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...
Related questions
Question
![**Problem:**
Find the slope of the tangent line to the curve
\[
-2x^2 - 2xy - 2y^3 = -38
\]
at the point \((-4, -1)\).
---
**Solution:**
To find the slope of the tangent line, we implicitly differentiate the equation with respect to \(x\) and then evaluate at the given point \((-4, -1)\).
1. Differentiate each term:
- For \(-2x^2\), the derivative is \(-4x\).
- For \(-2xy\), use the product rule: \(-2(y + x \frac{dy}{dx})\).
- For \(-2y^3\), the derivative is \(-6y^2 \frac{dy}{dx}\).
2. The equation becomes:
\[
-4x - 2(y + x \frac{dy}{dx}) - 6y^2 \frac{dy}{dx} = 0
\]
3. Simplify and solve for \(\frac{dy}{dx}\).
4. Substitute \(x = -4\) and \(y = -1\) into the differentiated equation to find the slope of the tangent.
5. Present the final result as the value of the slope at the given point.
Fill the solution in the provided box.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6d11dbd9-c0aa-4cd6-9283-77d349b104f3%2F1c37f219-b18d-48fe-ab5b-8e227740109d%2Fg7he8p7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find the slope of the tangent line to the curve
\[
-2x^2 - 2xy - 2y^3 = -38
\]
at the point \((-4, -1)\).
---
**Solution:**
To find the slope of the tangent line, we implicitly differentiate the equation with respect to \(x\) and then evaluate at the given point \((-4, -1)\).
1. Differentiate each term:
- For \(-2x^2\), the derivative is \(-4x\).
- For \(-2xy\), use the product rule: \(-2(y + x \frac{dy}{dx})\).
- For \(-2y^3\), the derivative is \(-6y^2 \frac{dy}{dx}\).
2. The equation becomes:
\[
-4x - 2(y + x \frac{dy}{dx}) - 6y^2 \frac{dy}{dx} = 0
\]
3. Simplify and solve for \(\frac{dy}{dx}\).
4. Substitute \(x = -4\) and \(y = -1\) into the differentiated equation to find the slope of the tangent.
5. Present the final result as the value of the slope at the given point.
Fill the solution in the provided box.
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