Questions: The cost of producing q items is given by the cost function f(q)=0.3 q^2-0.9 q+7 thousand dollars. What is the average rate of change of f from q=3 to q=15? Round your answer to two decimal places.

The cost of producing q items is given by the cost function f(q)=0.3 q^2-0.9 q+7 thousand dollars. What is the average rate of change of f from q=3 to q=15? Round your answer to two decimal places.

Transcript text: The cost of producing $q$ items is given by the cost function $f(q)=0.3 q^{2}-0.9 q+7$ thousand dollars. What is the average rate of change of $f$ from $q=3$ to $q=15$ ? Round your answer to two decimal places.

Solution

Solution Steps

To find the average rate of change of the function $ f(q) $ from $ q = 3 $ to $ q = 15 $, we need to calculate the difference in the function values at these points and divide by the difference in the $ q $ values. This is essentially finding the slope of the secant line between these two points on the graph of the function.

Step 1: Define the Cost Function

The cost function is given by: \[ f(q) = 0.3q^2 - 0.9q + 7 \]

Step 2: Calculate Function Values at Given Points

Evaluate the function at $ q = 3 $ and $ q = 15 $: \[ f(3) = 0.3(3)^2 - 0.9(3) + 7 = 0.3 \times 9 - 2.7 + 7 = 2.7 - 2.7 + 7 = 7 \] \[ f(15) = 0.3(15)^2 - 0.9(15) + 7 = 0.3 \times 225 - 13.5 + 7 = 67.5 - 13.5 + 7 = 61 \]

Step 3: Calculate the Average Rate of Change

The average rate of change of $ f $ from $ q = 3 $ to $ q = 15 $ is given by: \[ \text{Average Rate of Change} = \frac{f(15) - f(3)}{15 - 3} = \frac{61 - 7}{12} = \frac{54}{12} = 4.5 \]