Explanation Result used: The Taylor series at 0 for f ( x ) = e − x is e − x = 1 − x + 1 2 ! x 2 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! x n + ⋅ ⋅ ⋅ . Theorem used: Error Estimation for Alternating Series If x 0 is a number in the interval of convergence for f ( x ) = a 0 + a 1 x + a 2 x 2 + ⋅ ⋅ ⋅ + a k x k + ⋅ ⋅ ⋅ and the terms in the series f ( x 0 ) = a 0 + a 1 x 0 + a 2 x 0 2 + ⋅ ⋅ ⋅ + a k x 0 k + ⋅ ⋅ ⋅ are alternating in sign and decreasing in absolute value, then the error in the approximation f ( x 0 ) ≈ a 0 + a 1 x 0 + a 2 x 0 2 + ⋅ ⋅ ⋅ + a n x 0 n is strictly less than the absolute value of the next term. That is, | R n ( x 0 ) | < | a n + 1 x 0 n + 1 | . Calculation: The integral ∫ 0 0.4 x 2 e − x 2 d x . Let z = x 2 . Substitute z = x 2 in f ( x ) = e − x 2 , that is, f ( z ) = e − z . Since the Taylor series at 0 for e − x = 1 − x + 1 2 ! x 2 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! x n + ⋅ ⋅ ⋅ , then the value of f ( z ) is f ( z ) = 1 − z + 1 2 ! z 2 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! z n + ⋅ ⋅ ⋅ . Replace z by x 2 in f ( z ) = 1 − z + 1 2 ! z 2 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! z n + ⋅ ⋅ ⋅ and simplify it as follows. f ( x ) = 1 − x 2 + 1 2 ! ( x 2 ) 2 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! ( x 2 ) n + ⋅ ⋅ ⋅ = 1 − x 2 + 1 2 ! x 4 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! x 2 n + ⋅ ⋅ ⋅ Multiply each term of the series e − x 2 = 1 − x 2 + 1 2 ! x 4 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! x 2 n + ⋅ ⋅ ⋅ by x 2 and simplify as follows. x 2 e − x 2 = x 2 ( 1 − x 2 + 1 2 ! x 4 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! x 2 n + ⋅ ⋅ ⋅ ) = x 2 − x 2 + 2 + 1 2 ! x 4 + 2 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! x 2 n + 2 + ⋅ ⋅ ⋅ = x 2 − x 4 + 1 2 ! x 6 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! x 2 n + 2 + ⋅ ⋅ ⋅ Integrate on both sides of the equation x 2 e − x 2 = x 2 − x 4 + 1 2 ! x 6 − ⋅ ⋅ ⋅ + ( − 1 ) n n ! x 2 n + 2 + ⋅ ⋅ ⋅

14th Edition

ISBN: 9780135903896

Author: Barnett

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 10.4, Problem 37E

To determine

To approximate: The integral ∫00.4x2e−x2 dx with an error of no more than 0.0005.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.4, Problem 36E

Chapter 10.4, Problem 38E

Students have asked these similar questions

2) Perform the following index shifts: no anx as a series that starts at n = 2 i. Write y = ii. Write y = -2an (x + 1)" as a series that starts at n = 0 iii. Write y = E=3(−1)″ (x − 4)¹ as a series that starts at n = 0 1 iv. Write y = as a series that starts at n = 5 n=1 3n-1 v. Write y = Σo(-1)" as a series that starts at n = 2

If the specific problem does not include instruction in rounding off, express your answer (nonterminating or repeating decimal) using 10 digits as seen in your scientific calculator.

Chapter 10 Solutions

CALCULUS FOR BUSINESS...-MYLAB ACESS

Ch. 10.1 - Find the nth derivative of f(x)=lnx.Ch. 10.1 - Prob. 2MP Ch. 10.1 - Prob. 3MP Ch. 10.1 - Find the second-degree Taylor polynomial at a = 8...Ch. 10.1 - Prob. 5MP Ch. 10.1 - Prob. 1ED Ch. 10.1 - (A)Let p(x) be a polynomial of degree n 1....Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E

