a . Explanation Given: The function is f ( x ) = tanh x , where x = 0.5 . Calculation: Result used: The n th-order Taylor polynomial centered at a is p n ( x ) = ∑ k = 0 n f k ( a ) k ! ( x − a ) k (1) Calculation: Obtain the Taylor polynomial with n = 3 . Substitute n = 3 and a = 0 in equation (1) then the third order Taylor polynomial is as follows, p 3 ( x ) = ∑ k = 0 3 f k ( 0 ) k ! ( x − 0 ) k = f ( 0 ) + f ′ ( 0 ) ( x − 0 ) + ( 1 2 ! ) f ″ ( a ) ( x − 0 ) 2 + ( 1 3 ! ) f ‴ ( 0 ) ( x − 0 ) 3 p 3 ( x ) = f ( 0 ) + f ′ ( 0 ) ( x ) + ( 1 2 ! ) f ″ ( 0 ) ( x ) 2 + ( 1 3 ! ) f ‴ ( 0 ) ( x ) 3 (1) The value of f ( x ) = tanh x and its derivatives at x = 0 is computed as follows, f ( x ) = tanh x f ( 0 ) = tanh ( 0 ) = 0 Thus, f ( 0 ) = 0 . f ′ ( 0 ) computed as follows, f ′ ( x ) = sech 2 ( x ) = 1 − tanh 2 ( x ) f ′ ( 0 ) = 1 − tanh 2 ( 0 ) = 1 − 0 Thus, f ′ ( 0 ) = 1 . f ′ ′ ( 0 ) computed as follows, f ′ ′ ( x ) = 2 tanh ( x ) ⋅ sech 2 ( x ) = ( 2 tanh ( x ) 2 ( tanh 2 ( x ) − 1 ) ) f ′ ′ ( 0 ) = ( 2 tanh ( 0 ) 2 ⋅ ( tanh 2 ( 0 ) − 1 ) ) = 2 ⋅ 0 ⋅ ( 0 − 1 ) Thus, f ′ ′ ( 0 ) = 0 b . To determine To compute: The absolute error.

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 48E

To determine

To approximate: Taylor polynomial with n=3.

To determine

To compute: The absolute error.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.1, Problem 47E

Chapter 9.1, Problem 49E

Students have asked these similar questions

Numerical Method Evaluate x2/3 from x0=1.2 and xn=2.4 of order 3 using Simpson's 3/8 R.

please help me! i need help with part b. i really have been stuck and literally crying because I am terrible with approximation

Use Taylor's Theorem to obtain an upper bound for the error of the approximation. Then calculate the value of the error. (Round your answers to five decimal places.) π 2 R₁ ≤ R₁ = arccos (0.35)~ Need Help? Read It 0.35

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...

Additional Math Textbook Solutions

Find more solutions based on key concepts

In Example 1, what is the average velocity between t=2 and t=3? Example 1 Average Velocity A rock is launched v...

Calculus, Single Variable: Early Transcendentals (3rd Edition)

Which of the series in Exercises 13–46 converge, and which diverge? Give reasons for your answers. (When you ch...

University Calculus: Early Transcendentals (3rd Edition)

The ordered pair for the point R.

Glencoe Math Accelerated, Student Edition

Find the slopes of the following lines. The line going through the points (2,5)and(2,8).

Calculus & Its Applications (14th Edition)

The given statement is true or false: “A function is a relation between two sets D and R so that each element x...

Precalculus

If the upper bound theorem identifies 3 as the upper bound for the real zeros of fx=x4+8x3−4x2+7x−3

Precalculus

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.

To approximate: Taylor polynomial with n = 3 .

Solution Summary: The author explains the nth-order Taylor polynomial centered at a is mathrmtanh(0.5)approx 0.52083.