms of the Taylor series of ln(1 + sinremainder after these terms?^2 x) at x = π. What is the
- Given the polar curve r = θ^2 , 0 ≤ θ ≤ 3 /2, (a) sketch the curve;
(b) find the area swept out by the curve;
(c) find the arc length.
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