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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).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] e^(x²) dx using n = 50 subintervals.
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 8.8.53a
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Suppose ∫_a^b f(x) dx is approximated with Simpson’s Rule using n = 18 subintervals, where |f^(4)(x)| ≤ 1 on [a, b]. The absolute error E_S in approximating the integral satisfies E_S ≤ (Δx)^5 / 10.
Guida verificata passo dopo passo1
Recall the error bound formula for Simpson's Rule: the absolute error \( E_S \) satisfies \( |E_S| \leq \frac{(b - a)}{180} (\Delta x)^4 \max_{x \in [a,b]} |f^{(4)}(x)| \), where \( \Delta x = \frac{b - a}{n} \) and \( n \) is the number of subintervals (which must be even).
Identify the given values: \( n = 18 \), \( |f^{(4)}(x)| \leq 1 \) on \( [a,b] \), and \( \Delta x = \frac{b - a}{18} \).
Substitute \( \max |f^{(4)}(x)| = 1 \) and \( \Delta x = \frac{b - a}{18} \) into the error bound formula to express \( E_S \) in terms of \( (b - a) \) and \( \Delta x \).
Compare the given inequality \( E_S \leq \frac{(\Delta x)^5}{10} \) with the standard error bound formula to check if it holds for all \( (b - a) \) and \( n = 18 \).
Conclude whether the statement is true or false by analyzing if the given bound is consistent with the known error bound for Simpson's Rule, providing a counterexample if it is not.
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Simpson's Rule and its Error Bound
Simpson's Rule approximates definite integrals by fitting parabolas through subinterval points. The error bound depends on the fourth derivative of the function and the number of subintervals n. Specifically, the error is proportional to the maximum of |f⁽⁴⁾(x)| on [a, b] and inversely proportional to n⁴, reflecting how finer partitions improve accuracy.
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Intro to the Chain Rule
Role of the Fourth Derivative in Error Estimation
The fourth derivative of the function, f⁽⁴⁾(x), measures the function's smoothness and curvature changes. In Simpson's Rule, the error bound involves the maximum absolute value of f⁽⁴⁾(x) over the interval, as this derivative controls how well parabolas approximate the function. A smaller maximum leads to a smaller error.
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05:59Estimating the Area Under a Curve with Right Endpoints & Midpoint
Relationship Between Subinterval Width Δx and Number of Subintervals n
The subinterval width Δx is defined as (b - a)/n, linking the number of subintervals n to the partition size. Since the error bound in Simpson's Rule involves Δx raised to the fifth power, increasing n (thus decreasing Δx) significantly reduces the error, highlighting the importance of choosing an appropriate n for desired accuracy.
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06:11Introduction to Riemann Sums
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a. Find an expression for L.
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Domanda del libro di testo
60. Two Methods
a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.
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Domanda del libro di testo
57. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If x = 4 tanθ, then cscθ = 4/x.
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Domanda del libro di testo
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
66. Let f(x) = cos(x²).
a. Find a Midpoint Rule approximation to ∫[-1 to 1] cos(x²) dx using n = 30 subintervals.
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62. Electronic Chips Suppose the probability that a particular computer chip fails after a hours of operation is 0.00005 ∫(from a to ∞) e^(-0.00005t) dt.
a. Find the probability that the computer chip fails after 15,000 hr of operation.
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