Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
3rd Edition
ISBN: 9780134996103
Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher: PEARSON
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Chapter 11.4, Problem 44E

Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10−4.

44. ∫ 0 0.2 ln   ( 1 + t ) t d t

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Use series to approximate the definite integral I. (Give your answer correct to 3 decimal places.) I = 2x cos (x) dr = I
(a) Evaluate the integral: Hint: = Your answer should be in the form kn, where k is an integer. What is the value of k? d dx —arctan(r) a₁ = a2 = 2 16 x² + 4 · 6²³ a3 = (b) Now, let's evaluate the same integral using a power series. First, find the power series for the function Then, integrate it from 0 to 2, and call the result S. S should be an infinite series. 16 f(x) = x² + 4 What are the first few terms of S? ao= a4 = dr 1 I²+1 (c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by k (the answer to (a)), you have found an estimate for the value of in terms of an infinite series. Approximate the value of by the first 5 terms. (d) What is the upper bound for your error of your estimate if you use the first 12 terms? (Use the alternating series estimation.)
Use series to approximate the definite integral to within the indicated accuracy: sin(x) dx, with an error < 10 4 Note: The answer you derive here should be the partial sum of an appropriate series (the number of terms determined by an error estimate). This number is not necessarily the correct value of the integral truncated to the correct number of decimal places. 0.234

Chapter 11 Solutions

Single Variable 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 - Prob. 2QCCh. 11.1 - Prob. 3QCCh. 11.1 - Write out the next two Taylor polynomials p4 and...Ch. 11.1 - Prob. 5QCCh. 11.1 - Prob. 6QCCh. 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 - Prob. 4E
Ch. 11.1 - Suppose f(0) = 1, f(0) = 0, f"(0) = 2, and f(3)(0)...Ch. 11.1 - Prob. 6ECh. 11.1 - Prob. 7ECh. 11.1 - Suppose you want to estimate 26 using a...Ch. 11.1 - Linear and quadratic approximation a. Find the...Ch. 11.1 - Linear and quadratic approximation a. Find the...Ch. 11.1 - Linear and quadratic approximation a. Find the...Ch. 11.1 - Linear and quadratic approximation a. Find the...Ch. 11.1 - Linear and quadratic approximation a. Find the...Ch. 11.1 - Linear and quadratic approximation a. Find the...Ch. 11.1 - Linear and quadratic approximation a. Find the...Ch. 11.1 - Linear and quadratic approximation a. Find the...Ch. 11.1 - Find the Taylor polynomials p1, , p4 centered at a...Ch. 11.1 - Find the Taylor polynomials p1, , p5 centered at a...Ch. 11.1 - Find the Taylor polynomials p3, , p4 centered at a...Ch. 11.1 - Find the Taylor polynomials p4 and p5 centered at...Ch. 11.1 - Find the Taylor polynomials p1, p2, and p3...Ch. 11.1 - Find the Taylor polynomials p3 and p4 centered at...Ch. 11.1 - Find the Taylor polynomial p3 centered at a = e...Ch. 11.1 - Find the Taylor polynomial p2 centered at a = 8...Ch. 11.1 - Graphing Taylor polynomials a. Find the nth-order...Ch. 11.1 - Prob. 26ECh. 11.1 - Graphing Taylor polynomials a. Find the nth-order...Ch. 11.1 - Graphing Taylor polynomials a. Find the nth-order...Ch. 11.1 - Approximations with Taylor polynomials a. Use the...Ch. 11.1 - Approximations with Taylor polynomials a. Use the...Ch. 11.1 - Approximations with Taylor polynomials a. Use the...Ch. 11.1 - Approximations with Taylor polynomials a. Use the...Ch. 11.1 - Approximations with Taylor polynomials a....Ch. 11.1 - Prob. 34ECh. 11.1 - Approximations with Taylor polynomials a....Ch. 11.1 - Prob. 36ECh. 11.1 - Approximations with Taylor polynomials a....Ch. 11.1 - Prob. 38ECh. 11.1 - Prob. 39ECh. 11.1 - Prob. 40ECh. 11.1 - Prob. 41ECh. 11.1 - Prob. 42ECh. 11.1 - Prob. 43ECh. 11.1 - Prob. 44ECh. 11.1 - Prob. 45ECh. 11.1 - Prob. 46ECh. 11.1 - Prob. 47ECh. 11.1 - Estimating errors Use the remainder to find a...Ch. 11.1 - Estimating errors Use the remainder to find a...Ch. 11.1 - Prob. 50ECh. 11.1 - Prob. 51ECh. 11.1 - Prob. 52ECh. 11.1 - Prob. 53ECh. 11.1 - Prob. 54ECh. 11.1 - Prob. 55ECh. 11.1 - Prob. 56ECh. 11.1 - Prob. 57ECh. 11.1 - Prob. 58ECh. 11.1 - Prob. 59ECh. 11.1 - Prob. 60ECh. 11.1 - Prob. 61ECh. 11.1 - Prob. 62ECh. 11.1 - Prob. 63ECh. 11.1 - Prob. 64ECh. 11.1 - Prob. 65ECh. 11.1 - Prob. 66ECh. 11.1 - Prob. 67ECh. 11.1 - Prob. 68ECh. 11.1 - Prob. 69ECh. 11.1 - Prob. 70ECh. 11.1 - Prob. 71ECh. 11.1 - Prob. 72ECh. 11.1 - Prob. 73ECh. 11.1 - Prob. 74ECh. 11.1 - Prob. 75ECh. 11.1 - Prob. 76ECh. 11.1 - Prob. 77ECh. 11.1 - Prob. 78ECh. 11.1 - Prob. 79ECh. 11.1 - Best center point Suppose you wish to approximate...Ch. 11.1 - Prob. 81ECh. 11.1 - Prob. 82ECh. 11.1 - Prob. 83ECh. 11.1 - Prob. 84ECh. 11.1 - Prob. 85ECh. 11.1 - Prob. 86ECh. 11.1 - Prob. 87ECh. 11.1 - A different kind of approximation When...Ch. 11.2 - By substituting x = 0 in the power series for g,...Ch. 11.2 - What are the radius and interval of convergence of...Ch. 11.2 - Use the result of Example 4 to write a series...Ch. 11.2 - Verify that the power series in Example 5b does...Ch. 11.2 - Write the first four terms of a power series with...Ch. 11.2 - Prob. 2ECh. 11.2 - Prob. 3ECh. 11.2 - Is k=0x2ka power series? If so, find the center a...Ch. 11.2 - Prob. 5ECh. 11.2 - Prob. 6ECh. 11.2 - Prob. 7ECh. 11.2 - Prob. 8ECh. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Prob. 12ECh. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Prob. 30ECh. 11.2 - Radius and interval of convergence Determine the...Ch. 11.2 - Prob. 32ECh. 11.2 - Prob. 33ECh. 11.2 - Prob. 34ECh. 11.2 - Prob. 35ECh. 11.2 - Prob. 36ECh. 11.2 - Radius of convergence Find the radius of...Ch. 11.2 - Prob. 38ECh. 11.2 - Radius of convergence Find the radius of...Ch. 11.2 - Radius of convergence Find the radius of...Ch. 11.2 - Combining power series Use the geometric series...Ch. 11.2 - Combining power series Use the geometric series...Ch. 11.2 - Combining power series Use the geometric series...Ch. 11.2 - Combining power series Use the