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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