1. Warmup question: What is the largest value that | sin(0)| may take for any ? What is the largest value that |2 sin(0)| may take for any 0? What about |A sin(0)|, where A > 0 is a constant? This will help you find K below. 2. Let f(x): sin(2x + 1). The goal of this problem is to estimate the value of the definite +2.0 integral [20 sin(2x + 1) dx using the trapezoidal rule. Follow the directions below. (a) Find the second derivative f"(x). Make sure you carefully use the chain rule as you differentiate twice. (b) Using part (a) and Warmup Question #1, find an upper bound K for f"(x)| on [0.5, 2.0]. It doesn't need to be the least upper bound possible, just a usuable esti- mate of the magnitude of the second derivative. Explain your reasoning.
1. Warmup question: What is the largest value that | sin(0)| may take for any ? What is the largest value that |2 sin(0)| may take for any 0? What about |A sin(0)|, where A > 0 is a constant? This will help you find K below. 2. Let f(x): sin(2x + 1). The goal of this problem is to estimate the value of the definite +2.0 integral [20 sin(2x + 1) dx using the trapezoidal rule. Follow the directions below. (a) Find the second derivative f"(x). Make sure you carefully use the chain rule as you differentiate twice. (b) Using part (a) and Warmup Question #1, find an upper bound K for f"(x)| on [0.5, 2.0]. It doesn't need to be the least upper bound possible, just a usuable esti- mate of the magnitude of the second derivative. Explain your reasoning.
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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A) is -4sin(2x+1)
![1. Warmup question: What is the largest value that | sin(0)| may take for any 0? What is the
largest value that |2 sin(0)| may take for any ? What about |A sin(0)], where A > 0 is a
constant? This will help you find K below.
sin(2x + 1). The goal of this problem is to estimate the value of the definite
integral si
sin(2x + 1) dx using the trapezoidal rule. Follow the directions below.
2. Let f(x)
=
2.0
(a) Find the second derivative ƒ"(x). Make sure you carefully use the chain rule as you
differentiate twice.
(b) Using part (a) and Warmup Question #1, find an upper bound K for |f"(x)| on
[0.5, 2.0]. It doesn't need to be the least upper bound possible, just a usuable esti-
mate of the magnitude of the second derivative. Explain your reasoning.
(c) Use the theorem for error analysis of the trapezoidal rule EȚ
(b − a)³K
12n²
to de-
termine the fewest number n of trapezoids necessary to estimate 2.sin(2x + 1) dx
within 10-2. Your result must be a positive integer. Which way did you round? Up
or down? Why?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5bee61ad-dcf3-4b44-a661-7a89df673f47%2F097cb9aa-8d61-4d94-bd4e-d7f37831f239%2F1x2nu8p_processed.png&w=3840&q=75)
Transcribed Image Text:1. Warmup question: What is the largest value that | sin(0)| may take for any 0? What is the
largest value that |2 sin(0)| may take for any ? What about |A sin(0)], where A > 0 is a
constant? This will help you find K below.
sin(2x + 1). The goal of this problem is to estimate the value of the definite
integral si
sin(2x + 1) dx using the trapezoidal rule. Follow the directions below.
2. Let f(x)
=
2.0
(a) Find the second derivative ƒ"(x). Make sure you carefully use the chain rule as you
differentiate twice.
(b) Using part (a) and Warmup Question #1, find an upper bound K for |f"(x)| on
[0.5, 2.0]. It doesn't need to be the least upper bound possible, just a usuable esti-
mate of the magnitude of the second derivative. Explain your reasoning.
(c) Use the theorem for error analysis of the trapezoidal rule EȚ
(b − a)³K
12n²
to de-
termine the fewest number n of trapezoids necessary to estimate 2.sin(2x + 1) dx
within 10-2. Your result must be a positive integer. Which way did you round? Up
or down? Why?
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