Ch. 10.1 - Prob. 4E Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - Prob. 7E Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E Ch. 10.1 - Prob. 10E Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - Prob. 13E Ch. 10.1 - Prob. 14E Ch. 10.1 - In Problems 1316, find f(3)(x). 15.f(x)=ex Ch. 10.1 - In Problems 1316, find f(3)(x). 16.f(x)=x Ch. 10.1 - Prob. 17E Ch. 10.1 - In Problems 1720, find f4(x). 18.f(x)=e5x Ch. 10.1 - Prob. 19E Ch. 10.1 - In Problems 1720, find f4(x). 20.f(x)=12+x Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - In Problems 2128, find the indicated Taylor...Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - Prob. 32E Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Use the third-degree Taylor polynomial at 0 for...Ch. 10.1 - Prob. 41E Ch. 10.1 - Use the third-degree Taylor polynomial at 4 for...Ch. 10.1 - Prob. 43E Ch. 10.1 - Prob. 44E Ch. 10.1 - Prob. 45E Ch. 10.1 - Prob. 46E Ch. 10.1 - Prob. 47E Ch. 10.1 - Prob. 48E Ch. 10.1 - Prob. 49E Ch. 10.1 - Prob. 50E Ch. 10.1 - Prob. 51E Ch. 10.1 - Prob. 52E Ch. 10.1 - Prob. 53E Ch. 10.1 - Prob. 54E Ch. 10.1 - Prob. 55E Ch. 10.1 - Prob. 56E Ch. 10.1 - Prob. 57E Ch. 10.1 - Prob. 58E Ch. 10.1 - Prob. 59E Ch. 10.1 - Prob. 60E Ch. 10.1 - Prob. 61E Ch. 10.1 - Prob. 62E Ch. 10.1 - Prob. 63E Ch. 10.1 - Prob. 64E Ch. 10.1 - Prob. 65E Ch. 10.1 - Prob. 66E Ch. 10.1 - Prob. 67E Ch. 10.1 - Prob. 68E Ch. 10.1 - Prob. 69E Ch. 10.1 - Prob. 70E Ch. 10.1 - Prob. 71E Ch. 10.1 - Consider f(x) = ln (1 + x) and its third-degree...Ch. 10.1 - Prob. 73E Ch. 10.1 - Prob. 74E Ch. 10.1 - Prob. 75E Ch. 10.1 - Prob. 76E Ch. 10.1 - Prob. 77E Ch. 10.1 - Prob. 78E Ch. 10.1 - Prob. 79E Ch. 10.1 - Prob. 80E Ch. 10.1 - Prob. 81E Ch. 10.1 - Average price. Given the demand equation...Ch. 10.1 - Prob. 83E Ch. 10.1 - Prob. 84E Ch. 10.1 - Prob. 85E Ch. 10.1 - Prob. 86E Ch. 10.1 - Prob. 87E Ch. 10.1 - Prob. 88E Ch. 10.1 - Prob. 89E Ch. 10.1 - Prob. 90E Ch. 10.1 - Prob. 91E Ch. 10.1 - Prob. 92E Ch. 10.1 - Prob. 93E Ch. 10.1 - Prob. 94E Ch. 10.1 - Prob. 95E Ch. 10.1 - Prob. 96E Ch. 10.1 - Prob. 97E Ch. 10.1 - Prob. 98E Ch. 10.2 - Prob. 1MP Ch. 10.2 - Prob. 2MP Ch. 10.2 - Prob. 3MP Ch. 10.2 - Prob. 1ED Ch. 10.2 - (A)The six functions pn(x)=1+x++xn, n = 1, 2, , 6,...Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - Prob. 4E Ch. 10.2 - Prob. 5E Ch. 10.2 - Prob. 6E Ch. 10.2 - Prob. 7E Ch. 10.2 - Prob. 8E Ch. 10.2 - Prob. 9E Ch. 10.2 - Prob. 10E Ch. 10.2 - Prob. 11E Ch. 10.2 - Prob. 12E Ch. 10.2 - Prob. 13E Ch. 10.2 - Prob. 14E Ch. 10.2 - Prob. 15E Ch. 10.2 - Prob. 16E Ch. 10.2 - Prob. 17E Ch. 10.2 - Prob. 18E Ch. 10.2 - Prob. 19E Ch. 10.2 - Prob. 20E Ch. 10.2 - Prob. 21E Ch. 10.2 - Prob. 22E Ch. 10.2 - Prob. 23E Ch. 10.2 - Prob. 24E Ch. 10.2 - Prob. 25E Ch. 10.2 - Prob. 26E Ch. 10.2 - Prob. 27E Ch. 10.2 - Prob. 28E Ch. 10.2 - Prob. 29E