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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![The image is a handwritten note that seems to be related to calculus or mathematical analysis. Below is a transcription of the text:
---
**Determine where f(x) is increasing/decreasing**
- max/min
- concave up/down
- inflection points
- sketch
\[ f(x) = 5 \sin(x) e^{\cos(x)}, \quad \text{given } x = \left[ -\pi, \pi \right] \]
---
### Explanation:
This note outlines the steps needed to analyze the function \( f(x) = 5 \sin(x) e^{\cos(x)} \) over the interval \([ -\pi, \pi ]\). Key points include:
1. **Determine where \( f(x) \) is increasing or decreasing**:
- Calculate the first derivative \( f'(x) \) to find intervals where \( f(x) \) is increasing or decreasing.
2. **Maxima and Minima**:
- Identify critical points by setting \( f'(x) = 0 \) and determine whether they are local maxima or minima by using the first or second derivative test.
3. **Concave Up/Down**:
- Determine the concavity of the function by calculating the second derivative \( f''(x) \). Identify intervals where the function is concave up (\( f''(x) > 0 \)) or concave down (\( f''(x) < 0 \)).
4. **Inflection Points**:
- Find points where the concavity changes, i.e., where \( f''(x) = 0 \) and changes sign. These points are inflection points.
5. **Sketch**:
- Use the information about increasing/decreasing intervals, concavity, maxima, minima, and inflection points to sketch the graph of the function over the given interval.
By following these steps, one can thoroughly analyze the behavior of the function \( f(x) = 5 \sin(x) e^{\cos(x)} \) over the interval \([ -\pi, \pi ]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2133091-a753-411c-beef-444bf7f4574e%2F087e60cc-1b3b-4b2b-84c1-82b6c5e9aa77%2F4jkbyu5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image is a handwritten note that seems to be related to calculus or mathematical analysis. Below is a transcription of the text:
---
**Determine where f(x) is increasing/decreasing**
- max/min
- concave up/down
- inflection points
- sketch
\[ f(x) = 5 \sin(x) e^{\cos(x)}, \quad \text{given } x = \left[ -\pi, \pi \right] \]
---
### Explanation:
This note outlines the steps needed to analyze the function \( f(x) = 5 \sin(x) e^{\cos(x)} \) over the interval \([ -\pi, \pi ]\). Key points include:
1. **Determine where \( f(x) \) is increasing or decreasing**:
- Calculate the first derivative \( f'(x) \) to find intervals where \( f(x) \) is increasing or decreasing.
2. **Maxima and Minima**:
- Identify critical points by setting \( f'(x) = 0 \) and determine whether they are local maxima or minima by using the first or second derivative test.
3. **Concave Up/Down**:
- Determine the concavity of the function by calculating the second derivative \( f''(x) \). Identify intervals where the function is concave up (\( f''(x) > 0 \)) or concave down (\( f''(x) < 0 \)).
4. **Inflection Points**:
- Find points where the concavity changes, i.e., where \( f''(x) = 0 \) and changes sign. These points are inflection points.
5. **Sketch**:
- Use the information about increasing/decreasing intervals, concavity, maxima, minima, and inflection points to sketch the graph of the function over the given interval.
By following these steps, one can thoroughly analyze the behavior of the function \( f(x) = 5 \sin(x) e^{\cos(x)} \) over the interval \([ -\pi, \pi ]\).
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