Question

Question 2
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 average rate of change over the interval \([0, 1.1]\), we can calculate the average rate of change for both functions.

The average rate of change of a function \(f\) over an interval \([a, b]\) is given by the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

### Step 1: Calculate for the exponential function \(y = 4x\)

Let \(f(x) = 4x\).

- Evaluate \(f(0)\):
\[
f(0) = 4 \cdot 0 = 0
\]

- Evaluate \(f(1.1)\):
\[
f(1.1) = 4 \cdot 1.1 = 4.4
\]

Now, calculate the average rate of change:
\[
\text{Average Rate of Change for } 4x = \frac{f(1.1) - f(0)}{1.1 - 0} = \frac{4.4 - 0}{1.1} = \frac{4.4}{1.1} = 4
\]

### Step 2: Calculate for the quadratic function \(y = 4x^2\)

Let \(g(x) = 4x^2\).

- Evaluate \(g(0)\):
\[
g(0) = 4 \cdot 0^2 = 0
\]

- Evaluate \(g(1.1)\):
\[
g(1.1) = 4 \cdot (1.1)^2 = 4 \cdot 1.21 = 4.84
\]

Now, calculate the average rate of change:
\[
\text{Average Rate of Change for } 4x^2 = \frac{g(1.1) - g(0)}{1.1 - 0} = \frac{4.84 - 0}{1.1} = \frac{4.84}{1.1} \approx 4.4
\]

### Conclusion

- Average rate of change for \(y = 4x\) over \([0, 1.1]\): \(4\)
- Average rate of change for \(y = 4x^2\) over \([0, 1.1]\): \(\approx 4.4\)

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

The answer is: Option 2: quadratic function.