Determine the average rate of change of the function on the given interval. Express your answer in exact simplest form. \( f(x)=x^{2}+3 \) (a) on \( [-2,-1] \) (b) on \( [-1,1] \) (c) on \( [3,4] \) Part: \( 0 / 4 \) Part 1 of 4 (a) on \( [-2,-1] \) Find the average rate of change from \( x_{1}=-2 \) to \( x_{2}=-1 \) When \( x_{1}=-2, f\left(x_{1}\right)=f(-2)= \) When \( x_{2}=-1, f\left(x_{2}\right)=f(-1)= \)

To determine the average rate of change of the function \( f(x) = x^{2} + 3 \) on the given intervals, we will first calculate the function's values at the specified points. For part (a) on the interval \([-2, -1]\): 1. Calculate \( f(-2) \): \[ f(-2) = (-2)^{2} + 3 = 4 + 3 = 7 \] 2. Calculate \( f(-1) \): \[ f(-1) = (-1)^{2} + 3 = 1 + 3 = 4 \] Now, find the average rate of change: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{f(-1) - f(-2)}{-1 - (-2)} \] \[ = \frac{4 - 7}{-1 + 2} = \frac{-3}{1} = -3 \] So, the average rate of change of the function \( f(x) \) on the interval \([-2, -1]\) is \(-3\). Now let's move on to part (b) on the interval \([-1, 1]\): 1. Calculate \( f(-1) \) again (it's already calculated): \[ f(-1) = 4 \] 2. Calculate \( f(1) \): \[ f(1) = (1)^{2} + 3 = 1 + 3 = 4 \] Now, find the average rate of change: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{f(1) - f(-1)}{1 - (-1)} \] \[ = \frac{4 - 4}{1 + 1} = \frac{0}{2} = 0 \] So, the average rate of change of the function \( f(x) \) on the interval \([-1, 1]\) is \(0\). For part (c) on the interval \([3, 4]\), let's compute the necessary values: 1. Calculate \( f(3) \): \[ f(3) = (3)^{2} + 3 = 9 + 3 = 12 \] 2. Calculate \( f(4) \): \[ f(4) = (4)^{2} + 3 = 16 + 3 = 19 \] Now, calculate the average rate of change: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{f(4) - f(3)}{4 - 3} \] \[ = \frac{19 - 12}{1} = 7 \] Therefore, the average rate of change of the function \( f(x) \) on the interval \([3, 4]\) is \(7\). So to summarize: (a) The average rate of change on the interval \([-2, -1]\) is \(-3\). (b) The average rate of change on the interval \([-1, 1]\) is \(0\). (c) The average rate of change on the interval \([3, 4]\) is \(7\).