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Honors Calculus III Midterm Exam 1, April 19, 2008 - Prof. Hal Sadofsky, Exams of Advanced Calculus

A practice midterm exam for honors calculus iii, held on april 19, 2008. The exam covers topics such as limits, arctan function, logarithmic functions, taylor polynomials, and integration. Students are required to use the definition of limits to prove certain properties, find power series expansions, use induction, find taylor polynomials, and calculate integrals.

Typology: Exams

Pre 2010

Uploaded on 07/29/2009

koofers-user-27
koofers-user-27 🇺🇸

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Practice Midterm Exam 1, Honors Calculus III, April 19, 2008
(1) Prove using the definition of the limit that if limn→∞ an=l, then
lim
n→∞
(can) = cl.
(2) Let f(x) = arctan(x).
(a) Find P2n+1,0(x).
(b) Give a formula for f(k)(0).
(3) Let f(x) = log(x).
(a) Use induction to find a formula for f(k)(x).
(b) Find Pn,1(x).
(c) Use the remainder theorem to decide which nyou’ll need to use so
that Pn,1(2/3) is within .01 of log(2/3).
(4) Use Taylor polynomials around 0 to approximate cos(1) to within .001.
Make sure and explain using the remainder term why your answer is
within .001.
(5) Calculate the following integrals.
(a) Re2zsin(3x)dx.
(b) R1
xlog(x)dx.
(c) R1
0xexdx.
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Practice Midterm Exam 1, Honors Calculus III, April 19, 2008

(1) Prove using the definition of the limit that if limn→∞ an = l, then

lim n→∞ (can) = cl.

(2) Let f (x) = arctan(x). (a) Find P 2 n+1, 0 (x). (b) Give a formula for f (k)(0). (3) Let f (x) = log(x). (a) Use induction to find a formula for f (k)(x). (b) Find Pn, 1 (x). (c) Use the remainder theorem to decide which n you’ll need to use so that Pn, 1 (2/3) is within .01 of log(2/3). (4) Use Taylor polynomials around 0 to approximate cos(1) to within .001. Make sure and explain using the remainder term why your answer is within .001. (5) Calculate the following integrals. (a)

e^2 z^ sin(3x) dx. (b)

x log(x) dx. (c)

0 xe

x (^) dx.

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