A function is given. \( f(x)=5-3 x^{2} ; \quad x=4, x=4+h \) (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. (b)

Let's address each part of the problem step by step. Given the function: \[ f(x) = 5 - 3x^{2} \] and the two values of \( x \): \[ x = 4 \quad \text{and} \quad x = 4 + h \] --- ### **(a) Net Change Between the Given Values of the Variable** **Net change** refers to the difference in the function's value when the input changes from \( x = 4 \) to \( x = 4 + h \). Mathematically, it's expressed as: \[ \text{Net Change} = f(4 + h) - f(4) \] **Step-by-Step Calculation:** 1. **Calculate \( f(4) \):** \[ f(4) = 5 - 3(4)^2 = 5 - 3(16) = 5 - 48 = -43 \] 2. **Calculate \( f(4 + h) \):** \[ f(4 + h) = 5 - 3(4 + h)^2 \] First, expand \( (4 + h)^2 \): \[ (4 + h)^2 = 16 + 8h + h^2 \] Then substitute back: \[ f(4 + h) = 5 - 3(16 + 8h + h^2) = 5 - 48 - 24h - 3h^2 = -43 - 24h - 3h^2 \] 3. **Determine Net Change:** \[ \text{Net Change} = f(4 + h) - f(4) = (-43 - 24h - 3h^2) - (-43) = -24h - 3h^2 \] **Answer for Part (a):** \[ \text{Net Change} = -24h - 3h^2 \] --- ### **(b) Average Rate of Change Between the Given Values of the Variable** The **average rate of change** of the function between \( x = 4 \) and \( x = 4 + h \) is given by: \[ \text{Average Rate of Change} = \frac{f(4 + h) - f(4)}{h} \] Using the net change from part (a): \[ \text{Average Rate of Change} = \frac{-24h - 3h^2}{h} \] **Simplify the Expression:** \[ \text{Average Rate of Change} = -24 - 3h \] **Answer for Part (b):** \[ \text{Average Rate of Change} = -24 - 3h \] --- ### **Summary** - **(a)** The net change is \( -24h - 3h^2 \). - **(b)** The average rate of change is \( -24 - 3h \).