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:**
Compute the length of the curve:
\[ f(x) = x^2 - 2x, \quad 0 \leq x \leq 2 \]
**Explanation:**
This problem asks you to find the arc length of the function \( f(x) = x^2 - 2x \) over the interval from \( x = 0 \) to \( x = 2 \). The arc length can be calculated using the formula for the length of a curve given by a function \( y = f(x) \), which is:
\[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
In this scenario, you will need to
1. Differentiate \( f(x) \) with respect to \( x \) to find \( \frac{dy}{dx} \).
2. Substitute \( \frac{dy}{dx} \) into the arc length formula.
3. Evaluate the definite integral from \( x = 0 \) to \( x = 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d5fe7c4-bc88-41ab-8c75-523dd8842f64%2Fd93966e2-445a-4d32-8c0a-7f4bc393a01a%2Fnuiddij_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Compute the length of the curve:
\[ f(x) = x^2 - 2x, \quad 0 \leq x \leq 2 \]
**Explanation:**
This problem asks you to find the arc length of the function \( f(x) = x^2 - 2x \) over the interval from \( x = 0 \) to \( x = 2 \). The arc length can be calculated using the formula for the length of a curve given by a function \( y = f(x) \), which is:
\[ L = \int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx \]
In this scenario, you will need to
1. Differentiate \( f(x) \) with respect to \( x \) to find \( \frac{dy}{dx} \).
2. Substitute \( \frac{dy}{dx} \) into the arc length formula.
3. Evaluate the definite integral from \( x = 0 \) to \( x = 2 \).
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