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Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. x² + y² = 169 (a) Find dy/dt, given x = 5, y = 12, and dx/dt = 5. dy/dt = (b) Find dx/dt, given x = 12, y = 5, and dy/dt = -5. dx/dt =

Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt. x² + y² = 169 (a) Find dy/dt, given x = 5, y = 12, and dx/dt = 5. dy/dt = (b) Find dx/dt, given x = 12, y = 5, and dy/dt = -5. dx/dt =

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 Statement Assume that \( x \) and \( y \) are both differentiable functions of \( t \). Find the required values of \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \). Given equation: \[ x^2 + y^2 = 169 \] ### (a) Find \( \frac{dy}{dt} \), given \( x = 5 \), \( y = 12 \), and \( \frac{dx}{dt} = 5 \). \[ \frac{dy}{dt} = \_\_\_\_\_\_ \] ### (b) Find \( \frac{dx}{dt} \), given \( x = 12 \), \( y = 5 \), and \( \frac{dy}{dt} = -5 \). \[ \frac{dx}{dt} = \_\_\_\_\_\_ \] ### Explanation of Steps: 1. **Differentiate the given equation implicitly with respect to \( t \):** \[ x^2 + y^2 = 169 \] Differentiating both sides: \[ 2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 \] 2. **Solve for \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \):** #### For (a): Given \( x = 5 \), \( y = 12 \), and \( \frac{dx}{dt} = 5 \): \[ 2(5)\left(5\right) + 2(12)\left(\frac{dy}{dt}\right) = 0 \] \[ 50 + 24\left(\frac{dy}{dt}\right) = 0 \] \[ \frac{dy}{dt} = -\frac{50}{24} \] Simplify: \[ \frac{dy}{dt} = -\frac{25}{12} \] #### For (b): Given \( x = 12 \), \( y = 5 \), and \( \frac{dy}{dt} = -5 \): \[ 2(12)\left(\frac{dx}{dt}\right) + 2(5)\left(-5\right) = 0 \] \[ 24\left(\frac{dx}{
Transcribed Image Text:## Problem Statement Assume that \( x \) and \( y \) are both differentiable functions of \( t \). Find the required values of \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \). Given equation: \[ x^2 + y^2 = 169 \] ### (a) Find \( \frac{dy}{dt} \), given \( x = 5 \), \( y = 12 \), and \( \frac{dx}{dt} = 5 \). \[ \frac{dy}{dt} = \_\_\_\_\_\_ \] ### (b) Find \( \frac{dx}{dt} \), given \( x = 12 \), \( y = 5 \), and \( \frac{dy}{dt} = -5 \). \[ \frac{dx}{dt} = \_\_\_\_\_\_ \] ### Explanation of Steps: 1. **Differentiate the given equation implicitly with respect to \( t \):** \[ x^2 + y^2 = 169 \] Differentiating both sides: \[ 2x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0 \] 2. **Solve for \( \frac{dy}{dt} \) and \( \frac{dx}{dt} \):** #### For (a): Given \( x = 5 \), \( y = 12 \), and \( \frac{dx}{dt} = 5 \): \[ 2(5)\left(5\right) + 2(12)\left(\frac{dy}{dt}\right) = 0 \] \[ 50 + 24\left(\frac{dy}{dt}\right) = 0 \] \[ \frac{dy}{dt} = -\frac{50}{24} \] Simplify: \[ \frac{dy}{dt} = -\frac{25}{12} \] #### For (b): Given \( x = 12 \), \( y = 5 \), and \( \frac{dy}{dt} = -5 \): \[ 2(12)\left(\frac{dx}{dt}\right) + 2(5)\left(-5\right) = 0 \] \[ 24\left(\frac{dx}{
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