Questions: Find the average rate of change of g(x)=-2x^3+5x^2 from x=1 to x=3 Simplify your answer as much as possible.

Transcript text: Find the average rate of change of $g(x)=-2 x^{3}+5 x^{2}$ from $x=1$ to $x=3$ Simplify your answer as much as possible.

Solution

Solution Steps

To find the average rate of change of the function $ g(x) = -2x^3 + 5x^2 $ from $ x = 1 $ to $ x = 3 $, we need to calculate the difference in the function values at these points and divide by the difference in the $ x $-values. This is essentially finding the slope of the secant line between these two points on the graph of the function.

Evaluate $ g(x) $ at $ x = 1 $ and $ x = 3 $.
Calculate the difference in the function values: $ g(3) - g(1) $.
Divide this difference by the difference in $ x $-values: $ 3 - 1 $.

Step 1: Evaluate the Function at $ x = 1 $ and $ x = 3 $

To find the average rate of change, we first evaluate the function $ g(x) = -2x^3 + 5x^2 $ at the given points.

At $ x = 1 $: \[ g(1) = -2(1)^3 + 5(1)^2 = -2 \times 1 + 5 \times 1 = 3 \]
At $ x = 3 $: \[ g(3) = -2(3)^3 + 5(3)^2 = -2 \times 27 + 5 \times 9 = -54 + 45 = -9 \]

Step 2: Calculate the Difference in Function Values

Next, we calculate the difference in the function values at these points: \[ g(3) - g(1) = -9 - 3 = -12 \]

Step 3: Calculate the Difference in $ x $-Values

The difference in the $ x $-values is: \[ 3 - 1 = 2 \]

Step 4: Compute the Average Rate of Change

The average rate of change of the function from $ x = 1 $ to $ x = 3 $ is given by: \[ \frac{g(3) - g(1)}{3 - 1} = \frac{-12}{2} = -6.0 \]