Estimate the average rate of change for the function 𝑓(𝑥)=3𝑥+10 from [5,10].

The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is calculated using the formula:

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

Given the function \( f(x) = 3x + 10 \) and the interval \([5, 10]\):

1. Calculate \( f(5) \):
   \[
   f(5) = 3(5) + 10 = 15 + 10 = 25
   \]

2. Calculate \( f(10) \):
   \[
   f(10) = 3(10) + 10 = 30 + 10 = 40
   \]

3. Apply the average rate of change formula:
   \[
   \text{Average Rate of Change} = \frac{f(10) - f(5)}{10 - 5} = \frac{40 - 25}{10 - 5} = \frac{15}{5} = 3
   \]

So, the average rate of change of the function \( f(x) = 3x + 10 \) from \( x = 5 \) to \( x = 10 \) is \( \boxed{3} \).

Would you like more details or have any questions?

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Here are some related questions:

1. How does the average rate of change relate to the derivative of a function?
2. What is the geometric interpretation of the average rate of change for a linear function?
3. How would the average rate of change differ for a quadratic function?
4. Can the average rate of change be negative? If so, under what conditions?
5. How do you find the average rate of change for a non-linear function?
6. What is the average rate of change for \( f(x) = x^2 \) over the interval \([2, 5]\)?
7. What is the significance of the interval \([a, b]\) when calculating the average rate of change?
8. How does the average rate of change differ from the instantaneous rate of change?

**Tip:** When dealing with linear functions, the average rate of change over any interval will always equal the slope of the line.

Learn how to calculate the average rate of change for the function f(x)=3x+10 over the interval [5,10]. Understand the formula used, step-by-step calculations, and its application to linear functions. Suitable for high school students and learners interested in calculus and linear algebra concepts.

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