Explanation Given: The function f ( x ) = 1 4 + x 2 , a = 0 . Result used: The Taylor series centered at a is, f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) ( x − a ) 2 2 ! + f ‴ ( a ) ( x − a ) 3 3 ! + ⋯ = ∑ k = 0 ∞ f ( k ) ( a ) k ! ( x − a ) k Calculation: Obtain the first three nonzero terms of the Taylor series f ( x ) = 1 4 + x 2 centered at a = 0 . The value of f ( x ) = 1 4 + x 2 and its derivatives at a = 0 are computed in the below table, k f k ( x ) f k ( 0 ) 0 f ( x ) = 1 4 + x 2 f ( 0 ) = 1 4 1 f ′ ( x ) = − 2 x ( 4 + x 2 ) 2 f ′ ( 0 ) = 0 2 f ″ ( x ) = − 2 ( − 3 x 2 + 4 ) ( 4 + x 2 ) 3 f ″ ( 0 ) = − 1 8 3 f ‴ ( x ) = − 24 x ( − x 2 + 4 ) ( 4 + x 2 ) 4 f ‴ ( 0 ) = 0 4 f 4 ( x ) = 24 ( 5 x 4 − 40 x 2 + 16 ) ( 4 + x 2 ) 5 f 4 ( 0 ) = 3 8 5 f 5 ( x ) = 240 x

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

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

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 11, Problem 42RE

To determine

To write: The first three nonzero terms of the Taylor series for given function centered at a.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11, Problem 41RE

Chapter 11, Problem 43RE

Students have asked these similar questions

Evaluate the double integral ' √ √ (−2xy² + 3ry) dA R where R = {(x,y)| 1 ≤ x ≤ 3, 2 ≤ y ≤ 4} Double Integral Plot of integrand and Region R N 120 100 80- 60- 40 20 -20 -40 2 T 3 4 5123456 This plot is an example of the function over region R. The region and function identified in your problem will be slightly different. Answer = Round your answer to four decimal places.

Find Te²+ dydz 0 Write your answer in exact form.

xy² Find -dA, R = [0,3] × [−4,4] x²+1 Round your answer to four decimal places.

Chapter 11 Solutions

Calculus: Early Transcendentals (3rd Edition)

Ch. 11.1 - Verify that p3 satisfies p3(k)(a)=f(k)(a), for k =...Ch. 11.1 - Verify the following properties for f(x) = sin x...Ch. 11.1 - Why do the Taylor polynomials for sin x centered...Ch. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - At what point would you center the Taylor...Ch. 11.1 - In Example 7, find an approximate upper bound for...Ch. 11.1 - Suppose you use a second-order Taylor polynomial...Ch. 11.1 - Does the accuracy of an approximation given by a...Ch. 11.1 - The first three Taylor polynomials for f(x)=1+x...Ch. 11.1 - Suppose f(0) = 1, f(0) = 2, and f(0) = 1. Find the...

Additional Math Textbook Solutions

Find more solutions based on key concepts

Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard no...

Elementary Statistics (13th Edition)

Choose one of the answers in each case. In statistical inference, measurements are made on a (sample or popula...

Introductory Statistics

Consider a group of 20 people. If everyone shakes hands with everyone else, how many handshakes take place?

A First Course in Probability (10th Edition)

Find the point-slope form of the line passing through the given points. Use the first point as (x1, .y1). Plot ...

College Algebra with Modeling & Visualization (5th Edition)

The 37.5% of 89 .

Pre-Algebra Student Edition

Interpreting a P-Value In Exercises 3–8, the P-value for a hypothesis test is shown. Use the P-value to decide ...

Elementary Statistics: Picturing the World (7th Edition)

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 first three nonzero terms of the Taylor series for given function centered at a.

Solution Summary: The author calculates the first three nonzero terms of the Taylor series for given function centered at a.

Videos

To write: The first three nonzero terms of the Taylor series for given function centered at a.

Want to see the full answer?

Chapter 11 Solutions

Additional Math Textbook Solutions