Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problem 8.8.68d

Chapter 8, Problem 8.8.68d

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
68. Let f(x) = e^(x²).
d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

Verified step by step guidance

Step 1: Recall Theorem 8.1, which provides a method to estimate the error in numerical integration using the formula for the error bound. The error bound is given by:

E \leq \frac{K \cdot b - a^3}{12}

, where K is the maximum value of the second derivative of f(x) on the interval [a, b].

Step 2: Compute the second derivative of f(x). Start with f(x) = e^(x²). The first derivative is

f' (x) = 2 x \cdot e^{x^{2}}

. The second derivative is

f'' (x) = 2 e^{x^{2}} (4 x^{2} + 1)

Step 3: Determine the interval [a, b] from part (a) of the problem. This interval is where the numerical integration was performed. Identify the maximum value of the second derivative, f''(x), on this interval. This involves analyzing the expression

2 e^{x^{2}} (4 x^{2} + 1)

and finding its maximum value.

Step 4: Substitute the maximum value of f''(x) (denoted as K), the interval length (b - a), and the formula for the error bound into Theorem 8.1's error formula:

E \leq \frac{K \cdot b - a^3}{12}

Step 5: Simplify the expression to find the upper bound on the absolute error. This will provide the theoretical maximum error in the numerical integration estimate from part (a).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Theorem 8.1 (Taylor's Theorem)

Theorem 8.1, often referred to as Taylor's Theorem, provides a way to approximate a function using polynomials. It states that a function can be expressed as a Taylor series around a point, and the remainder term gives an estimate of the error involved in this approximation. Understanding this theorem is crucial for estimating the accuracy of polynomial approximations of functions like f(x) = e^(x²).

Absolute Error

Absolute error measures the difference between the true value of a function and its approximation. It is calculated as the absolute value of the difference between the actual function value and the estimated value. In the context of Theorem 8.1, finding an upper bound on the absolute error helps quantify how close the polynomial approximation is to the actual function.

Upper Bound

An upper bound in mathematics refers to a value that is greater than or equal to every value in a given set. In the context of estimating error using Theorem 8.1, determining an upper bound on the absolute error provides a limit on how large the error can be. This is essential for assessing the reliability of the approximation and ensuring that it meets desired accuracy criteria.

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Related Practice

Textbook Question

75. Exploring powers of sine and cosine

e. Repeat parts (a), (b), and (c) with sin²x replaced by sin⁴x. Comment on your observations.

101

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Textbook Question

87. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

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Textbook Question

81. Possible and impossible integrals

Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.

d. Show that, in general, if n is odd, then Iₙ = -½ e⁻ˣ² pₙ₋₁(x), where pₙ₋₁ is a polynomial of degree n - 1.

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Textbook Question

93. Three start-ups Three cars, A, B, and C, start from rest and accelerate along a line according to the following velocity functions:

vₐ(t) = 88t/(t + 1), v_B(t) = 88t²/(t + 1)², and v_C(t) = 88t²/(t² + 1).

d. Which car ultimately gains the lead and remains in front?

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Textbook Question

77. Tabular integration Consider the integral ∫ f(x)g(x) dx, where f can be differentiated repeatedly and g can be integrated repeatedly

Let Gₖ represent the result of calculating k indefinite integrals of g (omitting constants of integration).

d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts.

Evaluate ∫ x² e^(x/2) dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in.

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Textbook Question

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

e. Does this pattern continue? Is it true that A(1, ln b) = a² * A(a, (ln b)/a)?

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66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.68. Let f(x) = e^(x²).d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).

Verified video answer for a similar problem:

Key Concepts

Theorem 8.1 (Taylor's Theorem)

Absolute Error

Upper Bound

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
68. Let f(x) = e^(x²).
d. Use Theorem 8.1 to find an upper bound on the absolute error in the estimate found in part (a).