Textbook Question
1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
c. √x
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 7.1.67c
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:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3
Verified step by step guidance1
Identify the function given: \(y = \sqrt{3x - 2}\), and the point at which to find the tangent line is \(x_0 = 3\).
Calculate the value of the function at \(x_0\): find \(f(3) = \sqrt{3(3) - 2} = \sqrt{9 - 2}\) to get the point \((3, f(3))\) on the curve.
Find the derivative of the function \(f(x)\) to get the slope of the tangent line. Use the chain rule: \(f(x) = (3x - 2)^{1/2}\), so \(f'(x) = \frac{1}{2}(3x - 2)^{-1/2} \times 3\).
Evaluate the derivative at \(x_0 = 3\) to find the slope of the tangent line: \(m = f'(3)\).
Use the point-slope form of the line equation with the point \((3, f(3))\) and slope \(m\): \(y - f(3) = m(x - 3)\) to write the equation of the tangent line.
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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. Understanding inverses helps in analyzing how functions behave and is essential when exploring derivatives of inverse functions or their tangent lines.
Recommended video:
4:49
Inverse Cosine
Derivative and Differentiation
The derivative measures the rate of change or slope of a function at a given point. Calculating the derivative of f(x) at x_0 provides the slope of the tangent line, which is crucial for finding the tangent line equation.
Recommended video:
05:53Finding Differentials
Equation of a Tangent Line
The tangent line to a function at a point touches the curve without crossing it locally. Its equation is found using the point-slope form: y - f(x_0) = f'(x_0)(x - x_0), where f'(x_0) is the slope from the derivative.
Recommended video:
05:14Equations of Tangent Lines
Related Practice
Textbook Question
c. Find the slopes of the tangent lines to the graphs of f and g at (1, 1) and (−1, −1) (four tangent lines in all).
Textbook Question
10. True, or false? As x→∞,
c. 1/x - 1/x² = o(1/x)
Textbook Question
In Exercises 41–44:
c. Evaluate df/dx at x = a and df⁻¹/dx at x = f(a) to show that
(df⁻¹/dx)|ₓ₌f(a) = 1 / (df/dx)|ₓ₌a
44. f(x) = 2x², x ≥ 0, a = 5
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:
c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2
Textbook Question
88. Given that x>0, find the maximum value, if any, of
c. x^(1/x^n) (n a positive integer)