Calculus Maximus WS 11.2: Taylor Polynomials

Taylor Polynomials

On problems 1-5, find a Maclaurin polynomial of degree n for each of the following. 1. ( ) xf x e−= , 3n = 2. 2( ) xf x e= , 4n = 3. ( ) cosf x x= , 8n =

4. 2( ) xf x xe= , 4n = 5. 1( )1

f xx

=+

, 5n =

On problems 6-8, find a Taylor polynomial of degree n centered at =x c for each of the following.

6. f x( )x

= 1 , n = 5 , c = 1 7. f x( ) = ln x , n = 5 , c = 1 8. (f x) = sin x, n = 6 ,4

c = π

9. (Calculator Permitted) Use your answer from problem 1 to approximate f ⎜⎛ 1

2⎞⎟⎝ ⎠ to four decimal places.

10. (Calculator Permitted) Use your answer from problem 7 to approximate f (1.2) to four decimal places.

11. Suppose that function f x( ) is approximated near x = 0 by a sixth-degree Taylor polynomial

P6(x) = 3x − 4x3 +5x6 . Give the value of each of the following:

(a) f (0) (b) f ′(0) (c) f ′′′(0) (d) f (5) (0) (e) f (6) (0)

For problems 13-16, suppose that 22P x( ) a= +bx + cx is the second degree Taylor polynomial for the

function f about x = 0. What can you say about the signs of a, b, and c, if f has the graphs given below?

13. 14.

12. (Calculator Permitted) Suppose that g is a function which has continuous derivatives, and thatg(5) = 3, g ′(5) = −2, g ′′(5) =1, g ′′′(5) = −3

(a) What is the Taylor polynomial of degree 2 for g near 5? What is the Taylor polynomial of degree 3 near 5?

(b) Use the two polynomials that you found in part (a) to approximate g(4.9) .

15. 16.

Multiple Choice

20. If ( )0 0f = , ( )0 1f ′ = , ( )0 0f ′′ = , and ( )0 2f ′′′ = , then which of the following is the third-order

Taylor polynomial generated by ( )f x at 0x = ?

(A) 32x x+ (B) 31 13 2x x+ (C) 32

3x x+ (D) 32x x− (E) 31

3x x+

21. Which of the following is the coefficient of 4x in the Maclaurin polynomial generated by ( )cos 3x ?

(A) 278

(B) 9 (C) 124

(D) 0 (E) 278

−

22. Which of the following is the Taylor polynomial generated by ( ) cosf x x= at2

x π= ?

(A)

3 4

2 22 3! 4!

x xx

π ππ

⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎝ ⎠ ⎝ ⎠− − +⎜ ⎟⎝ ⎠ (B)

2 4

2 212! 4!

x xπ π⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠+ + (C)

2 4

2 212! 4!

x xπ π⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠− +

(D) 2 4

12 2

x xπ π⎛ ⎞ ⎛ ⎞− − + −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ (E)

3

22 6

xx

ππ

⎛ ⎞−⎜ ⎟⎛ ⎞ ⎝ ⎠− − +⎜ ⎟⎝ ⎠

17. Show how you can use the Taylor approximation3

sin3!

x x≈ − x for x near 0 to find0

lim sinx

x→ x

.

18. Use the fourth-degree Taylor approximation of2 4

cos x ≈ 12! 4!

− x x+ for x near 0 to find0

lim −1 cosx

xx→

.

19. Estimate the integral1

0∫sinttdt using a Taylor polynomial for sin t about t = 0 of degree 5.___

23. (Calculator Permitted) Which of the following gives the Maclaurin polynomial of order 5approximation to sin (1.5)?

(A) 0.965 (B) 0.985 (C) 0.997 (D) 1.001 (E) 1.005

24. Which of the following is the quadratic approximation for (f x) = e−x at x = 0?

2− +x x1(A) 12

21 (B) 12

2+ +x x1− −x x (C) 12

(D) 1+ x (E) 1− x

Lagrange Error Bound

1. (a) Find the fourth-degree Taylor polynomial for cos x about 0x = . Then use your polynomial toapproximate the value of cos0.8 , and use Taylor’s Theorem to determine the accuracy of theapproximation. Give three decimal places.

(b) Find the interval [a , b ] such that

(c) Could cos0.8 equal 0.695? Show why or why not.

2. (a) Write a fourth-degree Maclaurin polynomial for (f x) = ex . Then use your polynomial to

approximate e−1, and find a Lagrange error bound for the maximum error when x ≤1. Give three decimal places.

(b) Find an interval [ ,a b] such that a e−1≤ ≤ b.

3.

f ′( )5 8= , f ′′( ) =5 30, f ′′′( ) =5 48, and (4) (f x) ≤ 75 for all x in the interval [5,5.2].(a) Find the third-degree Taylor polynomial about x = 5 for (f x) .

(b) Use your answer to part (a) to estimate the value of f (5.2). What is the maximum possible error inmaking this estimate? Give three decimal places.

Let f be a function that has derivatives of all orders for all real numbers x. Assume that f ( )5 6= ,

(c) Find an interval [ ,a b] such that a f (5.≤ ≤2) b . Give three decimal places.

(d) Could f (5.2) equal 8.254? Show why or why not.

Review (Problems 4 - 7):

4. Find the first four nonzero terms of the power series fo (f x) = sin x centered at x = 34π .

5. Find the first four nonzero terms and the general term for the Maclaurin series for

(a) (f x) = xcos (x3 ) (b) (g x) 21

1 x=

+

7. Use the Maclaurin series for cos x to find0

1 coslimx

xx→

− .

8. The Taylor series about 3x = for a certain function f converges to ( )f x for all x in the interval ofconvergence. The nth derivative of f at 3x = is given by

( ) ( ) ( )( )1 !

35 3

nn

n

nf

n−

=+

and ( ) 133

f =

(a) Write the fourth-degree Taylor polynomial for f about 3x = .

(b) Find the radius of convergence of the Taylor series for f about 3x = .

(c) Show that the third-degree Taylor polynomial approximates ( )4f with an error less than 14000

.

9. Le f be a function that has derivatives of all orders on the interval (−1,1) . Assume f ( )0 1= ,

2f ′(0) = 1 , f ′′(0) = − 1

4, ( )

8f ′′′ 0 = 3 , and (4) (f x) ≤ 6 for all x in the interval (−1,1) .

(a) Find the third-degree Taylor polynomial about x = 0 for the function f.

(b) Use your answer to part (a) to estimate the value of f (0.5).

(d) What is the maximum possible error for the approximation made in part (b)?

6. Find the radius and interval of convergence for

(a) ( ) (x− −n )2

0

1 23

n

nn n

∞

=∑

∞(b) ∑ ( )2 !n ( − )

0 n5x

n=

10. Let f be the function defined by (f x) = x .(a) Find the second-degree Taylor polynomial about x = 4 for the function f.

(c) Find a bound on the error for the approximation in part (b). (b) Use your answer to part (a) to estimate the value of f (4.2) .

11. Let (f x)n=0 2

∞ xn=∑ n for all x for which the series converges.

(a) Find the interval of convergence of this series.

(b) Use the first three terms of this series to approximate 12

f −⎛ ⎞⎜ ⎟⎝ ⎠

.

(c) Estimate the error involved in the approximation in part (b). Show your reasoning.

polynomial for f about x = 0 .(a) Find P x( ) .

12. Let f be the function given by (f x) = cos⎛36

x π+ ⎞⎝⎜ ⎟⎠

and let (P x) be the fourth-degree Taylor

(b) Use the Lagrange error bound to show that 11 16 6 3000

⎛ ⎞f P− ⎛ ⎞ <⎜ ⎟⎝ ⎠ ⎜ ⎟⎝ ⎠.

13.1 2

0

= ∫I e−x dx that is within 0.001 of the actual value.(Review) Use series to find an estimate for

Justify.

14. The Taylor series about x = 5 for a certain function f converges to (f x) for all x in the interval ofconvergence. The nth derivative of f at x = 5 is given by

(n) ( ) = (n

− )(2 n + )

52

n1 !nf and ( )

2f 5 = 1 .

1000

Show that the sixth-degree Taylor polynomial for f about x = 5 approximates f (6) with an error less

than 1 .

15. Suppose a function f is approximated with a fourth-degree Taylor polynomial about x = 1. If the

maximum value of the fifth derivative between x = 1 and x = 3 is 0.01, that is, (5) (f x) < 0.01, then

the maximum error incurred using this approximation to compute f (3) is(A) 0.054 (B) 0.0054 (C) 0.26667 (D) 0.02667 (E) 0.00267

16. What are all the values of x for which the series1 n!

∞ n

n

x

=∑ converges?

(A) − ≤ x1 1≤ (B) − < x1 1< (C) − < x1 1≤ (D) − ≤ x1 1< (E) All real x

17.. The coefficient x6 in the Taylor series expansion about x = 0 for (f x) = sin (x2 ) is(A) − 1

6(B) 0 (C) 1

120(D) 1

6 (E) 1

of

18. The maximum error incurred by approximating the sum of the series 12! 3! 4! 5!

− +1 2 − 3 + 4 −L by the

sum of the first six terms is(A) 0.001190 (B) 0.006944 (C) 0.33333 (D) 0.125000 (E) None of these

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19. If f is a function such that ( ) ( )2sinf x x′ = , then the coefficient of 7x in the Taylor series for ( )f xabout 0x = is

(A) 17!

(B) 17

(C) 0 (D) 142

− (E) 17!

−

20. Now that you have finished the last question of the last “new concept” worksheet of your high schoolcareer, how do you feel? (Show your work)

(A) Relieved (B) Very Sad (C) Euphoric (D) Tired (E) All of these