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Midterm

MATH 166 Ball State nbd3SolutionExam


Department
Mathematical Sciences
Course Code
MATH 166
Professor
All
Study Guide
Midterm

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Math 166 Exam III Review
1. For the following power series, find the ra-
dius and interval of convergence.
a.)
P
n=0
(πx)n
n2+ 2
b.)
P
n=0
n(x+ 2)n1
5n1
c.)
P
n=0
(3)nxn
n+ 1
d.)
P
n=0
5nxn
(n+ 1)!
2. Find a power series with center c= 0 for
the following functions and find the corre-
sponding radius of convergence.
a.) f(x) = x
x5
b.) f(x) = (1 x)2Hint: f(x) is d
dx [1
1x].
c.) f(x) = arctan x
3. Write Z0.5
0
1
1 + x4dx as an infinite series.
4. Find a series representation for Zsin x2
xdx
5. Express Z1
0
ex2dx as an innite series. Use
the rst 3 terms of this series to approximate
the integral and estimate the error in this
approximation.
6. Find the coecient of (x4)3of the Taylor
Series for the function f(x) = xat a= 4.
7. Let f(x) = e2x.
a.) Find T3(x), the 3rd order Taylor poly-
nomial centered at a= 2.
b.) Use Taylors Remainder Theorem to es-
timate the accuracy of the approximation
T3(2.5).
8. Graph the parametric curve given by
x= 2 sin t, y = 3 + cos t, 0t2π. Be
sure to label the direction the curve is
traced.
9. Find the cartesian equation of the line tan-
gent to the parametric curve x= cos t,
y= 3 sin tat the point t=π/6.
10. Give the integral (you do not have to eval-
uate it) that represents the arclength of the
curve x= 2 + t2, y = cos t+t5,1t3.
11. Give the integral (you do not have to eval-
uate it) that represents the surface area of
the curve x= 2+t2, y = cos t+t5,1t3
rotated about the x-axis.
12. Give the integral (you do not have to eval-
uate it) that represents the surface area of
the curve x= 2+t2, y = cos t+t5,1t3
rotated about the y-axis.
13. Convert the polar equation r= 3 cos θto
cartesian form and then sketch the graph.
14. Sketch the graph of the polar equation
r= 1 + cos θfor 0 θ2π.
15. Find the area of the shape found in the
problem above.
16. Consider the polar equation r= 4 cos θfor
0θπ/4.
a) Give an integral (you do not have to eval-
uate it) for the length of arc by first con-
verting the polar equation into parametric
equations.
b) Give an integral (you do not have to eval-
uate it) for the surface area of the surface
obtained by rotating the curve about the x-
axis by rst converting the polar equation
into parametric equations.
c) Give an integral (you do not have to eval-
uate it) for the surface area of the surface
obtained by rotating the curve about the y-
axis by rst converting the polar equation
into parametric equations.
17. You should memorize the following series
along with the corresponding radius of con-
vergence for the indicated center a.
a.) exwith a= 0
b.) sin xwith a= 0
c.) cos xwith a= 0
d.) 1
1xwith a= 0
e.) 1
1+xwith a= 0
f.) Taylor series formula for any function
f(x) with center a.
1
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