Textbook Question
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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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.4.57a
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.
Verified step by step guidance1
Step 1: Begin by recalling the definitions of trigonometric functions. The tangent function is defined as \( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \), and the cosecant function is defined as \( \csc\theta = \frac{1}{\sin\theta} \).
Step 2: From the problem, we are given \( x = 4 \tan\theta \). Rearrange this equation to express \( \tan\theta \) in terms of \( x \): \( \tan\theta = \frac{x}{4} \).
Step 3: Use the Pythagorean identity \( \sin^2\theta + \cos^2\theta = 1 \) to relate \( \tan\theta \) to \( \sin\theta \) and \( \cos\theta \). Recall that \( \tan\theta = \frac{\sin\theta}{\cos\theta} \), so \( \sin\theta = \tan\theta \cdot \cos\theta \).
Step 4: Substitute \( \tan\theta = \frac{x}{4} \) into the expression for \( \sin\theta \). To find \( \csc\theta \), take the reciprocal of \( \sin\theta \): \( \csc\theta = \frac{1}{\sin\theta} \). Verify whether \( \csc\theta = \frac{4}{x} \) holds true by substituting \( \sin\theta \) and simplifying.
Step 5: Analyze the result. If \( \csc\theta \neq \frac{4}{x} \), provide a counterexample to show why the statement is false. If \( \csc\theta = \frac{4}{x} \), explain why the statement is true based on the trigonometric relationships derived.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities include the Pythagorean identities, reciprocal identities, and quotient identities. Understanding these identities is essential for manipulating and simplifying trigonometric expressions, which is crucial for solving problems in calculus.
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Verifying Trig Equations as Identities
Reciprocal Functions
Reciprocal functions are pairs of trigonometric functions that are defined as the reciprocal of each other. For example, the cosecant function (cscθ) is the reciprocal of the sine function (sinθ), meaning cscθ = 1/sinθ. Recognizing these relationships helps in transforming and solving trigonometric equations, particularly when verifying statements involving these functions.
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Substitution in Trigonometry
Substitution in trigonometry involves replacing one variable with another to simplify expressions or solve equations. In this context, substituting x = 4 tanθ allows us to express other trigonometric functions in terms of x. This technique is vital for analyzing relationships between different trigonometric functions and verifying the truth of statements involving them.
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Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. ∫(3/(x² + 4)) dx = ∫(3/x²) dx + ∫(3/4) dx.
Textbook Question
Arc length of a parabola Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.
a. Find an expression for L.
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Textbook Question
60. Two Methods
a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.
Textbook Question
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.
Textbook Question
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.