a . Explanation Given: The function f ( x ) = cos x , a = π 6 . Result used: The n th-order Taylor polynomial centered at a is shown below, p n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f n ( a ) n ! ( x − a ) n Calculation: The n th-order Taylor polynomial centered at π 6 is shown below, p n ( x ) = f ( π 6 ) + f ′ ( π 6 ) ( x − π 6 ) + f ″ ( π 6 ) 2 ! ( x − π 6 ) 2 + ⋯ + f n ( π 6 ) n ! ( x − π 6 ) n (1) Substitute π 6 for x in f ( x ) = cos x , f ( x ) = cos ( π 6 ) = 3 2 That is, f ( π 6 ) = 3 2 . The first derivative of f ( x ) = cos x is computed as follows, f ′ ( x ) = − sin x Substitute π 6 for x , f ′ ( π 6 ) = − sin ( π 6 ) = − ( 1 2 ) That is, f ′ ( π 6 ) = − 1 2 . The second derivative of f ( x ) = cos x is computed as follows, f ″ ( x ) = − cos x Substitute π 6 for x , f ″ ( π 6 ) = − cos ( π 6 ) = − 3 2 That is, f ″ ( π 6 ) = − 3 2 . The Taylor polynomial centered at π 6 for n = 0 , 1 and 2 is obtained as follows, Substitute 0 for n in equation (1) and obtain p 0 ( x ) , p 0 ( x ) = f ( a ) = f ( π 6 ) = 3 2 Thus, p 0 ( x ) = 3 2 b. To determine To sketch: The Taylor polynomial and the function.

Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 9.1, Problem 32E

To determine

To find: The nth-order Taylor polynomial centered at the point a for n=0,1,and 2.

To determine

To sketch: The Taylor polynomial and the function.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.1, Problem 31E

Chapter 9.1, Problem 33E

Students have asked these similar questions

Only 100% sure experts solve it correct complete solutions ok

rmine the immediate settlement for points A and B shown in figure below knowing that Aq,-200kN/m², E-20000kN/m², u=0.5, Depth of foundation (DF-0), thickness of layer below footing (H)=20m. 4m B 2m 2m A 2m + 2m 4m

sy = f(x) + + + + + + + + + X 3 4 5 7 8 9 The function of shown in the figure is continuous on the closed interval [0, 9] and differentiable on the open interval (0, 9). Which of the following points satisfies conclusions of both the Intermediate Value Theorem and the Mean Value Theorem for f on the closed interval [0, 9] ? (A A B B C D

Chapter 9 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 9.1 - Suppose you use a second-order Taylor polynomial...Ch. 9.1 - Does the accuracy of an approximation given by a...Ch. 9.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 9.1 - Prob. 4E Ch. 9.1 - How is the remainder Rn(x) in a Taylor polynomial...Ch. 9.1 - Explain how to estimate the remainder in an...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...Ch. 9.1 - Linear and quadratic approximation a. Find the...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The nth -order Taylor polynomial centered at the point a for n = 0 , 1 , and 2 .

Solution Summary: The nth-order Taylor polynomial centered at the point a=pi 6 and

Videos

To find: The nth-order Taylor polynomial centered at the point a for n=0,1,and 2.

To sketch: The Taylor polynomial and the function.

Want to see the full answer?

Chapter 9 Solutions