Questions: Find the average rate of change of the function g(x) = 5x^2 - 7 as x changes from -2 to 1. The average rate of the function g as x changes from -2 to 1 is (Type an integer or a simplified fraction.)

$Find the average rate of change of the function g(x) = 5x^2 - 7 as x changes from -2 to 1. The average rate of the function g as x changes from -2 to 1 is (Type an integer or a simplified fraction.)$

Transcript text: Find the average rate of change of the function $g(x)=5 x^{2}-7$ as $x$ changes from -2 to 1 . The average rate of the function g as $\times$ changes from -2 to 1 is (Type an integer or a simplified fraction.)

Solution

Solution Steps

To find the average rate of change of the function $ g(x) = 5x^2 - 7 $ from $ x = -2 $ to $ x = 1 $, we calculate the difference in the function values at these points and divide by the difference in $ x $-values. This is essentially finding the slope of the secant line between these two points on the graph of the function.

Step 1: Calculate Function Values

Evaluate the function $ g(x) = 5x^2 - 7 $ at $ x = -2 $ and $ x = 1 $.

\[ g(-2) = 5(-2)^2 - 7 = 5 \times 4 - 7 = 20 - 7 = 13 \]

\[ g(1) = 5(1)^2 - 7 = 5 \times 1 - 7 = 5 - 7 = -2 \]

Step 2: Calculate the Average Rate of Change

The average rate of change of $ g(x) $ from $ x = -2 $ to $ x = 1 $ is given by:

\[ \frac{g(1) - g(-2)}{1 - (-2)} = \frac{-2 - 13}{1 + 2} = \frac{-15}{3} = -5 \]