To determine which function has a greater estimated average rate of change over the interval \([-1.5, 3.5]\), we need to calculate the average rate of change for both functions.
The average rate of change of a function \(f(x)\) over the interval \([a, b]\) is given by the formula:
\[ \text{Average rate of change} = \frac{f(b) - f(a)}{b - a} \]
Let's apply this to both functions, \(f(x) = 3^x\) and \(g(x) = 3x^2\).
For the exponential function \(f(x) = 3^x\):
-
Calculate \(f(-1.5)\): \[ f(-1.5) = 3^{-1.5} = \frac{1}{3^{1.5}} \approx 0.192 \]
-
Calculate \(f(3.5)\): \[ f(3.5) = 3^{3.5} \approx 46.765 \]
-
Now calculate the average rate of change: \[ \text{Average rate of change} = \frac{f(3.5) - f(-1.5)}{3.5 - (-1.5)} = \frac{46.765 - 0.192}{5} \approx \frac{46.573}{5} \approx 9.3146 \]
For the quadratic function \(g(x) = 3x^2\):
-
Calculate \(g(-1.5)\): \[ g(-1.5) = 3(-1.5)^2 = 3(2.25) = 6.75 \]
-
Calculate \(g(3.5)\): \[ g(3.5) = 3(3.5)^2 = 3(12.25) = 36.75 \]
-
Now calculate the average rate of change: \[ \text{Average rate of change} = \frac{g(3.5) - g(-1.5)}{3.5 - (-1.5)} = \frac{36.75 - 6.75}{5} = \frac{30}{5} = 6 \]
Conclusion:
- The average rate of change for \(f(x) = 3^x\) is approximately \(9.3146\).
- The average rate of change for \(g(x) = 3x^2\) is \(6\).
Comparing these values, the exponential function \(f(x) = 3^x\) has a greater estimated average rate of change over the interval \([-1.5, 3.5]\).
Therefore, the correct response is:
The exponential function has a greater estimated average rate of change. Its rate of average rate of change is about 1.5 times larger than that of the quadratic function.