1
Taylor Series
For the next problems in this section, you may use the following known power series:
x
e =
∞
X
xn
n=0
sin(x) =
n!
for all x
∞
X
(−1)n x(2n+1)
n=0
(2n + 1)!
for all x
∞
X
1
=
xn for |x| < 1
1 − x n=0
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1. Write a power series expansion for e−x . State the center and radius of convergence of the series.
2. Write a power series expansion for x2 cos(2x). State the center and radius of convergence of the
series.
3. Write a power series expansion for tan−1 (x)/x. State the center and radius of convergence of the
series.
4. Write the second degree Taylor polynomial for tan(x) centered at 0.
5. Write the third degree Taylor polynomial for ln(x) centered at 1.
6. Write the third degree Taylor polynomial for x5 − 6x4 + 3x2 − 2 centered at 2.
centered at 0. Can you do this
7.
Write down the the third degree Taylor polynomial for sin(x)
1−x
problem without taking any derivatives?
8. Recall that the error of the n-th Taylor polynomial is calculated by:
f n+1 (z)(x − a)n+1
(n + 1)!
where a is the center of the polynomial and z is some number in between x and a. Find a bound for
the error from using the 3-rd Taylor polynomial centered at x = 2 to approximate ln(2.5).
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Parametric Equations
9.
Consider the curve S given by the parametric equations x = t3 , y = 3t2 + 3t − 3.
A. Calculate the arc length of S from (0, 1) to (1, 3).
B. Find the slope of the tangent line to S at the point (1, 3).
2
d y
C. Find dx
2 at the point (1, 3). Use this to determine whether the S is concave up or concave down
at that point.
10.
Consider the curve S given by the parametric equations x = et , y = sin(t) + 1.
A. Write down (but do not evaluate) an integral which calculates the arc length of S from (1, 1) to
(e2 , sin(2) + 1).
B. Find the slope of the tangent line to S at the point (e, sin(1) + 1).
1
2
d y
C. Find dx
2 at the point (e, sin(1) + 1). Use this to determine whether the S is concave up or concave
down at that point.
11.
Consider the curve S given by the parametric equations:
x = 3t − 1, y = t3 + 1
Write an equation for S in terms of the Cartesian coordinates x and y.
12. Consider the curve S given by the parametric equations:
x = ln t, y = t2 + 1, t > 0
Write an equation for S in terms of the Cartesian coordinates x and y.
13. Find all local minimums of the curve defined by the parametric equations:
x = t3 − 10t + 1, y = t2
14. The following three graphs correspond to the following three sets of parametric equations (in all
cases t varies from −3 to 3):
x = sin(t), y = t
x = cos(t), y = t1/3
x = t, y = (1/4)t4 − (1/3)t3 + (1/2)t2 − t + 1
x = t2 , y = t3
Identify which graph corresponds to which set of equations.
1.5
1.0
0.5
-1.0
-0.5
0.5
1.0
-0.5
-1.0
-1.5
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15.
Polar Coordinates
Consider the curve given by the polar equation r = sin(5θ).
A. Draw a picture of the polar curve.
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B. Find the area bounded by the polar curve.
16.
Consider the curve S given by the polar equation r = 3 sin(θ) + 2 cos(θ).
A. Write an equation for the curve in the Cartesian coordinates x and y.
B. Calculate the arc length of S for angles 0 ≤ θ ≤ π.
C. Find the slope of the tangent line to S at the angle π/2.
17.
Consider the curve S given by the polar equation r = θ + tan(θ).
A. Write down, but do not evaluate, an integral which calculates the arc length of S for angles 0 ≤
θ ≤ π.
B. Find the slope of the tangent line to S at the angle π/3.
18. Find the area of the region bounded by the polar curve r = sin(θ) + 32
19. Write an equation for the polar curve r = p
sec(θ) + csc(θ).
20. Write an equation for the polar curve r = tan(θ) with 0 < θ < π/2.
21.
Write an equation in polar coordinates which gives the same curve as the one given by the
equation xy = xy in Cartesian coordinates.
22. The following three graphs are of the following three polar curves (in all cases θ varies between
0 and π):
r = cos(θ) + sin(θ)
r = −θ2
r = 1 − sin(θ)
r=θ
0.25
0.20
0.15
0.10
0.05
-1.0
-0.5
0.0
0.5
1.0
Identify which graph corresponds to which curve.
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