Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
d. ln ∛9
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.67d
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:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Verified step by step guidance1
Identify the given function as \( y = f(x) = \sqrt{3x - 2} \) and the point \( x_0 = 3 \). The function \( g \) is the inverse of \( f \), so \( g = f^{-1} \). The point on \( g \) corresponding to \( x_0 \) is \( (f(x_0), x_0) \).
Calculate \( f(x_0) \) by substituting \( x_0 = 3 \) into \( f(x) \): \( f(3) = \sqrt{3 \cdot 3 - 2} \). This gives the coordinates of the point on \( g \) as \( (f(3), 3) \).
Use Theorem 1, which states that if \( g = f^{-1} \), then the derivative of \( g \) at \( f(x_0) \) is \( g'(f(x_0)) = \frac{1}{f'(x_0)} \). So, first find \( f'(x) \) by differentiating \( f(x) = \sqrt{3x - 2} \).
Evaluate \( f'(x) \) at \( x_0 = 3 \) to find \( f'(3) \). Then compute \( g'(f(3)) = \frac{1}{f'(3)} \), which is the slope of the tangent line to \( g \) at the point \( (f(3), 3) \).
Write the equation of the tangent line to \( g \) at \( (f(3), 3) \) using the point-slope form: \[ y - y_1 = m (x - x_1) \] where \( (x_1, y_1) = (f(3), 3) \) and \( m = g'(f(3)) \). This line is symmetric to the tangent line of \( f \) at \( x_0 = 3 \) across the line \( y = x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions and Their Graphs
An inverse function reverses the roles of inputs and outputs of the original function, swapping x and y. Graphically, the inverse function's graph is the reflection of the original function's graph across the line y = x (the 45° line). Understanding this symmetry is crucial for locating points like (f(x₀), x₀) on the inverse function.
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3:17
Inverse Tangent
Derivative of an Inverse Function (Theorem 1)
Theorem 1 states that if f is differentiable and invertible at x₀ with f'(x₀) ≠ 0, then the derivative of its inverse g at y₀ = f(x₀) is g'(y₀) = 1 / f'(x₀). This relationship allows us to find the slope of the tangent line to the inverse function using the derivative of the original function.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Equation of a Tangent Line
The tangent line to a function at a point provides the best linear approximation near that point. Its equation is y - y₁ = m(x - x₁), where m is the slope (derivative at the point) and (x₁, y₁) is the point of tangency. For the inverse function, the point and slope must be carefully identified using the inverse relationship.
Recommended video:
05:14Equations of Tangent Lines
Related Practice
Textbook Question
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:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2
Textbook Question
155. Which is bigger, πᵉ or e^π?
Calculators have taken some of the mystery out of this once-challenging question.
(Go ahead and check; you will see that it is a very close call.)
You can answer the question without a calculator, though.
d. Conclude that
xᵉ < eˣfor all positivex ≠ e.
Textbook Question
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:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2
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Textbook Question
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:
d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.
72. y= 2-x-x³, -2 ≤ x ≤ 2, x_0 = 3/2
Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
e. x - 2ln(x)
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