A function is given.
f(z) = 3 − 4z2; 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.
(a) To find the net change, we need to evaluate f(z) at both z = -2 and z = 0 and subtract the two values.
At z = -2:
f(-2) = 3 - 4(-2)^2
= 3 - 4(4)
= 3 - 16
= -13
At z = 0:
f(0) = 3 - 4(0)^2
= 3 - 4(0)
= 3 - 0
= 3
Net change = f(0) - f(-2)
= 3 - (-13)
= 3 + 13
= 16
Therefore, the net change between the given values of the variable is 16.
(b) To find the average rate of change, we need to divide the net change by the difference in the values of the variable.
Difference in values of the variable = 0 - (-2) = 2
Average rate of change = Net change / Difference in values of the variable
= 16 / 2
= 8
Therefore, the average rate of change between the given values of the variable is 8.
To find the net change between the given values of the variable, we need to subtract the function values at the two values of z.
(a) The net change is given by:
Net Change = f(z2) - f(z1)
Given that z1 = -2 and z2 = 0, we can substitute these values into the function:
f(z1) = 3 - 4z1^2
f(z2) = 3 - 4z2^2
Substituting the values, we get:
f(z1) = 3 - 4(-2)^2 = 3 - 4(4) = 3 - 16 = -13
f(z2) = 3 - 4(0)^2 = 3 - 4(0) = 3 - 0 = 3
Net Change = f(z2) - f(z1) = 3 - (-13) = 3 + 13 = 16
Therefore, the net change between the given values of the variable is 16.
(b) The average rate of change between the given values of the variable is given by:
Average Rate of Change = Net Change / Change in the variable
In this case, the change in the variable is the difference between the two values of z:
Change in the variable = z2 - z1 = 0 - (-2) = 2
Substituting the values, we get:
Average Rate of Change = Net Change / Change in the variable = 16 / 2 = 8
Therefore, the average rate of change between the given values of the variable is 8.