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Let y = 5x² + 8x + 4. If Ax = 0.4 at x = 2, use linear approximation to estimate Ay Ay a

Let y = 5x² + 8x + 4. If Ax = 0.4 at x = 2, use linear approximation to estimate Ay Ay a

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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# Linear Approximation Exercise **Problem Statement:** Let \( y = 5x^2 + 8x + 4 \). If \(\Delta x = 0.4\) at \( x = 2 \), use linear approximation to estimate \(\Delta y\). **Solution Approach:** 1. **Find the Derivative:** - Differentiate \( y \) with respect to \( x \) to find \( \frac{dy}{dx} \). 2. **Evaluate the Derivative at \( x = 2 \):** - Substitute \( x = 2 \) into \( \frac{dy}{dx} \) to find the slope of the tangent line at that point. 3. **Apply Linear Approximation Formula:** - Use \(\Delta y \approx \frac{dy}{dx} \cdot \Delta x\). 4. **Calculate \(\Delta y\):** - Substitute the slope and \(\Delta x = 0.4\) to find the estimated change in \( y \). This exercise provides an introduction to using derivatives for linear approximations, a useful tool for estimating changes in functions over small intervals.
Transcribed Image Text:# Linear Approximation Exercise **Problem Statement:** Let \( y = 5x^2 + 8x + 4 \). If \(\Delta x = 0.4\) at \( x = 2 \), use linear approximation to estimate \(\Delta y\). **Solution Approach:** 1. **Find the Derivative:** - Differentiate \( y \) with respect to \( x \) to find \( \frac{dy}{dx} \). 2. **Evaluate the Derivative at \( x = 2 \):** - Substitute \( x = 2 \) into \( \frac{dy}{dx} \) to find the slope of the tangent line at that point. 3. **Apply Linear Approximation Formula:** - Use \(\Delta y \approx \frac{dy}{dx} \cdot \Delta x\). 4. **Calculate \(\Delta y\):** - Substitute the slope and \(\Delta x = 0.4\) to find the estimated change in \( y \). This exercise provides an introduction to using derivatives for linear approximations, a useful tool for estimating changes in functions over small intervals.
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