Questions: A function is given. f(z)=5-2 z^2 ; z=-2, z=0 (a) Determine the net change between the given values of the variable. (b) Determine the average rate of change between the given values of the variable.

Solution Steps

To solve the given problem, we need to follow these steps:

(a) Calculate the net change in the function values at the given points \( z = -2 \) and \( z = 0 \).

(b) Calculate the average rate of change of the function between the given points \( z = -2 \) and \( z = 0 \).

We are given the function \( f(z) = 5 - 2z^2 \) and the points \( z = -2 \) and \( z = 0 \).

First, we calculate the function values at these points: \[ f(-2) = 5 - 2(-2)^2 = 5 - 2 \cdot 4 = 5 - 8 = -3 \] \[ f(0) = 5 - 2(0)^2 = 5 - 0 = 5 \]

The net change in the function values between \( z = -2 \) and \( z = 0 \) is: \[ \Delta f = f(0) - f(-2) = 5 - (-3) = 5 + 3 = 8 \]

The average rate of change of the function between \( z = -2 \) and \( z = 0 \) is: \[ \text{Average Rate of Change} = \frac{\Delta f}{\Delta z} = \frac{f(0) - f(-2)}{0 - (-2)} = \frac{8}{2} = 4.0 \]

Final Answer