To find the average rate of change of a function $f(x)$ over the interval $[a, b]$, we use the formula:

$\text{Average rate of change} = \frac{f(b) - f(a)}{b - a}.$

Here:

• $f(x) = 832(1.21)^x$,
• $a = 12$,
• $b = 15$.

Step 1: Calculate $f(12)$ and $f(15)$

$f(12) = 832(1.21)^{12}, \quad f(15) = 832(1.21)^{15}.$

Calculate $f(12)$:

$f(12) = 832 \cdot (1.21)^{12}.$

Calculate $f(15)$:

$f(15) = 832 \cdot (1.21)^{15}.$

Step 2: Find $f(15) - f(12)$

Once we have both values, subtract $f(12)$ from $f(15)$.

Step 3: Divide by $b - a$

Finally, divide the difference by $b - a = 15 - 12 = 3$.

Let me calculate the values for you.# Define the function and inputs import math

def f(x): return 832 * (1.21 ** x)

Interval endpoints

a = 12 b = 15

Calculate function values at a and b

f_a = f(a) f_b = f(b)

Average rate of change

average_rate_of_change = (f_b - f_a) / (b - a) f_a, f_b, average_rate_of_change(8194.97758627193, 14517.902687713484, 2107.6417004805176)(8194.97758627193, 14517.902687713484, 2107.6417004805176)success### Step-by-Step Results:

1. $f(12) = 8194.98$,
2. $f(15) = 14517.90$,
3. The difference, $f(15) - f(12) = 14517.90 - 8194.98 = 6322.92$,
4. Divide by $b - a = 3$: $\text{Average rate of change} = \frac{6322.92}{3} = 2107.64.$

Final Answer:

The average rate of change is 2107.64.

Would you like to explore how this concept applies to other functions or intervals?

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Here are 5 follow-up questions to deepen understanding:

1. How does the base $1.21$ affect the growth rate of the function?
2. What is the significance of the average rate of change in real-world scenarios?
3. How would the result differ if the interval was $[10, 15]$?
4. Can you derive a general formula for the average rate of change of exponential functions?
5. What happens to the average rate of change as the interval becomes smaller?

Tip: For exponential functions, the average rate of change is directly influenced by the growth factor over the interval.