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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.8.69d
Chapter 8, Problem 8.8.69d

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
69. Let f(x) = sin(eˣ).
d. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1.

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Step 1: Recall Theorem 8.1, which provides a method to estimate the error in numerical integration. The error bound is given by \( E \leq \frac{K(b-a)^3}{24n^2} \), where \( K \) is the maximum value of the second derivative of \( f(x) \) on the interval \([a, b]\), \( n \) is the number of subintervals, and \( [a, b] \) is the interval of integration.
Step 2: Compute the second derivative of \( f(x) = \sin(e^x) \). Start by finding the first derivative: \( f'(x) = \cos(e^x) \cdot e^x \). Then, differentiate again to find \( f''(x) = -\sin(e^x) \cdot (e^x)^2 + \cos(e^x) \cdot e^x \).
Step 3: Determine the maximum value of \( f''(x) \) on the interval \([a, b]\). This involves analyzing \( f''(x) \) and finding its critical points by setting \( f'''(x) = 0 \) and solving for \( x \). Evaluate \( f''(x) \) at these critical points and endpoints of the interval to find \( K \).
Step 4: Plug the values of \( K \), \( b-a \) (the length of the interval), and \( n \) (the number of subintervals) into the error formula \( E \leq \frac{K(b-a)^3}{24n^2} \). This will give the upper bound on the absolute error.
Step 5: Simplify the expression for \( E \) as much as possible without calculating the final numerical value. This provides the theoretical upper bound for the error in the estimate.

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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, commonly known 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 function approximations, particularly when dealing with functions like f(x) = sin(eˣ).
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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 defined as the absolute value of the difference between the actual function value and the estimated value. In the context of Theorem 8.1, calculating the upper bound on absolute error helps determine how close the approximation is to the actual function, which is essential for assessing the reliability of the estimate.
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Upper Bound

An upper bound is a value that serves as a limit on the size of a quantity, ensuring that the actual value does not exceed this limit. In the context of estimating errors using Theorem 8.1, finding an upper bound on the absolute error provides a way to quantify the worst-case scenario for the approximation's accuracy. This concept is vital for understanding the reliability of numerical estimates in calculus.
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Related Practice
Textbook Question

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project "The exponential Eiffel Tower")

d. Prove this property holds for any a ≥ 0 and c > 0:

The tangent lines to y = ±e^(-c·x) at x = a always intersect at R's center of mass

(Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique, 332, 571-584, 2004)

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Textbook Question

109. Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by W = ∫ from R to ∞ of F(x) dx, where R = 6370 km is the approximate radius of Earth, F(x) = (GMm)/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².

c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?

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Textbook Question

66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.

66. Let f(x) = cos(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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Textbook Question

101. Many methods needed Show that the integral from ∫(from 0 to ∞)(sqrt(x) * ln x) / (1 + x)^2 dx equals pi, following these steps

d. Evaluate the remaining integral using the change of variables z = sqrt(x)

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Textbook Question

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

d. ∫(1/eˣ) dx = ln eˣ + C.

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Textbook Question

91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis

on the interval [b, ∞).

c. Find the minimum value b* such that when b > b*, there exists some a > 0 where A(a,b) = 2.

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