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Quiz 21
Question 1
Find the Taylor polynomial P4 (x) centered at x = 0 for the function f(x) = x − 3 cos(x)
1 4
x
4
a)
−1 + 3x + x 2 +
b)
3−x−
c)
−3 + x +
3 2 1 4
x − x
2
8
d)
−3 + x +
3 2 1 4
x − x
4
2
e)
−3 − x +
3 2 1 4
x − x
4
4
3 2 1 4
x − x
8
2
Question 2
Find the Taylor polynomial P4 (x) centered at x = 0 for the function f(x) = 10 ln(cos(x))
5 4
x
6
a)
5x 2 +
b)
10 − 5x 2 +
c)
10 + 5x −
d)
−5x 2 −
e)
−10 + 5x 2 −
5 4
x
12
5 2 5 3 25 4
x + x −
x
4
8
64
5 4
x
6
5 4
x
12
Question 3
Find the Taylor polynomial Pn (x) centered at x = 0 for the function f(x) = e−6 x
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n
a)
∑
(−1)
b)
∑
k=0
n
c)
∑
k=0
n
d)
∑
k=0
n
e)
∑
k=1
6k x k
k!
k=1
n
k+1
6k+1 x k
k!
(−1) k 6k x k
k!
(−1) k x k
k!
6k x k
k!
Question 4
Find the Taylor polynomial Pn (x) centered at x = 0 for the function f(x) = cos(2 x)
n/2
a)
∑
k=0
n/2
b)
∑
k=0
n/2
c)
∑
k=0
n
d)
∑
k=0
n
e)
∑
k=0
(−1) k 22 k x 2 k
(2 k)!
(−1)
2k 2k 2k
2 x
(2 k)!
(−1) k 22 k+1 x 2 k+1
(2 k + 1)!
(−1) k 22 k−1 x 2 k−1
(2 k − 1)!
22 k x 2 k
(2 k)!
Question 5
Use the values in the table below and the formula for Taylor polynomials to give the 4th degree Taylor
polynomial for f centered at x = 0 .
f(0) f '(0) f ''(0) f '''(0) f (4) (0)
-5
-2
-2
4
-5
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a)
−5x 4 + 4x 3 − 2x 2 − 2x − 5
b)
−5 − 2x − x 2 +
4 3 5 4
x − x
4
3
c)
−5 − 2x − x 2 +
2 3
5 4
x −
x
24
3
d)
−5 − 2x − 2x 2 + 2x 3 −
e)
not enough information
5 4
x
6
Question 6
Let Pn be the n th Taylor Polynomial of the function f(x) centered at x = 0 . Assume that f is a function
such that ∣ f (n) (x) ∣≤ 1 for all n and x (the sine and cosine functions have this property.) Estimate the
1
1
error if P6 ( ) is used to approximate f( ) .
4
4
a)
1
( )
4
7!
b)
1
7!
c)
1
6!
d)
e)
7
1
( )
4
6!
6
1
( )
4
5!
5
Question 7
Assume that f(x) = ln(1 + x)
is the given function and that Pn represents the nth Taylor Polynomial
centered at x = 0 . Find the least integer n for which Pn (0.7) approximates ln(1.7) to within 0.01.
a)
11
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b)
10
c)
5
d)
7
e)
9
Question 8
−−−−
√
Use a Taylor polynomial centered at x = 0 to estimate e(1.6) to within 0.01.
a)
2.122
b)
2.522
c)
1.922
d)
2.322
e)
2.222
Question 9
Find the Lagrange form of the remainder R n for f(x) = e3 x
a)
27 x 5 e3 c
, |c| < |x|
40
b)
81 x 5 e3 c
, |c| < |x|
560
c)
81 x 6 e3 c
, |c| < |x|
80
d)
81 x 5 e3 c
, |c| < |x|
40
e)
27 x 6 e3 c
, |c| < |x|
80
centered at x = 0 when n = 5
Question 10
Expand g(x) = e−6 x in powers of (x + 1) .
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∞
a)
∑
(−1)
k+1 6 k+1
e 6 (x + 1)
k!
k=0
∞
b)
∑
k=0
∞
c)
∑
(−1) k e6 6k (x + 1)
k!
∞
d)
∑
k=0
∞
e)
∑
k=0
k
(−1)
k+1 6
e (x + 1)
k!
k
(−1)
k−1
6k (x + 1)
k!
k
k=0
(−1) k e6 (x + 1)
k!
k
k
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