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:**
Let \( y = 3x^2 + 5x + 4 \). If \(\Delta x = 0.1\) at \( x = 5\), use linear approximation to estimate \(\Delta y\).
**Solution:**
To solve this problem using linear approximation, we can use the formula:
\[
\Delta y \approx f'(x) \cdot \Delta x
\]
First, we find the derivative \( f'(x) \) of the function \( y = 3x^2 + 5x + 4 \):
\[
f'(x) = \frac{d}{dx}(3x^2 + 5x + 4) = 6x + 5
\]
Now, we evaluate the derivative at \( x = 5 \):
\[
f'(5) = 6(5) + 5 = 30 + 5 = 35
\]
Substitute \( f'(5) \) and \(\Delta x = 0.1\) into the linear approximation formula:
\[
\Delta y \approx 35 \cdot 0.1 = 3.5
\]
Thus, the estimated change in \( y \) is \(\Delta y \approx 3.5\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb8b0cd26-70d8-4d52-9819-bc6d6b2ac8f0%2F344e1813-a1a7-4a0f-b390-ee1a9812c2e2%2F9a2axi_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( y = 3x^2 + 5x + 4 \). If \(\Delta x = 0.1\) at \( x = 5\), use linear approximation to estimate \(\Delta y\).
**Solution:**
To solve this problem using linear approximation, we can use the formula:
\[
\Delta y \approx f'(x) \cdot \Delta x
\]
First, we find the derivative \( f'(x) \) of the function \( y = 3x^2 + 5x + 4 \):
\[
f'(x) = \frac{d}{dx}(3x^2 + 5x + 4) = 6x + 5
\]
Now, we evaluate the derivative at \( x = 5 \):
\[
f'(5) = 6(5) + 5 = 30 + 5 = 35
\]
Substitute \( f'(5) \) and \(\Delta x = 0.1\) into the linear approximation formula:
\[
\Delta y \approx 35 \cdot 0.1 = 3.5
\]
Thus, the estimated change in \( y \) is \(\Delta y \approx 3.5\).
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