Question

Which function has the greater estimated average rate of change over the interval [0,1], the exponential function y=4^x or the quadratic function y=4x^2
?



(1 point)

(?) is the function with the greater estimated average rate of change on the interval.

Answer 1

To find the estimated average rate of change of each function over the interval \([0, 1]\), we use the formula for the average rate of change:

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

where \( f(x) \) is the function, and \([a, b]\) is the interval.

### For the exponential function \(y = 4^x\):

1. Calculate \(f(0)\):
\[
f(0) = 4^0 = 1
\]

2. Calculate \(f(1)\):
\[
f(1) = 4^1 = 4
\]

3. Calculate the average rate of change:
\[
\text{Average rate of change} = \frac{f(1) - f(0)}{1 - 0} = \frac{4 - 1}{1} = 3
\]

### For the quadratic function \(y = 4x^2\):

1. Calculate \(f(0)\):
\[
f(0) = 4(0)^2 = 0
\]

2. Calculate \(f(1)\):
\[
f(1) = 4(1)^2 = 4
\]

3. Calculate the average rate of change:
\[
\text{Average rate of change} = \frac{f(1) - f(0)}{1 - 0} = \frac{4 - 0}{1} = 4
\]

### Compare the average rates of change:

- Average rate of change for \(y = 4^x\) over \([0, 1]\) is \(3\).
- Average rate of change for \(y = 4x^2\) over \([0, 1]\) is \(4\).

Thus, the quadratic function \(y = 4x^2\) has the greater estimated average rate of change over the interval \([0, 1]\).

The answer is:

\(4x^2\) is the function with the greater estimated average rate of change on the interval.