A function is given. \[ f(t)=4 t^{2} ; \quad t=3, t=3+h \] (a) Determine the net change between the given values of the variable. \( \square \) (b) Determine the average rate of change between the given values of the variable. \( \square \)

To find the net change of the function \( f(t) = 4t^2 \) between \( t = 3 \) and \( t = 3 + h \), we first need to calculate the function values at these points. (a) The net change is calculated as follows: \[ f(3 + h) - f(3) = 4(3 + h)^2 - 4(3)^2 \] Calculating each term: \[ f(3) = 4(3^2) = 4 \times 9 = 36 \] Now, expanding \( f(3 + h) \): \[ f(3 + h) = 4((3 + h)^2) = 4(9 + 6h + h^2) = 36 + 24h + 4h^2 \] Now, substituting back: \[ \text{Net change} = (36 + 24h + 4h^2) - 36 = 24h + 4h^2 \] (b) The average rate of change is given by the formula: \[ \frac{f(3 + h) - f(3)}{(3 + h) - 3} = \frac{(24h + 4h^2)}{h} \] This simplifies to: \[ 24 + 4h \] So, the average rate of change between the given values of the variable is \( 24 + 4h \).