Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Finding Tangent Lines**
**3. Problem Statement:**
Find an equation for the tangent line to the curve \( y = x^{-1/3} \) at the point \( (8, 1/2) \).
Explanation:
To find the tangent line to the curve \( y = x^{-1/3} \) at the given point \( (8, 1/2) \), follow these steps:
1. Calculate the derivative of the function \( y = x^{-1/3} \).
2. Evaluate the derivative at \( x = 8 \) to determine the slope of the tangent line.
3. Use the point-slope form of a line equation to find the equation of the tangent line.
**Detailed Steps:**
1. Derive \( y = x^{-1/3} \) with respect to \( x \).
2. Plug \( x = 8 \) into the derivative to find the slope.
3. Apply the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (8, 1/2) \) and \( m \) is the slope.
**4. Problem Statement:**
The question text is partially obscured.
What can be discerned is part of a mathematical expression:
\[
f(x) = \frac{x^2}{(1 + 1/(2))} \\
\]
Explanation:
The expression appears to involve finding an equation related to the given function. More information is needed to precisely determine the steps required.
**Important Note:**
The content of this transcription is intended for educational purposes. For detailed explanations, it is recommended to refer to calculus textbooks or academic resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6484a2ba-7b6c-4144-bd9b-1f2d784a131b%2F10d7f1fd-9da0-495e-bd7b-e1acb3ff6bf9%2Fwa69wk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding Tangent Lines**
**3. Problem Statement:**
Find an equation for the tangent line to the curve \( y = x^{-1/3} \) at the point \( (8, 1/2) \).
Explanation:
To find the tangent line to the curve \( y = x^{-1/3} \) at the given point \( (8, 1/2) \), follow these steps:
1. Calculate the derivative of the function \( y = x^{-1/3} \).
2. Evaluate the derivative at \( x = 8 \) to determine the slope of the tangent line.
3. Use the point-slope form of a line equation to find the equation of the tangent line.
**Detailed Steps:**
1. Derive \( y = x^{-1/3} \) with respect to \( x \).
2. Plug \( x = 8 \) into the derivative to find the slope.
3. Apply the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (8, 1/2) \) and \( m \) is the slope.
**4. Problem Statement:**
The question text is partially obscured.
What can be discerned is part of a mathematical expression:
\[
f(x) = \frac{x^2}{(1 + 1/(2))} \\
\]
Explanation:
The expression appears to involve finding an equation related to the given function. More information is needed to precisely determine the steps required.
**Important Note:**
The content of this transcription is intended for educational purposes. For detailed explanations, it is recommended to refer to calculus textbooks or academic resources.
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