WebAssign
Lesson 6-3 Taylor Series (Homework)
Current Score : – / 56
Due : Tuesday, August 5 2014 10:59 AM MDT
1. –/4 points
Consider the geometric series
∞
f(x) =
xn = 1 + x + x2 + x3 + x4 +
n=0
1. Find the following derivatives evaluated at 0.
f(0) =
f ' (0) =
f '' (0) =
f (3)(0) =
f (4)(0) =
f (5)(0) =
2. Look for a pattern and find the n-th order derivative at 0.
f (n)(0) =
Jaimos Skriletz
Math 175, section 31, Summer 2 2014
Instructor: Jaimos Skriletz
2. –/4 points
Consider a power series centered at 0
∞
f(x) =
an xn = a0 + a1 x + a2 x2 + a3 x3 + a4 x4 +
n=0
1. Find the following derivatives evaluated at 0.
f(0) =
f ' (0) =
f '' (0) =
f (3)(0) =
f (4)(0) =
f (5)(0) =
2. Look for a pattern and find the n-th order derivative at 0.
f (n)(0) =
Note: This gives a way to write the n-th coefficient of the Taylor Series in terms of the n-th
derivative of a function evaluated at x = 0.
3. –/3 points
A function is written as a Taylor series centered at 0
∞
f(x) =
Viewing Saved Work Revert to
Last Response
an xn = a0 + a1x + a2x2 + a3x3 +
n=0
1. if f(0) = −3 find the constant coefficient.
a0 =
2. if f '(0) = 6 find the coefficient of x
a1 =
3. If f ''(0) = 10 find the coefficient of x2.
a2 =
4. If f (3)(0) = −42 find the coefficent of x3.
a3 =
5. Write the 3rd order Taylor Polynomial for f centered at x = 0.
T 3(x) =
4. –/2 points
Suppose f(x) is a function such that
f(0) = 5
f '(0) = 0
f ''(0) = −4
(3)
f (0) = 0
f (4)(0) = 3
f (5)(0) = 0
f (6)(0) = −2
Write the 6th order Taylor Series approximate centered at x = 0 for this function
T 6(x) =
5. –/2 points
Suppose f(x) is a function such that
f(0) = −2
f '(0) = −4
f ''(0) = 3
f (3)(0) = 5
f (4)(0) = −6
f (5)(0) = −8
Write the 5th order Taylor Series approximate centered at x = 0 for this function
T 5(x) =
6. –/2 points
Consider the function f(x) = ex .
Find the 5th order Taylor series approximation centered at x = 0 for this function.
T 5(x) =
7. –/2 points
Consider the function f(x) = cos(x) .
Find the 6th order Taylor series approximation centered at x = 0 for this function.
T 6(x) =
8. –/2 points
Consider the function f(x) = sin(2x) .
Find the 7th order Taylor series approximation centered at x = 0 for this function.
T 7(x) =
9. –/2 points
Suppose the Taylor series centered at x = 0 for a function is
∞
f(x) =
(−1)n 2n
3n+1
n=0
xn =
1
2
4 2
8 3
− x+
x −
x +
3
9
27
81
Note: In the following you can enter in the formula (including powers and or factorials) for the
answer.
1. What is the 12th order coefficient?
a12 =
2. What is the exact value of
f (12)(0) =
10.–/2 points
Suppose the Taylor series centered at x = 0 for a function is
∞
f(x) =
n=0
(−1)n+1 4n x2n + 1 = −x + 2x3 − 2 x5 +
3
(2n)!
What is the exact value of f (9)(0) ?
Note: You can enter in the formula (including powers and/or factorials) for the answer.
f (9)(0) =
11.–/2 points
Consider the following Taylor Series centered at x = 0.
∞
A=
n=0
∞
B=
n=0
∞
C=
n=0
∞
D=
n=0
∞
E=
xn
(−1)n x2n
(2n)!
xn
n!
(−1)n x2n + 1
(2n + 1)!
(−1)n x2n
n=0
Match the following functions to their Taylor Series.
