Questions: Find the average rate of change for the given function. f(x)=0.4794 e^(1.462 sqrt(x)) between x=6 and x=6.5 The average rate of change is (Type an integer or decimal rounded to three decimal places as needed.)

Find the average rate of change for the given function.
f(x)=0.4794 e^(1.462 sqrt(x)) between x=6 and x=6.5

The average rate of change is
(Type an integer or decimal rounded to three decimal places as needed.)

Transcript text: Find the average rate of change for the given function. \[ f(x)=0.4794 e^{1.462 \sqrt{x}} \text { between } x=6 \text { and } x=6.5 \] The average rate of change is $\square$ (Type an integer or decimal rounded to three decimal places as needed.)

Solution

Solution Steps

Step 1: Define the Function and Interval

We are given the function $ f(x) = 0.4794 e^{1.462 \sqrt{x}} $ and the interval $[6, 6.5]$.

Step 2: Evaluate the Function at the Endpoints

First, we evaluate the function at the endpoints of the interval: \[ f(6) = 0.4794 e^{1.462 \sqrt{6}} \approx 17.2176 \] \[ f(6.5) = 0.4794 e^{1.462 \sqrt{6.5}} \approx 19.9287 \]

Step 3: Calculate the Average Rate of Change

The average rate of change of the function over the interval $[6, 6.5]$ is given by: \[ \text{Average Rate of Change} = \frac{f(6.5) - f(6)}{6.5 - 6} \] Substituting the values we found: \[ \text{Average Rate of Change} = \frac{19.9287 - 17.2176}{6.5 - 6} \approx \frac{2.7111}{0.5} \approx 5.4222 \]