Skip to main content
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.

Verified step by step guidance
1
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

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).
Recommended video:
06:11
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.
Recommended video:
04:57
Determining Error and Relative Error

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.
Recommended video:
05:06
Finding Area When Bounds Are Not Given
Related Practice
Textbook Question

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

d. If ∫(from 1 to ∞) x^(-p) dx exists, then ∫(from 1 to ∞) x^(-q) dx exists (where q > p).

1
views
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.

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).

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?

1
views
Textbook Question

2. Give an example of each of the following.

d. A repeated irreducible quadratic factor

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.