. Find the critical numbers and then the critical points= of the functions. de 1₁ Y = - 3 Sketch the graph of the following functions, by finding the local mins and maximums and the intervals of increase and decrease and making a sign diagram for the first derivative. ₁ f(x) = 2/2²³/₂2 5 H-2 f(x) = 2 2²-9
. Find the critical numbers and then the critical points= of the functions. de 1₁ Y = - 3 Sketch the graph of the following functions, by finding the local mins and maximums and the intervals of increase and decrease and making a sign diagram for the first derivative. ₁ f(x) = 2/2²³/₂2 5 H-2 f(x) = 2 2²-9
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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please help with these calculus questions.
![## Educational Content Transcription
### Problem 1
**Question:**
Find the critical numbers and then the critical points of the functions.
**Function:**
\[ y = \frac{2x}{2x - 3} \]
### Problem 2
**Question:**
Sketch the graph of the following functions by finding the local minimums and maximums and the intervals of increase and decrease, and making a sign diagram for the first derivative.
**Functions:**
1. \[ f(x) = \frac{5}{x - 2} \]
2. \[ f(x) = \frac{-x}{x^2 - 9} \]
### Explanation:
1. **Critical Numbers and Points:**
- To find the critical numbers of a function, take its derivative and set it equal to zero to solve for x.
- Critical points are the coordinates (x, y) on the graph where these critical numbers occur.
2. **Sketching the Graph:**
- Determine local minimums and maximums by finding where the first derivative changes sign.
- Identify intervals of increase and decrease using the sign of the first derivative.
- Construct a sign diagram for the first derivative to visualize these intervals.
The given functions in Problem 2 require applying these techniques to accurately sketch their graphs, determining the significant features such as peaks, valleys, and slopes.
### Additional Notes:
- A sign diagram is a visual tool to see where the derivative (first derivative) changes sign, indicating potential local extrema and the nature of the slope of the function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbd18e3e2-85c9-4b2f-a0fd-7822a7147daa%2F574168da-95de-4cf9-b6de-f32836931673%2F1ywkc7b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Educational Content Transcription
### Problem 1
**Question:**
Find the critical numbers and then the critical points of the functions.
**Function:**
\[ y = \frac{2x}{2x - 3} \]
### Problem 2
**Question:**
Sketch the graph of the following functions by finding the local minimums and maximums and the intervals of increase and decrease, and making a sign diagram for the first derivative.
**Functions:**
1. \[ f(x) = \frac{5}{x - 2} \]
2. \[ f(x) = \frac{-x}{x^2 - 9} \]
### Explanation:
1. **Critical Numbers and Points:**
- To find the critical numbers of a function, take its derivative and set it equal to zero to solve for x.
- Critical points are the coordinates (x, y) on the graph where these critical numbers occur.
2. **Sketching the Graph:**
- Determine local minimums and maximums by finding where the first derivative changes sign.
- Identify intervals of increase and decrease using the sign of the first derivative.
- Construct a sign diagram for the first derivative to visualize these intervals.
The given functions in Problem 2 require applying these techniques to accurately sketch their graphs, determining the significant features such as peaks, valleys, and slopes.
### Additional Notes:
- A sign diagram is a visual tool to see where the derivative (first derivative) changes sign, indicating potential local extrema and the nature of the slope of the function.
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