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Name: Student ID: Section: Instructor:
Math 113 (Calculus II)
Final Exam Form A
Apr 18, 7:00 p.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 10 10 7 11 10 12 7 13a 7 Sub 65
Possible Earned
13b 7 14 7 15 7 16 7 17 7
Sub 35 Total 100
Multiple Choice. In the grid below fill in the correct answer to each question.
- Here is a series โโ k=1 n(โ(n1)+1)n. Which of the following is true? a) The series diverges by the alternating series test. b) The series converges conditionally by the ratio test. c) The series converges absolutely by the ratio test. d) The series diverges by the integral test but converges by the ratio test. e) The series converges conditionally by a limit comparison test. f) The series converges absolutely by a limit comparison test. g) The series neither diverges nor converges. h) None of the above.
- Find the first four terms of the binomial series for (1 + x)^1 /^3. a) 1 + x โ 13 x^2 + 811 x^3 b) 1 + 13 x โ 19 x^2 + 815 x^3 c) 1 + 16 x + 19 x^2 + 812 x^3 d) 1 + 13 x + 14 x^2 + 15 x^3 e) None of the above.
- Find the interval of convergence of the power series โ^ โ k=1^ (โ1)
k (^2) k x k k.
a) [โ 1 , 1] b) [โ 2 , 2] c) (โ 2 , 2]
d) (โ^12 , 12 ] e) [โ^12 , 12 ) f) (โ 1 , 1] g) The series converges for all values of x.
h) None of the above.
- The length of the curve y = 23 x^3 /^2 for x โ [0, 2] is a) 53 โ 5 โ 13 b) โ 5 โ 12 ln (โ 5 โ 2 ) c) 2 โ 3 โ (^23)
d) 163 โ 43 โ 2 e) None of the above.
Short Answer. Fill in the blank with the appropriate answer.
- (10 points) a. What is limnโโ^ ln(1+ n n^2 )?
b. Find the first 4 terms of the power series of ex^2 centered at 0.
c. Let f (x) = x^2 + 1. Find the power series of f centered at 1.
d. What number equals โโ k=0 31 k?
e. Find limnโโ n^3 nโ 367
f. Identify โซ^ sec (x) dx.
g. What is the correct substitution to use in computing the integral, โซ^01 โ 1 โ x^2 dx?
h. Find the antiderivative, โซ^ x cos (x) dx.
i. What is the formula for the arc length of the graph of the function y = f (x) for
x โ [a, b]?
j. In the integral โซ^01 (1 + x^2 )^1 /^2 dx the substitution, u = 1 + x^2 is used. Write the integral
which results. Do not try to work the integral.
Free Response. For problems 10 - 17, write your answers in the space provided. Use the back of the page if needed, indicating that fact. Neatly show all work.
- (7 points) Determine whether the following series converges and explain your answer. โ^ โ n=
โ (^3) n (^2) + 1 3 n^2 + 2
- (10 points) Find the interval of convergence of the power series โ^ โ n=
(โ1)nn 4 n^ (x^ + 3)n
- (7 points) The graph ofis filled with water, (64 pounds per cubit foot) y = x^2 for x โ [0, 2] is rotated about the. Find the amount of work required to siphon y axis to form a tank that all the water to the top of the tank.
- (7 points) The region betweenrevolved about the line x = โ 1 .y Find the volume of the resulting solid of revolution. = ln x which lies between x = 1, x = 2, and the x axis is
- (7 points) The base of a solid is the inside of the circle,to the x axis are squares. Find the volume of the resulting solid. x^2 +y^2 โค 9. Cross sections perpendicular
- (7 points) Determine whether the integral โซ^01 โsin( 1 โxx) 2 dx converges.
- (7 points) Find โซ^1 โ (^) x(x^12 +1) dx
END OF EXAM