Ch. 10.2 - Prob. 30E Ch. 10.2 - Prob. 31E Ch. 10.2 - (A) Graph the nth-degree Taylor polynomials at 0...Ch. 10.2 - Prob. 33E Ch. 10.2 - Prob. 34E Ch. 10.2 - In Problems 3338, find the nth-degree Taylor...Ch. 10.2 - Prob. 36E Ch. 10.2 - Prob. 37E Ch. 10.2 - Prob. 38E Ch. 10.2 - Prob. 39E Ch. 10.2 - Prob. 40E Ch. 10.2 - Prob. 41E Ch. 10.2 - Prob. 42E Ch. 10.2 - (A) Find the interval of convergence of the Taylor...Ch. 10.2 - Prob. 44E Ch. 10.2 - Prob. 45E Ch. 10.2 - Prob. 46E Ch. 10.2 - Prob. 47E Ch. 10.2 - Prob. 48E Ch. 10.2 - Prob. 49E Ch. 10.2 - Problems 4750 require a basic knowledge of the...Ch. 10.3 - Prob. 1MP Ch. 10.3 - Find the Taylor series at 0 for f(x) = 3x3 ln(1 ...Ch. 10.3 - Prob. 3MP Ch. 10.3 - Prob. 4MP Ch. 10.3 - Prob. 5MP Ch. 10.3 - Prob. 6MP Ch. 10.3 - Prob. 7MP Ch. 10.3 - Prob. 8MP Ch. 10.3 - Prob. 1ED Ch. 10.3 - Prob. 2ED Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - Prob. 3E Ch. 10.3 - Prob. 4E Ch. 10.3 - Prob. 5E Ch. 10.3 - Prob. 6E Ch. 10.3 - Prob. 7E Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - Prob. 10E Ch. 10.3 - Prob. 11E Ch. 10.3 - Prob. 12E Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Solve the problems by performing operations on the...Ch. 10.3 - Prob. 18E Ch. 10.3 - Prob. 19E Ch. 10.3 - Prob. 20E Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - Prob. 33E Ch. 10.3 - Find the Taylor series at 0 for (A) f(x)=x1x2 (B)...Ch. 10.3 - Prob. 35E Ch. 10.3 - If f(x) satisfies f(x) = ln (1 + x2) and f(0) = 1,...Ch. 10.3 - Prob. 37E Ch. 10.3 - Prob. 38E Ch. 10.3 - Prob. 39E Ch. 10.3 - Prob. 40E Ch. 10.3 - Prob. 41E Ch. 10.3 - Prob. 42E Ch. 10.3 - Prob. 43E Ch. 10.3 - Prob. 44E Ch. 10.3 - Prob. 45E Ch. 10.3 - Prob. 46E Ch. 10.3 - Prob. 47E Ch. 10.3 - Prob. 48E Ch. 10.3 - Prob. 49E Ch. 10.3 - Prob. 50E Ch. 10.3 - Prob. 51E Ch. 10.3 - Prob. 52E Ch. 10.3 - Prob. 53E Ch. 10.3 - Prob. 54E Ch. 10.3 - Prob. 55E Ch. 10.3 - Prob. 56E Ch. 10.3 - Prob. 57E Ch. 10.3 - Prob. 58E Ch. 10.3 - Prob. 59E Ch. 10.3 - Prob. 60E Ch. 10.3 - Prob. 61E Ch. 10.3 - Prob. 62E Ch. 10.3 - Prob. 63E Ch. 10.3 - Prob. 64E Ch. 10.3 - Prob. 65E Ch. 10.3 - Prob. 66E Ch. 10.4 - Prob. 1MP Ch. 10.4 - Prob. 2MP Ch. 10.4 - Prob. 3MP Ch. 10.4 - Prob. 4MP Ch. 10.4 - Prob. 1ED Ch. 10.4 - Suppose you wish to use a Taylor series for...Ch. 10.4 - Prob. 1E Ch. 10.4 - Prob. 2E Ch. 10.4 - Prob. 3E Ch. 10.4 - Prob. 4E Ch. 10.4 - Prob. 5E Ch. 10.4 - Prob. 6E Ch. 10.4 - Prob. 7E Ch. 10.4 - Prob. 8E Ch. 10.4 - Prob. 9E Ch. 10.4 - Prob. 10E Ch. 10.4 - Prob. 11E Ch. 10.4 - Prob. 12E Ch. 10.4 - Prob. 13E Ch. 10.4 - Prob. 14E Ch. 10.4 - Prob. 15E Ch. 10.4 - Prob. 16E Ch. 10.4 - Prob. 17E Ch. 10.4 - Prob. 18E Ch. 10.4 - Prob. 19E Ch. 10.4 - Prob. 20E Ch. 10.4 - Prob. 21E Ch. 10.4 - Prob. 22E Ch. 10.4 - Prob. 23E Ch. 10.4 - Prob. 24E Ch. 10.4 - Prob. 25E Ch. 10.4 - Prob. 26E Ch. 10.4 - Prob. 