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

Answer 1

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

1. **Exponential function**: \( y = 4x \)

The average rate of change of the function from \( x = a \) to \( x = b \) is given by:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
For \( y = 4x \), we calculate:
- \( f(0) = 4 \cdot 0 = 0 \)
- \( f(1.1) = 4 \cdot 1.1 = 4.4 \)

Now we can compute the average rate of change:
\[
\text{Average Rate of Change}_{\text{exponential}} = \frac{f(1.1) - f(0)}{1.1 - 0} = \frac{4.4 - 0}{1.1 - 0} = \frac{4.4}{1.1} = 4
\]

2. **Quadratic function**: \( y = 4x^2 \)

Similarly, we calculate for the quadratic function:
- \( g(0) = 4 \cdot 0^2 = 0 \)
- \( g(1.1) = 4 \cdot (1.1)^2 = 4 \cdot 1.21 = 4.84 \)

Now we compute the average rate of change:
\[
\text{Average Rate of Change}_{\text{quadratic}} = \frac{g(1.1) - g(0)}{1.1 - 0} = \frac{4.84 - 0}{1.1 - 0} = \frac{4.84}{1.1} \approx 4.4
\]

Now we compare the results:
- The exponential function has an average rate of change of \( 4 \).
- The quadratic function has an average rate of change of approximately \( 4.4 \).

Since \( 4.4 > 4 \), we can conclude that:

**Option 2: quadratic function** has the greater estimated average rate of change over the interval \([0, 1.1]\).