https://file.aiquickdraw.com/mathbot/file/1726710482465mpqku4o5.jpg

To solve the problem of estimating the average rate of change from \( x = 4 \) to \( x = 7 \), we can follow these steps:

### Step 1: Recall the Formula for Average Rate of Change
The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is given by:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

Where:
- \( a = 4 \)
- \( b = 7 \)

### Step 2: Estimate \( f(4) \) and \( f(7) \)
From the graph:
- At \( x = 4 \), the corresponding \( y \)-value (or \( f(4) \)) appears to be approximately \( f(4) = 2 \).
- At \( x = 7 \), the corresponding \( y \)-value (or \( f(7) \)) is approximately \( f(7) = 5 \).

### Step 3: Apply the Formula
Now, applying the formula for the average rate of change:

\[
\text{Average Rate of Change} = \frac{f(7) - f(4)}{7 - 4} = \frac{5 - 2}{7 - 4} = \frac{3}{3} = 1
\]

### Conclusion:
The estimated average rate of change from \( x = 4 \) to \( x = 7 \) is **1**.

---

Would you like more details or have any further questions?

Here are some related questions to deepen your understanding:

1. What is the significance of the average rate of change in a real-world context?
2. How does the average rate of change relate to the slope of a secant line?
3. Can you estimate the instantaneous rate of change at \( x = 5 \) from the graph?
4. How would you calculate the average rate of change for a different interval, such as from \( x = 1 \) to \( x = 4 \)?
5. What does it mean if the average rate of change is negative over an interval?

**Tip:** When working with graphs, approximate the values carefully by identifying exact points or using interpolation for better accuracy.

Estimate the average rate of change from x = 4 to x = 7 based on the graph of f(x).

Learn how to estimate the average rate of change of a function f(x) from x = 4 to x = 7 using the formula (f(b) - f(a)) / (b - a). This example explores key calculus concepts like slope and secant lines, perfect for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation