https://file.aiquickdraw.com/mathbot/file/17271190512678zlh6odk.jpg

For question 022, we are tasked with determining the graph of the derivative \( f' \) given the graph of a function \( f \) on the interval \( (-2, 2) \).

### Analyzing the function \( f \):
1. From the graph of \( f \), we see that it is piecewise linear. The slope of each segment will give us the derivative \( f' \).
2. On \( (-2, -1) \), the graph of \( f \) is decreasing, meaning \( f' \) will be negative here.
3. On \( (-1, 0) \), the graph of \( f \) is increasing, meaning \( f' \) will be positive here.
4. On \( (0, 1) \), the graph of \( f \) is constant, so the derivative \( f' \) will be zero here.
5. On \( (1, 2) \), the graph of \( f \) is decreasing again, so \( f' \) will be negative here.

### Identifying the correct derivative:
1. For the interval \( (-2, -1) \), the slope is negative, so \( f' \) should be below the x-axis in this region.
2. For \( (-1, 0) \), the slope is positive, so \( f' \) should be above the x-axis in this region.
3. For \( (0, 1) \), since the function is constant, \( f' = 0 \).
4. For \( (1, 2) \), the slope is negative again, so \( f' \) should be below the x-axis.

Looking at the provided options, **option 2** is the correct graph of the derivative. This option shows the correct changes in sign (positive and negative) and zero values corresponding to the flat section of \( f \).

---

For question 023, we are asked to find the graph of the derivative \( f' \) of a function on \( (-7, 7) \) based on its provided graph.

### Analyzing the function \( f \):
1. The function starts increasing, meaning \( f' \) is positive.
2. Then, the function reaches a maximum at \( x = -4 \), where \( f' = 0 \).
3. Between \( x = -4 \) and \( x = 0 \), the function decreases, meaning \( f' \) is negative.
4. At \( x = 0 \), the function reaches another minimum, so \( f' = 0 \).
5. Between \( x = 0 \) and \( x = 2 \), the function increases, meaning \( f' \) is positive.
6. After \( x = 2 \), the function decreases again, meaning \( f' \) is negative.

The derivative should capture these sign changes and critical points. Based on this reasoning, you can analyze the provided options and select the correct graph of \( f' \).

---

Would you like more details on how to interpret these graphs or any specific part of the process explained further?

Here are some related questions for practice:
1. How does the slope of the graph of \( f \) relate to the graph of its derivative \( f' \)?
2. What happens to the derivative \( f' \) at points where \( f \) has local maxima or minima?
3. How can you graphically identify intervals where \( f \) is increasing or decreasing?
4. What is the relationship between the zeroes of \( f' \) and the turning points of \( f \)?
5. How does the concavity of \( f \) relate to the graph of \( f' \)?

**Tip**: Always examine how the slope of a function behaves across intervals when determining its derivative.

Given the graph of a function, determine the graph of its derivative.

Learn how to determine the graph of the derivative of a piecewise function. This calculus problem covers key concepts like the derivative as the slope of the tangent, and how changes in slope affect the derivative graph. Suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation