x + 2x – 3, that is also parallel to Find the equation of the normal line to the parabola y – 2x + 2y = 9. 15 y = x + 4

Calculus: Early Transcendentals
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ISBN:9781285741550
Author:James Stewart
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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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**Problem Statement:**

Find the equation of the normal line to the parabola \( y = x^2 + 2x - 3 \) that is also parallel to \( -2x + 2y = 9 \).

**Solution Attempt:**

\[ y = x + \frac{15}{4} \]

**Feedback:**

The given solution is incorrect, as marked by the red cross. 

---

**Explanation:**

This problem involves finding the equation of a normal line to a given parabola that is parallel to a specific line equation. 

1. **Understanding the Parabola:**
   - The parabola equation is \( y = x^2 + 2x - 3 \).
   - The derivative, \( \frac{dy}{dx} = 2x + 2 \), gives the slope of the tangent at any point on the parabola.
   
2. **Normal Line Slope:**
   - The slope of the normal line is the negative reciprocal of the tangent slope.
   
3. **Parallel Line Condition:**
   - Line \( -2x + 2y = 9 \) can be rearranged to standard form: \( y = x + \frac{9}{2} \).
   - The slope is 1. So, the normal line must parallel a line with slope 1, meaning its slope is the same.

4. **Conclusion:**
   - The attempted solution did not correctly align with these criteria.
Transcribed Image Text:**Problem Statement:** Find the equation of the normal line to the parabola \( y = x^2 + 2x - 3 \) that is also parallel to \( -2x + 2y = 9 \). **Solution Attempt:** \[ y = x + \frac{15}{4} \] **Feedback:** The given solution is incorrect, as marked by the red cross. --- **Explanation:** This problem involves finding the equation of a normal line to a given parabola that is parallel to a specific line equation. 1. **Understanding the Parabola:** - The parabola equation is \( y = x^2 + 2x - 3 \). - The derivative, \( \frac{dy}{dx} = 2x + 2 \), gives the slope of the tangent at any point on the parabola. 2. **Normal Line Slope:** - The slope of the normal line is the negative reciprocal of the tangent slope. 3. **Parallel Line Condition:** - Line \( -2x + 2y = 9 \) can be rearranged to standard form: \( y = x + \frac{9}{2} \). - The slope is 1. So, the normal line must parallel a line with slope 1, meaning its slope is the same. 4. **Conclusion:** - The attempted solution did not correctly align with these criteria.
Expert Solution
Step 1

  Definition used -

                        Slope of normal - The slope of normal at a point is the negative reciprocal of the slope of the tangent at that point.

                       The slope of normal for function f(x) at x = a can be found by the given formula -

                                                      m=-1f'(a)

                       

                                    

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