Maclaurin series
The function f(x) = sin(2x) has a Maclaurin series. Find the
first 4 nonzero terms in the series, that is write down the Taylor
polynomial with 4 nonzero terms.
Answer(s) submitted:
• 2x-(4xˆ3)/3+(4xˆ5)/15-(8xˆ7)/315
(correct)
2. (1 point) Find the Maclaurin series of the function
f(x) = (9x
2
)e
−4x
(f(x) =
∞
∑
n=0
cnx
n
)
c1 =
c2 =
c3 =
c4 =
c5 =
Answer(s) submitted:
•
•
•
•
•
(incorrect)
3. (1 point) Match the series with the right expression. (Use
the Maclaurin series.)
1.
∞
∑
n=0
(−1)
n
1
3
2n+1
(2n+1)!
2.
∞
∑
n=0
1
3
n
n!
3.
∞
∑
n=0
(−1)
n
1
3
2n+1
2n+1
4.
∞
∑
n=0
(−1)
n
1
3
2n
(2n)!
A. cos
1
3
B. e
1/3
C. arctan
1
3
D. sin
1
3
Answer(s) submitted:
•
•
•
•
(incorrect)
4. (1 point) Match each of the Maclaurin series with right
function.
1.
∞
∑
n=0
(−1)
n2x
2n+1
(2n+1)!
2.
∞
∑
n=0
(−1)
n2
2n
x
2n
(2n)!
3.
∞
∑
n=0
2
n
x
n
n!
4.
∞
∑
n=0
(−1)
n2x
2n+1
2n+1
A. e
2x
B. cos(2x)
C. 2 arctan(x)
D. 2 sin(x)
Answer(s) submitted:
•
•
•
•
(incorrect)
5. (1 point)
Write out the first four terms of the Maclaurin series of f(x)
if
f(0) = −7, f
0
(0) = 1, f
00(0) = −4, f
000(0) = 4
f(x) = +···
Answer(s) submitted:
•
(incorrect)
6. (1 point) Find the Maclaurin series of the function
f(x) = 10x
3 −6x
2 −8x+6
(f(x) =
∞
∑
n=0
cnx
n
)
c0 =
c1 =
c2 =
1
c3 =
c4 =
Find the radius of convergence R = Enter INF if
the radius of covergence is infinity .
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
7. (1 point) The Taylor series for f(x) = x
3
at -3 is
∞
∑
n=0
cn(x+3)
n
.
Find the first few coefficients.
c0 =
c1 =
c2 =
c3 =
c4 =
Answer(s) submitted:
•
•
•
•
•
(incorrect)
8. (1 point) The Taylor series for f(x) = e
x
at a = −3 is
∞
∑
n=0
cn(x+3)
n
.
Find the first few coefficients.
c0 =
c1 =
c2 =
c3 =
c4 =
Answer(s) submitted:
•
•
•
•
•
(incorrect)
9. (1 point)
The function f(x) = lnx has a Taylor series at a = 3. Find the
first 4 nonzero terms in the series, that is write down the Taylor
polynomial with 4 nonzero terms.
Answer(s) submitted:
•
(incorrect)
10. (1 point) The Taylor series for f(x) = cos(x) at a =
π
4
is
∞
∑
n=0
cn(x−
π
4
)
n
.
Find the first few coefficients.
c0 =
c1 =
c2 =
c3 =
c4 =
Answer(s) submitted:
•
•
•
•
•
(incorrect)
11. (1 point) The Taylor series for f(x) = √
100+x at a = 0
is
∞
∑
n=0
cn(x)
n
.
Find the first few coefficients.
c0 =
c1 =
c2 =
c3 =
Find the error in approximating √
101 = f(1) using the third
degree Taylor polynomial of f at a = 0.
That is, find the error of the approximation √
101 ≈ T3(1).
The absolute value of the error is
Answer(s) submitted:
2
•
•
•
•
•
(incorrect)
12. (1 point) Use the binomial series to expand the function
f(x) = 1
(1−5x)
1/4
as a power series
∞
∑
n=0
cnx
n
Compute the following coefficients.
c0 =
c1 =
c2 =
c3 =
c4 =
Answer(s) submitted:
•
•
•
•
•
(incorrect)
13. (1 point)
The function f(x) = x
−8 has a Taylor series at a = 1. Find the
first 4 nonzero terms in the series, that is write down the Taylor
polynomial with 4 nonzero terms.
Answer(s) submitted:
•
(incorrect)
14. (1 point) Find the degree 3 Taylor polynomial T3(x) of
function
f(x) = (−7x+92)
4/3
at a = 4.
T3(x) =
Answer(s) submitted:
•
(incorrect)
15. (1 point) Compute the 9th derivative of
f(x) = arctan
x
3
2
at x = 0.
f
(9)
(0) =
Hint: Use the MacLaurin series for f(x).
Answer(s) submitted:
•
(incorrect)
16. (1 point)
Consider the function
f(x) = e
15x −1
x
.
a) Write the first 3 non zero terms of the MacLaurin series for
the function.
b) Use part a) to write the first 3 non zero terms of the
MacLaurin series for
g(x) = Z
e
15x −1
x
dx, g(0) = 0
Answer(s) submitted:
•
•
(incorrect)
17. (1 point) Let F(x) = Zx
0
sin(2t
2
) dt.
Find the MacLaurin polynomial of degree 7 for F(x).
Use this polynomial to estimate the value of
Z0.76
0
sin(2x
2
) dx.
Note: your answer to the last part needs to be correct to 9
decimal places.
Answer(s) submitted:
•
•
(incorrect)
3
18. (1 point)
Use a Maclaurin series derived in the text to derive the
Maclaurin series for the function f(x) = Z
sin(x)
x
dx, f(0) = 0.
Find the first 4 nonzero terms in the series, that is write down
the Taylor polynomial with 4 nonzero terms.
Answer(s) submitted:
•
(incorrect)
19. (1 point) Evaluate
lim
x→0
cos(x)−1+
x
2
2
4x
4
Hint: Using power series.
Answer(s) submitted:
•
(incorrect)
20. (1 point) Evaluate
lim
x→0
e
−3x
3
−1+3x
3 −
9
2
x
6
14x
9
Hint: Using power series.
Answer(s) submitted:
•
(incorrect)
21. (1 point)
a) Consider the function arctan(x
2
).
Write a partial sum for the power series which represents
this function consisting of the first 4 nonzero terms.
For example, if the series were ∑
∞
n=0
3
n
x
2n
, you would write
1+3x
2 +3
2
x
4 +3
3
x
6
. Also indicate the radius of convergence.
Partial Sum:
Radius of Convergence:
b) Use part a) to write the partial sum for the power series
which represents R
arctan(x
2
)dx.
Write the first 4 nonzero terms. Also indicate the radius of convergence.
Partial Sum:
Radius of Convergence:
c) Use part b) to approximate the integral R0.4
0
arctan(x
2
)dx.
Answer(s) submitted:
•
•
•
•
•
(incorrect)
22. (1 point) Represent the function f(x) = x
0.4
as a power
series:
∞
∑
n=0
cn(x−4)
n
Find the following coefficients:
c0 =
c1 =
c2 =
c3 =