Which function has the greater estimated average rate of change over the interval [0,1.1]

Question

, 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.