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Name:
Math 106C: Fall 2012
Final Exam
Read directions carefully and show all your work. Partial credit will be assigned based upon the correctness, completeness, and clarity of your answers. Correct answers without proper justification or those that use unapproved short-cut methods will not receive full credit. If you use a calculator to help find an answer, you must write down enough information on what you have done to make your method understandable.
Good Luck!
- (21 points) Evaluate the following integrals. If an integral is improper, determine whether or not it converges and if it does, find the value to which it converges.
(a)
x^2012 ln x dx
(b)
1
xeโx 2 dx
(c)
1
x + 1 x^2 โ 2 x dx
- (1 point) What is your favorite technique of integration?
- (8 points) Find the interval of convergence of the power series: S(x) =
โ^ โ
n=
(x + 1)n n 4 n^
. (Remember to
check the endpoints of the interval and explain your work.)
- (1 point) What is your favorite convergence test?
- Let f (x) = cos(
x).
(a) (6 points) Write a power series equal to f (x) = cos(
x). Your answer should contain at least 5 terms OR be written in sigma notation.
(b) (6 points) Use your answer in (a) to find the third order Taylor polynomial based at x 0 = 0 for f (x) = cos(
x). Then use the Taylor polynomial to estimate cos(
(c) (6 points) Use the power series you found in (a) to help you estimate
1
cos(
x) dx. (Use at least 3 terms.)
(f) (7 points) Compute
1
cos(
x) dx using algebraic integration techniques. (Hint: Start with a u-substitution.)
- (1 point) Which technique for estimating integrals do you prefer? Power series or Riemann sums?
- (12 points) Determine whether each of the following statements is TRUE or FALSE. Briefly explain your answer.
(a) x^2 โ 4 x(x^2 + 4) can be put in the form
A
x
B
x^2 + 4
(b) The series
โ^ โ
k=
(โ1)k โ (^4) k converges absolutely.
(c) If S(x) =
โ^ โ
k=
ak(x โ 2)k^ has interval of convergence [โ 1 , 5), then D(x) =
โ^ โ
k=
akk(x โ 2)kโ^1 must
also have interval of convergence [โ 1 , 5).
(d) If f (x) is continuous, positive, and decreasing on the interval [2, โ) and
0
f (x) dx converges,
then
โ^ โ
n=
f (n) converges.
- (1 point) Where are you spending your winter break?
