Explanation Theorem used: Estimate of the Remainder: Let n be a fixed positive integer. Suppose there exists a number M such that | f ( n + 1 ) ( c ) | ≤ M , for all c between a and x inclusive. The remainder in the n th-order Taylor polynomial for f centered at a satisfies | R n ( x ) | = | f ( x ) − p n ( x ) | ≤ M | x − a | n + 1 ( n + 1 ) ! . Formula used: The n th-order Taylor Polynomial centered at a is, p n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f n ( a ) n ! ( x − a ) n . Calculation: Consider the function is f ( x ) = cos x . The aim is to determine whether approximating cos ( π 12 ) is more accurate by using the Taylor polynomials centered at 0 or π 6 or not. Use the above formula and obtain the Taylor polynomial for cos x centered at x = 0 as follows: p n ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + ⋯ + f n ( a ) n ! ( x − a ) n = cos ( 0 ) + ( − sin ( 0 ) ) ( x − 0 ) + ( − cos ( 0 ) 2 ) ( x − 0 ) 2 + ( sin ( 0 ) 6 ) ( x − 0 ) 3 + ⋯ = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + ⋯ That is, p n ( x ) = 1 − x 2 2 ! + x 4 4 ! − x 6 6 ! + ⋯ . Consider different values for n and obtain the remainder in the n th-order Taylor polynomial for f centered at 0, where x = π 12 . n p n ( π 12 ) | p n ( π 12 ) − cos π 12 | 1 1.0000000000 3.41 × 10 − 2 2 0.9657305403 1.95 × 10 − 4 3 0.9657305403 1.95 × 10 − 4 4 0.9659262729 4.47 × 10 − 7 5 0.9659262729 4.47 × 10 − 7 6 0.9659258257 5.47 × 10 − 10 Table 1 From Table 1 observe that, the remainder in the n th-order Taylor polynomial for f centered at 0, where x = π 12 is accurate to 0. Thus, approximating cos ( π 12 ) is more accurate by using the Taylor polynomials centered at 0

Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996684

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

Publisher: PEARSON

See similar textbooks

Concept explainers

Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.

Videos

Question

Chapter 11.1, Problem 80E

To determine

Whether the approximation of cos(π12) is more accurate by using the Taylor polynomial centered at 0 or at π6, check the consistency with Theorem 11.2 and also to find whether the answer depends on the order of the polynomial or not.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 11.1, Problem 79E

Chapter 11.1, Problem 81E

Students have asked these similar questions

show step by step answer

Write the given third order linear equation as an equivalent system of first order equations with initial values. Use Y1 = Y, Y2 = y', and y3 = y". - - √ (3t¹ + 3 − t³)y" — y" + (3t² + 3)y' + (3t — 3t¹) y = 1 − 3t² \y(3) = 1, y′(3) = −2, y″(3) = −3 (8) - (888) - with initial values Y = If you don't get this in 3 tries, you can get a hint.

Question 2 1 pts Let A be the value of the triple integral SSS. (x³ y² z) dV where D is the region D bounded by the planes 3z + 5y = 15, 4z — 5y = 20, x = 0, x = 1, and z = 0. Then the value of sin(3A) is -0.003 0.496 -0.408 -0.420 0.384 -0.162 0.367 0.364

Chapter 11 Solutions

Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (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

Sample Size. In Exercises 29–36, find the sample size required to estimate the population mean. 29. Mean IQ of ...

Elementary Statistics (13th Edition)

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1-26. 1.

University Calculus: Early Transcendentals (4th Edition)

Mathematical Connections Explain why a number and a numeral are considered different.

A Problem Solving Approach To Mathematics For Elementary School Teachers (13th Edition)

Fill in each blank so that the resulting statement is true. If n is a counting number, bn, read ______, indicat...

College Algebra (7th Edition)

Whether the ‘Physicians Committee for Responsible Medicine’ has the potential to create a bias in a statistical...

Elementary Statistics

Classifying Types of Probability In Exercises 53–58, classify the statement as an example of classical probabil...

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.

Whether the approximation of cos ( π 12 ) is more accurate by using the Taylor polynomial centered at 0 or at π 6 , check the consistency with Theorem 11.2 and also to find whether the answer depends on the order of the polynomial or not.