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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