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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.
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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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