Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Problem Statement
Given the function \( f \) and the point \( Q \), find all points \( P \) on the graph of \( f \) such that the line tangent to \( f \) at \( P \) passes through \( Q \). Check your work by graphing \( g \) and the tangent lines.
\[ f(x) = x^2 - 5; \quad Q(3,0) \]
---
### Solution
**The point(s) is/are:**
[Answer Box]
(Type an ordered pair. Use a comma to separate answers as needed.)
---
### Explanation of Graph
The graph provided is a standard Cartesian coordinate grid with the \( x \)-axis ranging from -8 to 8, and the \( y \)-axis ranging approximately from -8 to 28. Each axis is labeled with units, and grid lines mark increments of 1 unit.
- The \( x \)-axis is horizontal at \( y = 0 \).
- The \( y \)-axis is vertical at \( x = 0 \).
This graph will be used to plot the function \( f(x) = x^2 - 5 \) and the tangent lines that pass through the given point \( Q(3,0) \).
### Steps to Solve the Problem
1. **Calculate the Derivative of \( f(x) \):**
\[ f'(x) = 2x \]
2. **Equation of the Tangent Line:**
The equation of the tangent line to \( f \) at point \( P(a, a^2 - 5) \) can be written using the point-slope form:
\[ y - (a^2 - 5) = f'(a)(x - a) \]
\[ y - (a^2 - 5) = 2a(x - a) \]
3. **Find where the Tangent Line Passes through Point \( Q(3,0) \):**
Substitute \( Q(3,0) \) into the tangent line equation:
\[ 0 - (a^2 - 5) = 2a(3 - a) \]
\[ -a^2 + 5 = 6a - 2a^2 \]
\[ -a^2 + 2a^2 - 6a + 5 =](https://content.bartleby.com/qna-images/question/156ea66c-c160-41b2-923a-d004818b7703/44e3a543-4fb3-48fb-a8bd-809aec13d1ba/vr5hkaa_processed.png)
Transcribed Image Text:### Problem Statement
Given the function \( f \) and the point \( Q \), find all points \( P \) on the graph of \( f \) such that the line tangent to \( f \) at \( P \) passes through \( Q \). Check your work by graphing \( g \) and the tangent lines.
\[ f(x) = x^2 - 5; \quad Q(3,0) \]
---
### Solution
**The point(s) is/are:**
[Answer Box]
(Type an ordered pair. Use a comma to separate answers as needed.)
---
### Explanation of Graph
The graph provided is a standard Cartesian coordinate grid with the \( x \)-axis ranging from -8 to 8, and the \( y \)-axis ranging approximately from -8 to 28. Each axis is labeled with units, and grid lines mark increments of 1 unit.
- The \( x \)-axis is horizontal at \( y = 0 \).
- The \( y \)-axis is vertical at \( x = 0 \).
This graph will be used to plot the function \( f(x) = x^2 - 5 \) and the tangent lines that pass through the given point \( Q(3,0) \).
### Steps to Solve the Problem
1. **Calculate the Derivative of \( f(x) \):**
\[ f'(x) = 2x \]
2. **Equation of the Tangent Line:**
The equation of the tangent line to \( f \) at point \( P(a, a^2 - 5) \) can be written using the point-slope form:
\[ y - (a^2 - 5) = f'(a)(x - a) \]
\[ y - (a^2 - 5) = 2a(x - a) \]
3. **Find where the Tangent Line Passes through Point \( Q(3,0) \):**
Substitute \( Q(3,0) \) into the tangent line equation:
\[ 0 - (a^2 - 5) = 2a(3 - a) \]
\[ -a^2 + 5 = 6a - 2a^2 \]
\[ -a^2 + 2a^2 - 6a + 5 =
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