Find the average rate of change of 𝑓(𝑥)=𝑥3 between x=0.6 and x=3.5

*Round your answer to the nearest tenth place

To find the average rate of change of the function \( f(x) = x^3 \) between \( x = 0.6 \) and \( x = 3.5 \), use the formula:

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

where \( a = 0.6 \) and \( b = 3.5 \).

1. Calculate \( f(0.6) \):
   \[
   f(0.6) = (0.6)^3 = 0.216
   \]

2. Calculate \( f(3.5) \):
   \[
   f(3.5) = (3.5)^3 = 42.875
   \]

3. Apply the average rate of change formula:
   \[
   \text{Average Rate of Change} = \frac{42.875 - 0.216}{3.5 - 0.6} = \frac{42.659}{2.9} \approx 14.710
   \]

Rounding to the nearest tenth:

\[
\text{Average Rate of Change} \approx \boxed{14.7}
\]

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 for \( f(x) = x^3 \) compare to that of a linear function?
2. What is the significance of the interval length in the calculation of the average rate of change?
3. How would the average rate of change differ for \( f(x) = x^2 \) over the same interval?
4. Can you estimate the instantaneous rate of change at \( x = 2.05 \) for \( f(x) = x^3 \)?
5. What is the geometric interpretation of the average rate of change for a cubic function?
6. How does changing the interval affect the average rate of change?
7. How would you calculate the average rate of change for a function like \( f(x) = \sin(x) \) over a given interval?
8. How is the average rate of change used in real-world applications, such as physics or economics?

**Tip:** When calculating average rates of change, pay close attention to rounding instructions as they can affect the precision of your final answer.

Learn how to calculate the average rate of change for the function f(x) = x^3 between x = 0.6 and x = 3.5. Step-by-step solution provided with rounding to the nearest tenth. Suitable for students in Grades 11-12 studying calculus and polynomial functions.

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