School

Ball State UniversityDepartment

Mathematical SciencesCourse Code

MATH 166Professor

AllStudy Guide

MidtermThis

**preview**shows half of the first page. to view the full**1 pages of the document.**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)n−1

5n−1

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

x−5

b.) f(x) = (1 −x)−2Hint: f(x) is d

dx [1

1−x].

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

e−x2dx as an inﬁnite series. Use

the ﬁrst 3 terms of this series to approximate

the integral and estimate the error in this

approximation.

6. Find the coeﬃcient of (x−4)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, 0≤t≤2π. 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+t5,1≤t≤3.

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+t5,1≤t≤3

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+t5,1≤t≤3

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 ≤θ≤2π.

15. Find the area of the shape found in the

problem above.

16. Consider the polar equation r= 4 cos θfor

0≤θ≤π/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 ﬁrst 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 ﬁrst 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

1−xwith 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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