geometric series...Ch. 11.2 - Combining power series Use the geometric series...Ch. 11.2 - Combining power series Use the geometric series...Ch. 11.2 - Combining power series Use the power series...Ch. 11.2 - Prob. 48ECh. 11.2 - Prob. 49ECh. 11.2 - Prob. 50ECh. 11.2 - Differentiating and integrating power series Find...Ch. 11.2 - Differentiating and integrating power series Find...Ch. 11.2 - Differentiating and integrating power series Find...Ch. 11.2 - Differentiating and integrating power series Find...Ch. 11.2 - Differentiating and integrating power series Find...Ch. 11.2 - Differentiating and integrating power series Find...Ch. 11.2 - Functions to power series Find power series...Ch. 11.2 - Functions to power series Find power series...Ch. 11.2 - Functions to power series Find power series...Ch. 11.2 - Prob. 60ECh. 11.2 - Prob. 61ECh. 11.2 - Prob. 62ECh. 11.2 - Prob. 63ECh. 11.2 - Prob. 64ECh. 11.2 - Prob. 65ECh. 11.2 - Prob. 66ECh. 11.2 - Prob. 67ECh. 11.2 - Series to functions Find the function represented...Ch. 11.2 - Prob. 69ECh. 11.2 - Prob. 70ECh. 11.2 - Prob. 71ECh. 11.2 - Prob. 72ECh. 11.2 - Exponential function In Section 11.3, we show that...Ch. 11.2 - Prob. 74ECh. 11.2 - Prob. 75ECh. 11.2 - Prob. 76ECh. 11.2 - Prob. 77ECh. 11.2 - Prob. 78ECh. 11.3 - Verify that if the Taylor series for f centered at...Ch. 11.3 - Prob. 2QCCh. 11.3 - Verify that the series k=0(1)k+1(x5)k4k+1 from...Ch. 11.3 - Find the first three terms of the Maclaurin series...Ch. 11.3 - Evaluate the binomial coefficients (32) and (123).Ch. 11.3 - Prob. 6QCCh. 11.3 - Prob. 1ECh. 11.3 - Prob. 2ECh. 11.3 - Prob. 3ECh. 11.3 - Prob. 4ECh. 11.3 - Suppose you know the Maclaurin series for f and...Ch. 11.3 - For what values of p does the Taylor series for...Ch. 11.3 - In terms of the remainder, what does it mean for a...Ch. 11.3 - Find the Maclaurin series for sin(x) using the...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Prob. 12ECh. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Prob. 20ECh. 11.3 - Prob. 21ECh. 11.3 - Prob. 22ECh. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Taylor series and interval of convergence a. Use...Ch. 11.3 - Prob. 27ECh. 11.3 - Prob. 28ECh. 11.3 - Prob. 29ECh. 11.3 - Taylor series centered at a 0 a. Find the first...Ch. 11.3 - Taylor series centered at a 0 a. Find the first...Ch. 11.3 - Taylor series centered at a 0 a. Find the first...Ch. 11.3 - Prob. 33ECh. 11.3 - Taylor series a. Use the definition of a Taylor...Ch. 11.3 - Prob. 35ECh. 11.3 - Prob. 36ECh. 11.3 - Manipulating Taylor series Use the Taylor series...Ch. 11.3 - Prob. 38ECh. 11.3 - Prob. 39ECh. 11.3 - Prob. 40ECh. 11.3 - Manipulating Taylor series Use the Taylor series...Ch. 11.3 - Prob. 42ECh. 11.3 - Prob. 43ECh. 11.3 - Prob. 44ECh. 11.3 - Binomial series a. Find the first four nonzero...Ch. 11.3 - Prob. 46ECh. 11.3 - Prob. 47ECh. 11.3 - Prob. 48ECh. 11.3 - Binomial series a. Find the first four nonzero...Ch. 11.3 - Binomial series a. Find the first four nonzero...Ch. 11.3 - Working with binomial series