Unformatted text preview: Math 408d Spring 2007 —— Sato
Exam 2 —- March 9, 2007
Do all your work in your bluebook. Answers with insufﬁcient justiﬁcation will receive no credit. Support ALL of your answers. Calculators are allowed. 1. (10 points) (
Find a power series representation for the function f(x) = _
2 ﬁt 2. a. (10 points) Find a Maclaurin series for the function f(x) = x sin(x3). Show the steps you used.
b. (10 points) Use the Maclaurin series of part a to approximate the deﬁnite integral to an error of less than
j x sin(x3) dx
0 3. (10 points)
Find the center and radius of the circle given in polar coordinates by r = 2cos 6 - 25in 9 4. a. (10 points) Find the terms of the Taylor series for the function f(x) = ln(x+l) centered at a = 1 through
the 4th power. This would be the terms through n = 4.
b. (10 points) Use the third Taylor polynomial T3(x) for the function of part a, centered at a = l, to
approximate f(l .1). Estimate the error in the approximation. 6. Suppose that a curve in the plane is given by the parametric equations x = 12 + l, y = 1‘3 — 3.
a. (5 points) Find dy/dx and determine where the graph has horizontal and vertical tangents.
b. (.5 points) Find dzy/ctrz.
c. (5 points) Find the length of the curve from I: 0 to t = l.
d. (5 points) Set up, but do not evaluate, an integral to ﬁnd the surface area generated if the portion of the
curve in part a is rotated about the y-axis. 7. (10 points) Find the interval of convergence of the power series:
00 mm V‘
? ~— <x~1>
V120 8. (10 points) Find the ﬁrst 5 terms of the Taylor series for ﬁx) = ln(1+x) centered at a = l. ...
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- Spring '07