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#40 Find yll Hf 2x²+y²=4

#40 Find yll Hf 2x²+y²=4

Calculus: Early Transcendentals
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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**Problem 40:** Find \( y'' \) if \( 2x^2 + y^2 = 4 \). This problem involves finding the second derivative \( y'' \) of the given equation using implicit differentiation. ### Explanation: To solve this, follow these steps: 1. **Implicit Differentiation:** - Differentiate both sides of the equation \( 2x^2 + y^2 = 4 \) with respect to \( x \). - Remember to apply the chain rule to \( y^2 \). 2. **First Derivative:** - For \( 2x^2 \), the derivative is \( 4x \). - For \( y^2 \), the derivative is \( 2y \frac{dy}{dx} \) (or \( 2yy' \)). - Set the derivative of the right side (constant) to zero. 3. **Solve for \( y' \):** - Combine terms to form an equation in terms of \( y' \). - Solve for \( y' \). 4. **Second Derivative:** - Differentiate \( y' \) again implicitly to find \( y'' \). This process will yield the second derivative \( y'' \) with respect to \( x \), giving insight into the concavity of the curve defined by the equation \( 2x^2 + y^2 = 4 \).
Transcribed Image Text:**Problem 40:** Find \( y'' \) if \( 2x^2 + y^2 = 4 \). This problem involves finding the second derivative \( y'' \) of the given equation using implicit differentiation. ### Explanation: To solve this, follow these steps: 1. **Implicit Differentiation:** - Differentiate both sides of the equation \( 2x^2 + y^2 = 4 \) with respect to \( x \). - Remember to apply the chain rule to \( y^2 \). 2. **First Derivative:** - For \( 2x^2 \), the derivative is \( 4x \). - For \( y^2 \), the derivative is \( 2y \frac{dy}{dx} \) (or \( 2yy' \)). - Set the derivative of the right side (constant) to zero. 3. **Solve for \( y' \):** - Combine terms to form an equation in terms of \( y' \). - Solve for \( y' \). 4. **Second Derivative:** - Differentiate \( y' \) again implicitly to find \( y'' \). This process will yield the second derivative \( y'' \) with respect to \( x \), giving insight into the concavity of the curve defined by the equation \( 2x^2 + y^2 = 4 \).
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