27E Ch. 10.4 - Prob. 28E Ch. 10.4 - Prob. 29E Ch. 10.4 - Prob. 30E Ch. 10.4 - Prob. 31E Ch. 10.4 - Prob. 32E Ch. 10.4 - In Problems 938, use Theorem 1 to perform the...Ch. 10.4 - Prob. 34E Ch. 10.4 - Prob. 35E Ch. 10.4 - Prob. 36E Ch. 10.4 - Prob. 37E Ch. 10.4 - Prob. 38E Ch. 10.4 - Prob. 39E Ch. 10.4 - Prob. 40E Ch. 10.4 - Prob. 41E Ch. 10.4 - Prob. 42E Ch. 10.4 - Prob. 43E Ch. 10.4 - Prob. 44E Ch. 10.4 - In Problems 4548, use the second-degree Taylor...Ch. 10.4 - Prob. 46E Ch. 10.4 - In Problems 4548, use the second-degree Taylor...Ch. 10.4 - Prob. 48E Ch. 10.4 - Prob. 49E Ch. 10.4 - Prob. 50E Ch. 10.4 - Prob. 51E Ch. 10.4 - To estimate 01.511+x2dx a student takes the first...Ch. 10.4 - There are different ways to approximate a function...Ch. 10.4 - There are different ways to approximate a function...Ch. 10.4 - In Problems 5566, use Theorem 1 to perform the...Ch. 10.4 - Prob. 56E Ch. 10.4 - Prob. 57E Ch. 10.4 - Prob. 58E Ch. 10.4 - Useful life. A computer store rents time on...Ch. 10.4 - Average price. Given the demand equation...Ch. 10.4 - Temperature. The temperature (in degrees Celsius)...Ch. 10.4 - Temperature. Repeat Problem 61 for...Ch. 10.4 - Prob. 63E Ch. 10.4 - Prob. 64E Ch. 10.4 - Prob. 65E Ch. 10.4 - Prob. 66E Ch. 10 - Prob. 1RE Ch. 10 - Prob. 2RE Ch. 10 - Prob. 3RE Ch. 10 - Prob. 4RE Ch. 10 - Prob. 5RE Ch. 10 - Prob. 6RE Ch. 10 - Prob. 7RE Ch. 10 - Use Theorem 1 of Section 10.2 to find the interval...Ch. 10 - Prob. 9RE Ch. 10 - Prob. 10RE Ch. 10 - In Problems 10 and 11, use the formula an =...Ch. 10 - Prob. 12RE Ch. 10 - Prob. 13RE Ch. 10 - Prob. 14RE Ch. 10 - Prob. 15RE Ch. 10 - Prob. 16RE Ch. 10 - Prob. 17RE Ch. 10 - Prob. 18RE Ch. 10 - Prob. 19RE Ch. 10 - Prob. 20RE Ch. 10 - Prob. 21RE Ch. 10 - Prob. 22RE Ch. 10 - Prob. 23RE Ch. 10 - Prob. 24RE Ch. 10 - In Problems 25 and 26, use the second-degree...Ch. 10 - Prob. 26RE Ch. 10 - Prob. 27RE Ch. 10 - In Problems 27 and 28, use a Taylor polynomial at...Ch. 10 - Prob. 29RE Ch. 10 - Prob. 30RE Ch. 10 - Prob. 31RE Ch. 10 - Prob. 32RE Ch. 10 - Prob. 33RE Ch. 10 - Prob. 34RE Ch. 10 - Prob. 35RE Ch. 10 - Prob. 36RE Ch. 10 - Prob. 37RE Ch. 10 - Prob. 38RE Ch. 10 - Prob. 39RE Ch. 10 - Prob. 40RE Ch. 10 - Medicine. The rate of healing for a skin wound (in...Ch. 10 - Prob. 42RE Ch. 10 - Prob. 43RE

Knowledge Booster

Learn more about

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

To approximate: The integral ∫ 0 0.4 x 2 e − x 2 d x with an error of no more than 0.0005.

Solution Summary: The author explains the error estimation for Alternating Series. If x_0 is a number in the interval of convergence for the Taylor series, and the terms are alternating in sign and decreasing in

Videos

To approximate: The integral ∫00.4x2e−x2 dx with an error of no more than 0.0005.

Want to see the full answer?

Chapter 10 Solutions