Use properties of...Ch. 11.3 - Prob. 52ECh. 11.3 - Prob. 53ECh. 11.3 - Prob. 54ECh. 11.3 - Prob. 55ECh. 11.3 - Prob. 56ECh. 11.3 - Working with binomial series Use properties of...Ch. 11.3 - Working with binomial series Use properties of...Ch. 11.3 - Prob. 59ECh. 11.3 - Prob. 60ECh. 11.3 - Prob. 61ECh. 11.3 - Prob. 62ECh. 11.3 - Prob. 63ECh. 11.3 - Prob. 64ECh. 11.3 - Prob. 65ECh. 11.3 - Prob. 66ECh. 11.3 - Prob. 67ECh. 11.3 - Prob. 68ECh. 11.3 - Prob. 69ECh. 11.3 - Prob. 70ECh. 11.3 - Any method a. Use any analytical method to find...Ch. 11.3 - Prob. 72ECh. 11.3 - Prob. 73ECh. 11.3 - Prob. 74ECh. 11.3 - Prob. 75ECh. 11.3 - Prob. 76ECh. 11.3 - Prob. 78ECh. 11.3 - Prob. 80ECh. 11.3 - Prob. 81ECh. 11.3 - Prob. 82ECh. 11.3 - Prob. 83ECh. 11.3 - Prob. 84ECh. 11.3 - Prob. 85ECh. 11.3 - Composition of series Use composition of series to...Ch. 11.3 - Prob. 87ECh. 11.3 - Prob. 88ECh. 11.3 - Prob. 89ECh. 11.3 - Prob. 90ECh. 11.3 - Prob. 91ECh. 11.4 - Use the Taylor series sin x = x - x3/6+ to verify...Ch. 11.4 - Prob. 2QCCh. 11.4 - Prob. 3QCCh. 11.4 - Prob. 1ECh. 11.4 - Prob. 2ECh. 11.4 - Prob. 3ECh. 11.4 - Prob. 4ECh. 11.4 - Prob. 5ECh. 11.4 - Prob. 6ECh. 11.4 - Prob. 7ECh. 11.4 - Prob. 8ECh. 11.4 - Prob. 9ECh. 11.4 - Prob. 10ECh. 11.4 - Limits Evaluate the following limits using Taylor...Ch. 11.4 - Prob. 12ECh. 11.4 - Prob. 13ECh. 11.4 - Prob. 14ECh. 11.4 - Prob. 15ECh. 11.4 - Prob. 16ECh. 11.4 - Limits Evaluate the following limits using Taylor...Ch. 11.4 - Prob. 18ECh. 11.4 - Limits Evaluate the following limits using Taylor...Ch. 11.4 - Prob. 20ECh. 11.4 - Prob. 21ECh. 11.4 - Prob. 22ECh. 11.4 - Prob. 23ECh. 11.4 - Prob. 24ECh. 11.4 - Power series for derivatives a. Differentiate the...Ch. 11.4 - Power series for derivatives a. Differentiate the...Ch. 11.4 - Prob. 27ECh. 11.4 - Prob. 28ECh. 11.4 - Power series for derivatives a. Differentiate the...Ch. 11.4 - Prob. 30ECh. 11.4 - Power series for derivatives a. Differentiate the...Ch. 11.4 - Prob. 32ECh. 11.4 - Differential equations a. Find a power series for...Ch. 11.4 - Prob. 34ECh. 11.4 - Prob. 35ECh. 11.4 - Differential equations a. Find a power series for...Ch. 11.4 - Approximating definite integrals Use a Taylor...Ch. 11.4 - Approximating definite integrals Use a Taylor...Ch. 11.4 - Approximating definite integrals Use a Taylor...Ch. 11.4 - Prob. 40ECh. 11.4 - Approximating definite integrals Use a Taylor...Ch. 11.4 - Prob. 42ECh. 11.4 - Prob. 43ECh. 11.4 - Approximating definite integrals Use a Taylor...Ch. 11.4 - Approximating real numbers Use an appropriate...Ch. 11.4 - Approximating real numbers Use an appropriate...Ch. 11.4 - Prob. 47ECh. 11.4 - Approximating real numbers Use an appropriate...Ch. 11.4 - Approximating real numbers Use an appropriate...Ch. 11.4 - Prob. 50ECh. 11.4 - Prob. 51ECh. 11.4 - Prob. 52ECh. 11.4 - Evaluating an infinite series Write the Taylor...Ch. 11.4 - Prob. 54ECh. 11.4 - Representing functions by power series Identify...Ch. 11.4 - Prob. 56ECh. 11.4 - Prob. 57ECh. 11.4 - Prob. 58ECh. 11.4 - Prob. 59ECh. 11.4 - Prob. 60ECh. 11.4 - Prob. 61ECh. 11.4 - Prob. 62ECh. 11.4 - Prob. 63ECh. 11.4 - Prob. 64ECh. 11.4 - Prob. 65ECh. 11.4 - Limits with a parameter Use Taylor series to...Ch. 11.4 - Prob. 67ECh. 11.4 - Prob. 68ECh. 11.4 - A limit by Taylor series Use Taylor series to...Ch. 11.4 - Prob. 70ECh. 11.4 - Prob. 71ECh. 11.4 - Prob. 72ECh. 11.4 - Prob. 73ECh. 11.4 - Prob. 74ECh. 11.4 - Prob. 75ECh. 11.4 - Prob. 76ECh. 11.4 - Prob. 77ECh. 11.4 - Sine integral function The function...Ch. 11.4 - Fresnel integrals The theory of optics gives rise...Ch. 11.4 - Prob. 80ECh. 11.4 - Prob. 81ECh. 11.4 - Prob. 83ECh. 11.4 - Prob. 84ECh. 11 - Explain why or why not Determine whether the...Ch. 11 - Taylor polynomials Find the nth-order Taylor...Ch. 11 - Taylor polynomials Find the nth-order Taylor...Ch. 11 - Taylor polynomials Find the nth-order Taylor...Ch. 11 - Taylor polynomials Find the nth-order Taylor...Ch. 11 - Taylor polynomials Find the nth-order Taylor...Ch. 11 - Taylor polynomials Find the nth-order Taylor...Ch. 11 - Taylor polynomials Find the nth-order Taylor...Ch. 11 - Prob. 9RECh. 11 - Approximations a. Find the Taylor polynomials of...Ch. 11 - Prob. 11RECh. 11 - Prob. 12RECh. 11 - Approximations a. Find the Taylor polynomials of...Ch. 11 - Prob. 14RECh. 11 - Prob. 15RECh. 11 - Prob. 16RECh. 11 - Prob. 17RECh. 11 - Prob. 18RECh. 11 - Prob. 19RECh. 11 - Prob. 20RECh. 11 - Radius and interval of convergence Use the Ratio...Ch. 11 - Prob. 22RECh. 11 - Radius and interval of convergence Use the Ratio...Ch. 11 - Radius and interval of convergence Use the Ratio...Ch. 11 - Radius and interval of convergence Use the Ratio...Ch. 11 - Radius and interval of convergence Use the Ratio...Ch. 11 - Radius of convergence Find the radius of...Ch. 11 - Radius of convergence Find the radius of...Ch. 11 - Prob. 29RECh. 11 - Prob. 30RECh. 11 - Prob. 31RECh. 11 - Prob. 32RECh. 11 - Prob. 33RECh. 11 - Power series from the geometric series Use the...Ch. 11 - Prob. 35RECh. 11 - Taylor series Write out the first three nonzero...Ch. 11 - Prob. 37RECh. 11 - Taylor series Write out the first three nonzero...Ch. 11 - Taylor series Write out the first three nonzero...Ch. 11 - Prob. 40RECh. 11 - Taylor series Write out the first three nonzero...Ch. 11 - Prob. 42RECh. 11 - Prob. 43RECh. 11 - Prob. 44RECh. 11 - Prob. 45RECh. 11 - Prob. 46RECh. 11 - Convergence Write the remainder term Rn(x) for the...Ch. 11 - Convergence Write the remainder term Rn(x) for the...Ch. 11 - Limits by power series Use Taylor series to...Ch. 11 - Prob. 50RECh. 11 - Limits by power series Use Taylor series to...Ch. 11 - Prob. 52RECh. 11 - Prob. 53RECh. 11 - Prob. 54RECh. 11 - Prob. 55RECh. 11 - Prob. 56RECh. 11 - Definite integrals by power series Use a Taylor...Ch. 11 - Prob. 58RECh. 11 - Approximating real numbers Use an appropriate...Ch. 11 - Prob. 60RECh. 11 - Approximating real numbers Use an appropriate...Ch. 11 - Prob. 62RECh. 11 - Prob. 63RECh. 11 - Prob. 64RECh. 11 - Prob. 65RECh. 11 - Prob. 66RE
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