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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![**Task: Calculating a Left-Endpoint Riemann Sum**
Given the function \( f(x) = \frac{1}{x^2 + 3} \), compute the left-endpoint (Riemann) sum using \( n = 5 \) on the interval \([-1, 1]\).
1. Calculate the width of each subinterval, \(\Delta x\):
\[
\Delta x = \frac{1 - (-1)}{5} = \frac{2}{5} = 0.4
\]
2. Determine the subintervals and left endpoints:
\[
[-1, -0.6), [-0.6, -0.2), [-0.2, 0.2), [0.2, 0.6), [0.6, 1]
\]
Left endpoints: \(-1, -0.6, -0.2, 0.2, 0.6\)
3. Compute the Riemann sum:
\[
L_5 = \sum_{i=1}^{5} f(x_i) \cdot \Delta x
\]
where \(x_i\) are the left endpoints of each subinterval.
4. Evaluate \( f(x) \) at each left endpoint:
\[
f(-1), f(-0.6), f(-0.2), f(0.2), f(0.6)
\]
5. Calculate \( L_5 \):
\[
L_5 = f(-1) \cdot 0.4 + f(-0.6) \cdot 0.4 + f(-0.2) \cdot 0.4 + f(0.2) \cdot 0.4 + f(0.6) \cdot 0.4
\]
**Hint:**
To simplify calculations, first find \(f(x_i)\) for each endpoint, multiply by the subinterval width (\(\Delta x = 0.4\)), then sum the results for the approximation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4551f2ef-4c78-4d27-9320-7cb594875026%2Fdd2c0b10-2c08-47bb-af5c-9e254a5cc4cb%2F9vj4ra_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task: Calculating a Left-Endpoint Riemann Sum**
Given the function \( f(x) = \frac{1}{x^2 + 3} \), compute the left-endpoint (Riemann) sum using \( n = 5 \) on the interval \([-1, 1]\).
1. Calculate the width of each subinterval, \(\Delta x\):
\[
\Delta x = \frac{1 - (-1)}{5} = \frac{2}{5} = 0.4
\]
2. Determine the subintervals and left endpoints:
\[
[-1, -0.6), [-0.6, -0.2), [-0.2, 0.2), [0.2, 0.6), [0.6, 1]
\]
Left endpoints: \(-1, -0.6, -0.2, 0.2, 0.6\)
3. Compute the Riemann sum:
\[
L_5 = \sum_{i=1}^{5} f(x_i) \cdot \Delta x
\]
where \(x_i\) are the left endpoints of each subinterval.
4. Evaluate \( f(x) \) at each left endpoint:
\[
f(-1), f(-0.6), f(-0.2), f(0.2), f(0.6)
\]
5. Calculate \( L_5 \):
\[
L_5 = f(-1) \cdot 0.4 + f(-0.6) \cdot 0.4 + f(-0.2) \cdot 0.4 + f(0.2) \cdot 0.4 + f(0.6) \cdot 0.4
\]
**Hint:**
To simplify calculations, first find \(f(x_i)\) for each endpoint, multiply by the subinterval width (\(\Delta x = 0.4\)), then sum the results for the approximation.
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