Review for the Third Midterm of Math 150 B 11/24/2014 Problem 1 Recall that 1 1−x = ∑∞ n=0 x n for |x| < 1. Find a power series representation for the following functions and state the radius of convergence for the power series a) f(x) = x 2 (1+x)2 . b) f(x) = 2 1+4x2 . c) f(x) = x 4 2−x. d) f(x) = x 1+x2 . e) f(x) = 1 6+x . f) f(x) = x 2 27−x3 . Problem 2 Find a Taylor series with a = 0 for the given function and state the radius of conver- gence. You may use either the direct method (definition of a Taylor series) or known series. a) f(x) = ln(1 + x) b) f(x) = sin x x c) f(x) = x sin(3x). Problem 3 Find the radius of convergence and interval of convergence for the series ∑∞ n=1 (x+2)n n4n . Ans. Radius r=2, √ 2 − 2 < x < √ 2 + 2. Problem 4 Find the interval of convergence of the following power series. You must justify your answers. ∑∞ n=0 n2(x+4)n 23n . Ans. −12 < x < 4. Problem 5 For the function f(x) = 1/ √ x, find the fourth order Taylor polynomial with a=1. Problem 6 A curve has the parametric equations x = cos t, y = 1 + sin t, 0 ≤ t ≤ 2π a) Find dy dx when t = π 4 . b) Find the equation of the line tangent to the curve at t = π/4. Write it in y = mx+b form. c) Eliminate the parameter t to find a cartesian (x, y) equation of the curve. d) Using (c), or otherwise identify the curve. Problem 7 State whether the given series converges or diverges a) ∑∞ n=0 (−1) n+1 n22n n! . b) ∑∞ n=0 n(−3)n 4n−1 . c) ∑∞ n=1 sin n 2n2+n . Problem 8 1 Approximate the value of the integral ∫ 1 0 e−x 2 dx with an error no greater than 5×10−4. Ans. ∫ 1 0 e−x 2 dx = 1 − 1 3 + 1 5.2! − 1 7.3! + ... + (−1)n (2n+1)n! + .... n ≥ 5, for n=5 ∫ 1 0 e−x 2 dx ≈ 1 − 1 3 + 1 5.2! − 1 7.3! + 1 9.4! − 1 11.5! ≈ 0.747. Problem 9 Find the radius of convergence for the series ∑∞ n=1 nn(x−2)2n n! . Ans. R = 1√ e . Problem 10 Let f(x) = ∑∞ n=0 (x−1)n n2+1 . a) Calculate the domain of f. b) Calculate f ′(x). c) Calculate the domain of f ′. Problem 11 Let f(x) = ∑∞ n=0 cos n n! xn. a) Calculate the domain of f. b) Calculate f ′(x). c) Calculate ∫ f(x)dx. Problem 12 Using properties of series, known Maclaurin expansions of familiar functions and their arithmetic, calculate Maclaurin series for the following. a) ex 2 b) sin 2x c) ∫ x5 sin xdx d) cos x−1 x2 e) d((x+1) tan−1(x)) dx Problem 13 Calculate the Taylor polynomial T5(x), expanded at a=0, for f(x) = ∫ x 0 ln |sect + tan t|dt. Ans. T5(x) = x2 2 + x 4 4! . Problem 14 Suppose we only consider |x| ≤ 0.8. Find the best upper bound or maximum value you can for∣∣∣sin x − (x − x33! + x55! )∣∣∣ Same question: If ( x − x 3 3! + x 5 5! ) is used to approximate sin x for |x| ≤ 0.8. What is the maximum error? Explain what method you are using. Problem 15 The Taylor polynomial T5(x) of degree 5 for (4 + x) 3/2 is (4 + x)3/2 ≈ 8 + 3x + 3 16 x2 − 1 128 x3 + 3 4096 x4 − 3 32768 x5. a) Use this polynomial to find Taylor polynomials for (4 + ...

1. Review for the Third Midterm of Math 150 B 11/24/2014
Problem 1
Recall that 1
1−x =
∑∞
n=0 x
n for |x| < 1.
Find a power series representation for the following functions
and state the radius of
convergence for the power series
a) f(x) = x
2
(1+x)2
.
b) f(x) = 2
1+4x2
.
c) f(x) = x
4
2−x.
d) f(x) = x

2. 1+x2
.
e) f(x) = 1
6+x
.
f) f(x) = x
2
27−x3 .
Problem 2
Find a Taylor series with a = 0 for the given function and state
the radius of conver-
gence. You may use either the direct method (definition of a
Taylor series) or known
series.
a) f(x) = ln(1 + x)
b) f(x) = sin x
x
c) f(x) = x sin(3x).
Problem 3
Find the radius of convergence and interval of convergence for
the series
∑∞
n=1
(x+2)n
n4n
.

3. Ans. Radius r=2,
√
2 − 2 < x <
√
2 + 2. Problem 4
Find the interval of convergence of the following power series.
You must justify your
answers.
∑∞
n=0
n2(x+4)n
23n
.
Ans. −12 < x < 4.
Problem 5
For the function f(x) = 1/
√
x, find the fourth order Taylor polynomial with a=1.
Problem 6
A curve has the parametric equations
x = cos t, y = 1 + sin t, 0 ≤ t ≤ 2π
a) Find dy
dx
when t = π

4. 4
.
b) Find the equation of the line tangent to the curve at t = π/4.
Write it in y = mx+b
form.
c) Eliminate the parameter t to find a cartesian (x, y) equation
of the curve.
d) Using (c), or otherwise identify the curve.
Problem 7
State whether the given series converges or diverges
a)
∑∞
n=0 (−1)
n+1 n22n
n!
.
b)
∑∞
n=0
n(−3)n
4n−1
.
c)
∑∞
n=1
sin n

5. 2n2+n
.
Problem 8
1
Approximate the value of the integral
∫ 1
0
e−x
2
dx with an error no greater than 5×10−4.
Ans.
∫ 1
0
e−x
2
dx = 1 − 1
3
+ 1
5.2!
− 1
7.3!
+ ... +
(−1)n
(2n+1)n!

6. + .... n ≥ 5,
for n=5
∫ 1
0
e−x
2
dx ≈ 1 − 1
3
+ 1
5.2!
− 1
7.3!
+ 1
9.4!
− 1
11.5!
≈ 0.747.
Problem 9
Find the radius of convergence for the series
∑∞
n=1
nn(x−2)2n
n!
.

7. Ans. R = 1√
e
.
Problem 10
Let f(x) =
∑∞
n=0
(x−1)n
n2+1
.
a) Calculate the domain of f.
b) Calculate f ′(x).
c) Calculate the domain of f ′.
Problem 11
Let f(x) =
∑∞
n=0
cos n
n!
xn.
a) Calculate the domain of f.
b) Calculate f ′(x).
c) Calculate
∫
f(x)dx.
Problem 12

8. Using properties of series, known Maclaurin expansions of
familiar functions and their
arithmetic, calculate Maclaurin series for the following.
a) ex
2
b) sin 2x
c)
∫
x5 sin xdx
d) cos x−1
x2
e)
d((x+1) tan−1(x))
dx
Problem 13
Calculate the Taylor polynomial T5(x), expanded at a=0, for
f(x) =
∫ x
0
ln |sect + tan t|dt.
Ans. T5(x) =
x2
2
+ x
4

9. 4!
.
Problem 14
Suppose we only consider |x| ≤ 0.8. Find the best upper bound
or maximum value
you can for∣ ∣ ∣ sin x − (x − x33! + x55! )∣ ∣ ∣
Same question: If
(
x − x
3
3!
+ x
5
5!
)
is used to approximate sin x for |x| ≤ 0.8. What is
the maximum error? Explain what method you are using.
Problem 15
The Taylor polynomial T5(x) of degree 5 for (4 + x)
3/2 is
(4 + x)3/2 ≈ 8 + 3x + 3
16
x2 − 1
128

10. x3 + 3
4096
x4 − 3
32768
x5.
a) Use this polynomial to find Taylor polynomials for (4 + x)1/2
and (4 + x)5/2. b)
Also replace x by 4x to find a polynomial for (1 + x)3/2.
Problem 16
With regard to the power series
∑∞
n=1 (−1)
n+1n−
1
4 (2x)n.
2
a) Does this series converge or not at x = 1
2
? Why so?
b) Does this series converge or not at x = −1
2
? Why so?
c) Determine the radius of convergence of this series.

11. Problem 17
a) Find the arc length function for the curve y = x2 − 1
8
ln x starting at the point
(1, 1).
b) Find the arc length for the curve y = x
5
6
+ 1
10x3
over the interval [1, 2].
Problem 18
Evaluate the arc length of the curve y = xe−x
2
over the interval [0, 1]. (Do not eval-
uate the integral).
Problem 19
Set up the arc length of the ellipse x
2
a2
+ y
2
b2

