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Find at æ = 3 if y=-5x² - 2 and da dt dy dt = 4.

Find at æ = 3 if y=-5x² - 2 and da dt dy dt = 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: Differentiating Implicit Functions** Consider the function \( y = -5x^2 - 2 \) with \(\frac{dy}{dt} = 4\). We want to find \(\frac{dx}{dt}\) when \( x = 3 \). ### Steps: 1. Differentiate the given function implicitly with respect to \( t \). 2. Plug in the known values to solve for \(\frac{dx}{dt}\). ### Solution: Given: \[ y = -5x^2 - 2 \] Differentiate both sides with respect to \( t \): \[ \frac{dy}{dt} = \frac{d}{dt}(-5x^2 - 2) \] \[ \frac{dy}{dt} = -10x \cdot \frac{dx}{dt} \] We know \(\frac{dy}{dt} = 4\), so substitute these values: \[ 4 = -10x \cdot \frac{dx}{dt} \] Now, solve for \(\frac{dx}{dt}\) when \( x = 3 \): \[ 4 = -10(3) \cdot \frac{dx}{dt} \] \[ 4 = -30 \cdot \frac{dx}{dt} \] \[ \frac{dx}{dt} = -\frac{4}{30} \] \[ \frac{dx}{dt} = -\frac{2}{15} \] ### Conclusion: The value of \(\frac{dx}{dt}\) at \( x = 3 \) is \(-\frac{2}{15}\).
Transcribed Image Text:**Problem: Differentiating Implicit Functions** Consider the function \( y = -5x^2 - 2 \) with \(\frac{dy}{dt} = 4\). We want to find \(\frac{dx}{dt}\) when \( x = 3 \). ### Steps: 1. Differentiate the given function implicitly with respect to \( t \). 2. Plug in the known values to solve for \(\frac{dx}{dt}\). ### Solution: Given: \[ y = -5x^2 - 2 \] Differentiate both sides with respect to \( t \): \[ \frac{dy}{dt} = \frac{d}{dt}(-5x^2 - 2) \] \[ \frac{dy}{dt} = -10x \cdot \frac{dx}{dt} \] We know \(\frac{dy}{dt} = 4\), so substitute these values: \[ 4 = -10x \cdot \frac{dx}{dt} \] Now, solve for \(\frac{dx}{dt}\) when \( x = 3 \): \[ 4 = -10(3) \cdot \frac{dx}{dt} \] \[ 4 = -30 \cdot \frac{dx}{dt} \] \[ \frac{dx}{dt} = -\frac{4}{30} \] \[ \frac{dx}{dt} = -\frac{2}{15} \] ### Conclusion: The value of \(\frac{dx}{dt}\) at \( x = 3 \) is \(-\frac{2}{15}\).
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