Question

Which function has the greater estimated average rate of change over the interval [0,1.1]
, the exponential function y=4x
or the quadratic function y=4x2
?

Option 1: exponential function

Option 2: quadratic function

(1 point)
The function with the greater estimated average rate of change on the interval is option
.

Answer 1

To determine which function has the greater estimated average rate of change over the interval \([0, 1.1]\), we can compute the average rate of change for both functions and compare the results.

1. **Exponential function**: \(y = 4^x\)
- Evaluating at the endpoints:
- \(f(0) = 4^0 = 1\)
- \(f(1.1) = 4^{1.1} \approx 4^1 \cdot 4^{0.1} \approx 4 \cdot 1.1487 \approx 4.5948\)
- Average rate of change:
\[
\text{Average Rate of Change} = \frac{f(1.1) - f(0)}{1.1 - 0} = \frac{4.5948 - 1}{1.1} \approx \frac{3.5948}{1.1} \approx 3.2635
\]

2. **Quadratic function**: \(y = 4x^2\)
- Evaluating at the endpoints:
- \(g(0) = 4(0^2) = 0\)
- \(g(1.1) = 4(1.1^2) = 4 \cdot 1.21 = 4.84\)
- Average rate of change:
\[
\text{Average Rate of Change} = \frac{g(1.1) - g(0)}{1.1 - 0} = \frac{4.84 - 0}{1.1} = \frac{4.84}{1.1} \approx 4.4
\]

Now, comparing the two average rates of change calculated:

- Average rate of change for the exponential function \( \approx 3.2635 \)
- Average rate of change for the quadratic function \( \approx 4.4 \)

Therefore, the function with the greater estimated average rate of change on the interval is the **quadratic function**.

**The function with the greater estimated average rate of change on the interval is option 2: quadratic function.**