Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
69. ∫(from 5/4 to 2)dx/(1-x²)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.67b
In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Verified step by step guidance1
Start with the given function: \(y = \sqrt{3x - 2}\), where \(\frac{2}{3} \leq x \leq 4\).
To find the inverse function, first express the equation in terms of \(x\): square both sides to eliminate the square root, giving \(y^2 = 3x - 2\).
Next, solve for \(x\) by isolating it on one side: add 2 to both sides to get \(y^2 + 2 = 3x\), then divide both sides by 3 to obtain \(x = \frac{y^2 + 2}{3}\).
Define the inverse function \(g\) as \(g(y) = \frac{y^2 + 2}{3}\), which expresses \(x\) as a function of \(y\).
Verify the domain and range of \(g\) to ensure it matches the original function's range and domain, considering the original domain \(\frac{2}{3} \leq x \leq 4\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse g(y) satisfies g(f(x)) = x. Finding the inverse involves solving y = f(x) for x in terms of y, ensuring the function is one-to-one on the given domain.
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Domain and Range Restrictions
To have an inverse, a function must be one-to-one, often requiring domain restrictions. The given interval 2/3 ≤ x ≤ 4 ensures the function is invertible by limiting its domain. Understanding these restrictions is crucial to correctly define and find the inverse function.
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5:10Finding the Domain and Range of a Graph
Derivative and Tangent Line Approximation
The derivative of a function at a point gives the slope of the tangent line there, which can approximate the function near that point. For inverse functions, the derivative of the inverse at a point relates to the reciprocal of the original function's derivative, aiding in tangent line calculations.
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05:13Slopes of Tangent Lines
Related Practice
Textbook Question
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
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Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
8. b. arccot(√3)
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
2. b. tan^(-1)(√3)
Textbook Question
Verify the integration formulas in Exercises 37–40.
37. b. ∫sech(x)dx = sin⁻¹(tanh x) + C
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Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
6. b. arccsc(-2/√3)