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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.1.67c
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 guidance
1
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.
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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.
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Finding 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.
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Equations of Tangent Lines
Related Practice
Textbook Question

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

1. c. tan^(-1)(1/√3)

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

23. Human evolution continues The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years.

c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?

Textbook Question

88. Given that x>0, find the maximum value, if any, of

c. x^(1/x^n) (n a positive integer)