Which statement about f(x) = 1000(1.3)* is true?
a) instantaneous rate of change is the same for all values of x
b) instantaneous rate of change is positive for all values of x
c) instantaneous rate of change is negative for all values of x
d) instantaneous rate of change is 0 for some values of x

The given function is \( f(x) = 1000(1.3)^x \), which is an exponential function with a base greater than 1.

To analyze the instantaneous rate of change, we look at the derivative \( f'(x) \), which represents the instantaneous rate of change.

1. The derivative of \( f(x) = 1000(1.3)^x \) is:

\[
f'(x) = 1000 \cdot (1.3)^x \cdot \ln(1.3)
\]

Since \( \ln(1.3) \) is positive (as 1.3 > 1), and \( 1000(1.3)^x \) is positive for all values of \( x \), we know that:

- The derivative \( f'(x) \) is **positive** for all values of \( x \).
- This implies the instantaneous rate of change is positive for all values of \( x \).

### Therefore, the correct answer is:
**b) The instantaneous rate of change is positive for all values of \( x \).**

Would you like further details or have any questions?

Here are 5 related questions:
1. How is the derivative of exponential functions calculated?
2. What does the sign of the derivative tell us about the behavior of the function?
3. How would the rate of change differ if the base were less than 1?
4. How do we interpret the natural logarithm in the derivative of exponential functions?
5. What happens to the rate of change as \( x \) increases in this function?

**Tip:** For exponential growth functions like this one, the rate of change is always positive and increasing exponentially!

Which statement about f(x) = 1000(1.3)^x is true? a) instantaneous rate of change is the same for all values of x b) instantaneous rate of change is positive for all values of x c) instantaneous rate of change is negative for all values of x d) instantaneous rate of change is 0 for some values of x

Analyze the instantaneous rate of change for the exponential function f(x) = 1000(1.3)^x. Learn how to calculate the derivative and interpret its behavior for all values of x. This detailed guide covers exponential growth and calculus concepts suitable for students in Grades 11-12.

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