To calculate the percentage rate of change of the function $f(x) = 6 \cdot 3^x$ between $x = 3$ and $x = 4$, we can follow these steps:

1. Evaluate $f(3)$: $f(3) = 6 \cdot 3^3 = 6 \cdot 27 = 162$

2. Evaluate $f(4)$: $f(4) = 6 \cdot 3^4 = 6 \cdot 81 = 486$

3. Find the absolute change in $f(x)$: $\Delta f = f(4) - f(3) = 486 - 162 = 324$

4. Find the percentage rate of change: The formula for percentage rate of change is: $\text{Percentage change} = \frac{\Delta f}{f(3)} \times 100 = \frac{324}{162} \times 100 = 200\%$

Thus, the percentage rate of change of $f(x)$ between $x = 3$ and $x = 4$ is 200%.

Do you need more details or have any questions?

Here are 5 related questions for deeper understanding:

1. How would the percentage rate of change differ for smaller intervals, say between $x = 2$ and $x = 3$?
2. What is the derivative of $f(x) = 6 \cdot 3^x$, and how does it relate to the rate of change at any $x$?
3. How does the base of the exponential function, 3, influence the growth rate of the function?
4. What would happen if the base were a fraction, say $f(x) = 6 \cdot \left(\frac{1}{3}\right)^x$?
5. How can we interpret the 200% growth in real-world applications?

Tip: In exponential functions, the base dictates how quickly the function grows or decays.