Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.8.68d

Capitolo 8, Problema 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).

Guida verificata passo dopo passo

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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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).

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Fundamental Theorem of Calculus Part 1

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.

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Determining Error and Relative Error

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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Finding Area When Bounds Are Not Given

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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).

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Concetti chiave

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).