12. = 1 as an integral. (Do not evaluate the
integral).
Problem 20
a) Find the Taylor Polynomial of degree 3 centered at a=1 for
f(x) =
√
x + 3.
b) Use your polynomial to approximate
√
4.8.
Problem 21
Find the equation of the tangent line to the curve x = 3 + ln t, y
= t2 + 2 at the
point (3, 3).
Problem 22
Find the arc length of the curve given in parametric form for −1
≤ t ≤ 1, x = t y =
t sin t
Problem 23
Find the arc length of the curve given in parametric form for 0 ≤
t ≤ π/4
x = cos 3t sin 4t, y = sin 3t cos 4t.
Problem 24
Find the arc length of the curve given in parametric form for 0 ≤
t ≤ π/2
x = 5 cos t − cos 5t, y = 5 sin t − sin 5t.
Ans. L=10.
Problem 25

13. Consider the curve parametrically as x = t3 − t, y = t4 − 5t2 + 4.
a) Find dy
dx
.
b) Find all the points where tangent lines are horizontal.
c) Find all the points where tangent line are vertical.
d) Find the slope of the tangent line at the point on the graph of
the curve when t = 1.
Problem 26
Consider the power series W(x) =
∑∞
n=1
(−n)n
n!
xn.
3
a) Determine the interval (open interval: you do not need to
decide on the behavior
at the end points) of the convergence for W(x). You may use
without proof that
lim→x∞(
n
n+1
)n = 1

14. e
.
b) Calculate a power series for W ′(x).
c) Let I = [0, 0.1]. You are given the information, that for the
secon derivative
|W (2)(x)| ≤ 2 for x in I. Using the formula error term of a
Taylor polynomial, show
that |W(x) − x| ≤ x2 for x in I.
Problem 27
a) Find the equations of the tangent lines to the cycloid x = rθ−r
sin θ, y = r−r cos θ
at θ = π
2
.
b) Find all the points on the graph of the cycloid at which the
tangent line is either
horizontal or vertical, and find the equations of the horizontal
lines.
c) Analyze the concavity of the cycloid (compute d
2y
dx2
).
d) Find the arc length of one arch of the cycloid traced by a
point on the circumfer-
ence of a wheel with radius r units (i.e. θ ∈ [0, 2π].
Ans. a) y = x + r(2 − π
2
). b) equation of tangent line y = 2r, c) d

15. 2y
dx2
= − 1
r(1−cos θ),
d) L = 8r.
Problem 28
Consider the polar curve r = θ2.
a) Find parametric equations x = f(θ), y = g(θ) for this curve.
b) Find the slope of the line tangent to this curve at the point (r,
θ) = (π2, π).
Problem 29
Let f(x) = cos x
a) Find the Taylor polynomial of degree 4 centered at a = π for
f(x).
b) Use this polynomial to approximate cos 9π
7
c) Find a bound on the error in using this polynomial to
estimate f(x) if π − 1
3
≤
x ≤ π + 1
3
.
Problem 30
Find the slope of the tangent line to the curve that satisfies
r = d

16. 1+e sin θ
,
at the point θ = π
3
, where the equation is in polar coordinates with d > 0 and
0 < e < 1.
Problem 31
a) Sketch the curve r =
√
θ, in polar coordinates, for 0 ≤ θ ≤ 2π.
Find the area enclosed by this curve.
Problem 32
4
Find a polar equation for the circle x2 + (y − 3)2 = 9.
Problem 33
Replace the following polar equations by equivalent Cartesian
equations.
a) r = 1 + 2r cos θ
b) r = 1 − cosθ
c) r = 4
2 cos θ−sin θ.
Problem 34

17. Find the area of the surface generated by revolving the curve C
a) y = 2
√
x, 1 ≤ x ≤ 2, about the x-axis.
b) x = y
3
3
, 0 ≤ x ≤ 1, about the y-axis.
Ans. a) S = 8π
3
(3
√
3 − 2
√
2). b)
π(
√
8−1)
9
.
Problem 35
Find surface area of the solid generated by revolving the smooth
curve C represented
by the parametric equations x(t) = 2t3, y(t) = 3t2, 0 ≤ t ≤ 1,
about the x-axis.
Ans. S = 24π
5

18. (
√
2 + 1).
Problem 36
I) Find the rectangular coordinates of each point whose polar
coordinates are:
a) (4, π
3
), b) (−2, 3π
4
).
II) Find Four polar coordinates (2 with r > 0 and 2 with r < 0) of
each point
whose rectangular coordinates are:
a) (4, −4), b) (−1, −
√
3).
Problem 37
Find the arc length of the logarithmic spiral represented by r =
f(θ) = e3θ from θ = 0
to θ = 2.
Ans. s =
√
10
3
(e6 − 1).
Problem 38
Find the area of the region enclosed by the the limacon r = 2 +

19. cos θ.
Ans. 9π
2
.
Problem 39
Find the area of the region that lies outside the cardioid r = 1 +
cos θ and inside of
the circle r = 3 cos θ.
Ans. π.
5