1
1 −x
­
ex
­
sin(x)
­
cos(x)
­
12.–/3 points
Using the Taylor series for ex
ex =
∞
n=0
xn = 1 + x + 1 x2 + 1 x3 + 1 x4 +
2
3!
4!
n!
2
Find the Taylor series for the function f(x) = e−x /2 .
1. Find the 6th order Taylor series approximation centered at x = 0.
T 6(x) =
2. Find the Taylor Series for this function centered at x = 0.
∞
2
e−x /2 =
n=0
13.–/2 points
Using the Taylor series for sin(x)
∞
sin(x) =
n=0
(−1)n x2n + 1 = x − 1 x3 + 1 x5 − 1 x7 +
3!
5!
7!
(2n + 1)!
Find the Taylor series for the function f(x) = 2sin(2x) .
1. Find the 5th order Taylor series approximation centered at x = 0.
T 5(x) =
2. Find the Taylor Series for this function centered at x = 0.
∞
2sin(2x) =
n=0
14.–/2 points
Using the geometric series
1
=
1 −x
∞
xn = 1 + x + x2 + x3 + x4 +
n=0
Find the Taylor series for the function ln(1 + x) .
d
1
Hint:
ln(1 + x) =
dx
1 +x
1. Find the 4th order Taylor series approximation centered at x = 0.
T 4(x) =
2. Find the Taylor Series for this function centered at x = 0.
∞
ln(1 + x) =
n=0
15.–/2 points
Using the geometric series
1
=
1 −x
∞
xn = 1 + x + x2 + x3 + x4 +
n=0
Find the Taylor series for the function tan−1(x) .
1
d
Hint:
tan−1(x) =
dx
1 + x2
1. Find the 7th order Taylor series approximation centered at x = 0.
T 7(x) =
2. Find the Taylor Series for this function centered at x = 0.
tan−1(x) =
∞
n=0
16.–/1 points
The error function is defined as
x
2
erf(x) =
π
2
e−t dt
0
Find the Taylor series centered at x = 0 for this function.
∞
2
erf(x) =
π
n=0
17.–/1 points
The sine integral is defined as
x
Si(x) =
0
sin(t)
dt
t
Find the Taylor series centered at x = 0 for this function.
∞
Si(x) =
n=0
18.–/2 pointsRogaC alcET2 10.7.005.
Find the Maclaurin series for f(x).
f(x) = cos 8x
∞
f(x) =
n=0
19.–/2 pointsRogaC alcET2 10.7.008.
Find the Maclaurin series for f(x).
f(x) = e6x
∞
f(x) =
n=0
20.–/2 pointsRogaC alcET2 10.7.004.
Find the Maclaurin series for f(x).
f(x) =
x
1 − 3x5
∞
f(x) =
n=0
21.–/2 pointsRogaC alcET2 10.7.019.
Find the terms through degree four of the Maclaurin series of f(x). Use multiplication and
substitution as necessary.
f(x) = ex sin(3x)
22.–/2 pointsRogaC alcET2 10.7.025.
Find the terms through degree four of the Maclaurin series of f(x). Use multiplication and
substitution as necessary.
f(x) = ex tan−1(5x)
23.–/2 pointsRogaC alcET2 10.7.026.
Find the terms through degree four of the Maclaurin series of f(x). Use multiplication and
substitution as necessary.
f(x) = sin(x3 − 3x)
24.–/2 pointsRogaC alcET2 10.7.039.
Find the Maclaurin series for f(x). Hint: Use the identity cos2x =
f(x) = cos2(7x)
∞
f(x) = 1 +
n=1
25.–/2 points
Consider the function
f(x) = x3 e−x
2
Find the exact value of the following
(Hint: Use the Taylor Series for f.)
f (9)(0) =
26.–/2 points
Consider the function
f(x) = x2sin(2x)
Find the exact value of the following
(Hint: Use the Taylor Series for f.)
f (7)(0) =
1
1 + cos(2x) .
2