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.

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.

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

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

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

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

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

Final Answer