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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.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.4: Multiple-angle Formulas
Problem 70E
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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?
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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