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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![**Determine the Validity of the Statement**
**Problem Statement:**
Determine whether the following statement is true or false:
An equation of the tangent line to the parabola \( y = x^2 \) at \( (-3, 9) \) is \( y - 9 = 3x(x + 3) \).
**Analysis:**
To determine the validity of the statement, we need to follow these steps:
1. **Identify the Point of Tangency:**
The point given is \( (-3, 9) \).
2. **Find the Derivative:**
The derivative of \( y = x^2 \) is \( y' = 2x \).
3. **Evaluate the Slope at \( x = -3 \):**
For \( x = -3 \):
\[
y'(-3) = 2(-3) = -6
\]
So, the slope of the tangent line at \( (-3, 9) \) is \( -6 \).
4. **Equation of the Tangent Line:**
Using the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is \( (-3, 9) \) and \( m \) is the slope \( -6 \):
\[
y - 9 = -6(x + 3)
\]
Simplifying:
\[
y - 9 = -6x - 18
\]
\[
y = -6x - 9
\]
**Comparison:**
The given equation of the tangent line in the problem statement is \( y - 9 = 3x(x + 3) \). Simplifying the right-hand side:
\[
y - 9 = 3x^2 + 9x
\]
However, it should be a linear equation representing the tangent line, not a quadratic equation.
**Conclusion:**
The given equation \( y - 9 = 3x(x + 3) \) does not represent the tangent line to the parabola \( y = x^2 \) at point \((-3, 9) \). Therefore, the statement is false.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc1c90340-a748-4160-8298-b9e689632892%2Fa3829804-1718-4902-a372-313ee13ec772%2Foe5iihg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Determine the Validity of the Statement**
**Problem Statement:**
Determine whether the following statement is true or false:
An equation of the tangent line to the parabola \( y = x^2 \) at \( (-3, 9) \) is \( y - 9 = 3x(x + 3) \).
**Analysis:**
To determine the validity of the statement, we need to follow these steps:
1. **Identify the Point of Tangency:**
The point given is \( (-3, 9) \).
2. **Find the Derivative:**
The derivative of \( y = x^2 \) is \( y' = 2x \).
3. **Evaluate the Slope at \( x = -3 \):**
For \( x = -3 \):
\[
y'(-3) = 2(-3) = -6
\]
So, the slope of the tangent line at \( (-3, 9) \) is \( -6 \).
4. **Equation of the Tangent Line:**
Using the point-slope form of the equation of a line:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is \( (-3, 9) \) and \( m \) is the slope \( -6 \):
\[
y - 9 = -6(x + 3)
\]
Simplifying:
\[
y - 9 = -6x - 18
\]
\[
y = -6x - 9
\]
**Comparison:**
The given equation of the tangent line in the problem statement is \( y - 9 = 3x(x + 3) \). Simplifying the right-hand side:
\[
y - 9 = 3x^2 + 9x
\]
However, it should be a linear equation representing the tangent line, not a quadratic equation.
**Conclusion:**
The given equation \( y - 9 = 3x(x + 3) \) does not represent the tangent line to the parabola \( y = x^2 \) at point \((-3, 9) \). Therefore, the statement is false.
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