## Calculus: Analyzing a Function### Problem SetConsider the function $ f(x) $ defined as:\[ f(x) = \frac{x}{1 + x^2} \]#### Tasks:1. **Determine $ f'(x) $**: - Find the first derivative of the function $ f(x) $.2. **Determine all critical values of the function $ f(x) $**: - Identify the points at which the derivative $ f'(x) $ is zero or undefined.3. **Use the first derivative test**: - Use the first derivative test to determine whether the critical values are associated with local maxima or local minima.Explanation for educational purposes: When analyzing the behavior of functions, finding the derivative is crucial as it provides information about the rate of change of the function. The critical points, where this rate of change is zero or undefined, can indicate potential locations of local extremum (maximum or minimum values). The first derivative test helps in classifying these critical points.### Detailed Steps**1. Calculating $ f'(x) $:** Given $ f(x) = \frac{x}{1 + x^2} $, To find the derivative $ f'(x) $, we use the quotient rule $ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} $ where $ u = x $ and $ v = 1 + x^2 $. **2. Identifying Critical Values:** Set the first derivative $ f'(x) $ to zero and solve for $ x $. **3. Applying the First Derivative Test:** - Evaluate the sign changes of $ f'(x) $ around the critical values to determine the nature of the extremum.Remember, analyzing the behavior of functions using their derivatives provides significant insight into their graph, helping us understand where and how the function reaches its peak values (maxima) and lowest values (minima).

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Answered: let fla)=_X 1+ x2 a) determine f'(*)

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let fla)=_X 1+ x2 a) determine f'(*)

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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

## Calculus: Analyzing a Function

### Problem Set

Consider the function $ f(x) $ defined as:

\[ f(x) = \frac{x}{1 + x^2} \]

#### Tasks:

1. **Determine $ f'(x) $**:
- Find the first derivative of the function $ f(x) $.

2. **Determine all critical values of the function $ f(x) $**:
- Identify the points at which the derivative $ f'(x) $ is zero or undefined.

3. **Use the first derivative test**:
- Use the first derivative test to determine whether the critical values are associated with local maxima or local minima.

Explanation for educational purposes: When analyzing the behavior of functions, finding the derivative is crucial as it provides information about the rate of change of the function. The critical points, where this rate of change is zero or undefined, can indicate potential locations of local extremum (maximum or minimum values). The first derivative test helps in classifying these critical points.

### Detailed Steps

**1. Calculating $ f'(x) $:**

Given $ f(x) = \frac{x}{1 + x^2} $,

To find the derivative $ f'(x) $, we use the quotient rule $ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} $ where $ u = x $ and $ v = 1 + x^2 $.

**2. Identifying Critical Values:**

Set the first derivative $ f'(x) $ to zero and solve for $ x $.

**3. Applying the First Derivative Test:**

- Evaluate the sign changes of $ f'(x) $ around the critical values to determine the nature of the extremum.

Remember, analyzing the behavior of functions using their derivatives provides significant insight into their graph, helping us understand where and how the function reaches its peak values (maxima) and lowest values (minima).

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