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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Question
How do I solve this derivative equation when it is equal to 0
![(a) **What is the derivative of \( y(t) \)?**
\[ y'(t) = \frac{1}{2} \cos(t) - \frac{t}{2} \sin(t) \]
(b) **Places where a derivative is 0 are places where the function has a horizontal tangent line. Set the derivative equal to 0 and solve for \( t \). There are infinitely many solutions. Ask WolframAlpha to solve the equation that you have. What are the solutions with \( t < 10 \) (round to the nearest tenth)?**
\[ 0 = -\frac{1}{2} \cos(t) - \frac{t}{2} \sin(t) \]
\[ 0 = \cos(t) - t \sin(t) \]
(c) **Suppose this is a model of a building experiencing resonance as it sways back and forth in an earthquake. What is the building doing at the times you found in part (b)?**
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**Graph Explanation:**
There is a small graph visible at the top left corner of the image. It appears to be a plot of a function over a certain domain, likely representing the solution or behavior described in the problem. The x-axis likely represents time \( t \) while the y-axis represents the value of \( y(t) \) or its derivative. However, without more detail, we cannot specify the exact function plotted.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F051d9a8d-fe28-4947-87a7-657149e37dc8%2F5842bda4-be35-4209-a461-914ff392e967%2F43mxwpc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(a) **What is the derivative of \( y(t) \)?**
\[ y'(t) = \frac{1}{2} \cos(t) - \frac{t}{2} \sin(t) \]
(b) **Places where a derivative is 0 are places where the function has a horizontal tangent line. Set the derivative equal to 0 and solve for \( t \). There are infinitely many solutions. Ask WolframAlpha to solve the equation that you have. What are the solutions with \( t < 10 \) (round to the nearest tenth)?**
\[ 0 = -\frac{1}{2} \cos(t) - \frac{t}{2} \sin(t) \]
\[ 0 = \cos(t) - t \sin(t) \]
(c) **Suppose this is a model of a building experiencing resonance as it sways back and forth in an earthquake. What is the building doing at the times you found in part (b)?**
---
**Graph Explanation:**
There is a small graph visible at the top left corner of the image. It appears to be a plot of a function over a certain domain, likely representing the solution or behavior described in the problem. The x-axis likely represents time \( t \) while the y-axis represents the value of \( y(t) \) or its derivative. However, without more detail, we cannot specify the exact function plotted.
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