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...
Related questions
Question
100%
![**Problem Statement:**
Let \( y = 5x^2 + 5x + 2 \). If \(\Delta x = 0.2\) at \(x = 5\), use linear approximation to estimate \(\Delta y\).
**Solution:**
To estimate \(\Delta y\) using linear approximation, we need to find the derivative \(\frac{dy}{dx}\) and use it to approximate the change in \(y\) for this small \(\Delta x\).
1. **Find the derivative:**
\[
\frac{dy}{dx} = \frac{d}{dx}(5x^2 + 5x + 2) = 10x + 5
\]
2. **Evaluate the derivative at \(x = 5\):**
\[
\frac{dy}{dx} \bigg|_{x=5} = 10(5) + 5 = 50 + 5 = 55
\]
3. **Estimate \(\Delta y\):**
\[
\Delta y \approx \frac{dy}{dx} \bigg|_{x=5} \cdot \Delta x = 55 \cdot 0.2 = 11
\]
Thus, the estimated change in \(y\) is:
\[ \Delta y \approx 11 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff0123591-fed7-4568-91c4-d5c32b5e453b%2F60357cff-3864-4b71-b924-44bbb49c3da0%2Fb92hs92_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( y = 5x^2 + 5x + 2 \). If \(\Delta x = 0.2\) at \(x = 5\), use linear approximation to estimate \(\Delta y\).
**Solution:**
To estimate \(\Delta y\) using linear approximation, we need to find the derivative \(\frac{dy}{dx}\) and use it to approximate the change in \(y\) for this small \(\Delta x\).
1. **Find the derivative:**
\[
\frac{dy}{dx} = \frac{d}{dx}(5x^2 + 5x + 2) = 10x + 5
\]
2. **Evaluate the derivative at \(x = 5\):**
\[
\frac{dy}{dx} \bigg|_{x=5} = 10(5) + 5 = 50 + 5 = 55
\]
3. **Estimate \(\Delta y\):**
\[
\Delta y \approx \frac{dy}{dx} \bigg|_{x=5} \cdot \Delta x = 55 \cdot 0.2 = 11
\]
Thus, the estimated change in \(y\) is:
\[ \Delta y \approx 11 \]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning