Questions: A function is given. f(x) = 3 - 3x^2 ; x = 5, x = 5 + h (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) Net Change: Calculate the function values at \( x = 5 \) and \( x = 5 + h \), then find the difference between these values.

(b) Average Rate of Change: Use the net change calculated in part (a) and divide it by the change in \( x \), which is \( h \).

Given the function \( f(x) = 3 - 3x^2 \), we need to calculate the function values at \( x = 5 \) and \( x = 5 + h \) where \( h = 1 \).

\[ f(5) = 3 - 3(5)^2 = 3 - 75 = -72 \]

\[ f(5 + 1) = f(6) = 3 - 3(6)^2 = 3 - 108 = -105 \]

The net change between the values of the function at \( x = 5 \) and \( x = 5 + h \) is given by:

\[ \Delta f = f(5 + h) - f(5) = -105 - (-72) = -105 + 72 = -33 \]

The average rate of change of the function between \( x = 5 \) and \( x = 5 + h \) is given by:

\[ \text{Average Rate of Change} = \frac{\Delta f}{\Delta x} = \frac{-33}{1} = -33.0 \]

Final Answer