Quiz 21
Question 1
Find the Taylor polynomial centered at for the function
a)
b)
c)
d)
e)
Question 2
Find the Taylor polynomial centered at for the function
a)
b)
c)
d)
e)
Question 3
Find the Taylor polynomial centered at for the function
(x)P4 x = 0 f(x) = x − 3 cos(x)
−1 + 3x + +x2 14
x4
3 − x − −32
x2 18
x4
−3 + x + −32
x2 18
x4
−3 + x + −32
x2 14
x4
−3 − x + −34
x2 14
x4
(x)P4 x = 0 f(x) = 10 ln(cos(x))
5 +x2 56
x4
10 − 5 +x2 512
x4
10 + 5x − + −54
x2 58
x3 2564
x4
−5 −x2 56
x4
−10 + 5 −x2 512
x4
(x)Pn x = 0 f(x) = e−6 x
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a)
b)
c)
d)
e)
Question 4
Find the Taylor polynomial centered at for the function
a)
b)
c)
d)
e)
Question 5
Use the values in the table below and the formula for Taylor polynomials to give the 4th degree Taylorpolynomial for centered at .
f(0) f '(0) f ''(0) f '''(0) f (4) (0)
-5 -2 -2 4 -5
∑k=1
n (−1) k+1 6k x k
k!
∑k=0
n 6k+1 xk
k!
∑k=0
n (−1) k 6k xk
k!
∑k=0
n (−1) k x k
k!
∑k=1
n 6k xk
k!
(x)Pn x = 0 f(x) = cos(2 x)
∑k=0
n/2 (−1) k 22 k x2 k
(2 k)!
∑k=0
n/2 (−1) 2 k 22 k x2 k
(2 k)!
∑k=0
n/2 (−1) k 22 k+1 x2 k+1
(2 k + 1)!
∑k=0
n (−1) k 22 k−1 x2 k−1
(2 k − 1)!
∑k=0
n 22 k x2 k
(2 k)!
f x = 0
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a)
b)
c)
d)
e) not enough information
Question 6
Let be the th Taylor Polynomial of the function centered at . Assume that is a functionsuch that for all and (the sine and cosine functions have this property.) Estimate the
error if is used to approximate .
a)
b)
c)
d)
e)
Question 7
Assume that is the given function and that represents the nth Taylor Polynomialcentered at . Find the least integer for which approximates to within 0.01.
a) 11
−5 + 4 − 2 − 2x − 5x4 x3 x2
−5 − 2x − + −x2 43
x3 54
x4
−5 − 2x − + −x2 23
x3 524
x4
−5 − 2x − 2 + 2 −x2 x3 56
x4
Pn n f(x) x = 0 f∣ (x) ∣≤ 1f (n) n x
( )P614
f( )14
( )14
7
7!
17!
16!
( )14
6
6!
( )14
5
5!
f(x) = ln(1 + x) Pn
x = 0 n (0.7)Pn ln(1.7)
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b) 10
c) 5
d) 7
e) 9
Question 8
Use a Taylor polynomial centered at to estimate 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 for centered at when
a) ,
b) ,
c) ,
d) ,
e) ,
Question 10
Expand in powers of .
x = 0 e(1.6)− −−−
√
R n f(x) = e3 x x = 0 n = 5
27 x5e3 c
40|c| < |x|
81 x5e3 c
560|c| < |x|
81 x6e3 c
80|c| < |x|
81 x5e3 c
40|c| < |x|
27 x6e3 c
80|c| < |x|
g(x) = e−6 x (x + 1)
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a)
b)
c)
d)
e)
∑k=0
∞ (−1) k+1 e66k+1 (x + 1) k
k!
∑k=0
∞ (−1) k e66k (x + 1) k
k!
∑k=0
∞ (−1) k+1 e6(x + 1) k
k!
∑k=0
∞ (−1) k−1 6k (x + 1) k
k!
∑k=0
∞ (−1) k e6(x + 1) k
k!
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