*Homework 1.3 - Average rates of change of functions Score: \( 12 / 19 \) Answered: \( 12 / 19 \) Question 13 Find the average rate of change of \( f(x)=5 x^{2}-7 \) on the interval \( [3, a] \). Your answer will be an expression involving \( a \) Question Help: Video Submit Question

To find the average rate of change of the function \( f(x) = 5x^2 - 7 \) on the interval \([3, a]\), we can use the formula for the average rate of change: \[ \text{Average Rate of Change} = \frac{f(a) - f(3)}{a - 3} \] ### Step 1: Calculate \( f(3) \) First, we need to evaluate \( f(3) \): \[ f(3) = 5(3)^2 - 7 \] ### Step 2: Calculate \( f(a) \) Next, we evaluate \( f(a) \): \[ f(a) = 5(a)^2 - 7 \] ### Step 3: Substitute into the average rate of change formula Now we can substitute \( f(3) \) and \( f(a) \) into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{(5a^2 - 7) - (5(3^2) - 7)}{a - 3} \] ### Step 4: Simplify the expression Now, let's simplify the expression step by step. 1. Calculate \( f(3) \): \[ f(3) = 5(3^2) - 7 = 5(9) - 7 = 45 - 7 = 38 \] 2. Substitute \( f(3) \) into the average rate of change formula: \[ \text{Average Rate of Change} = \frac{(5a^2 - 7) - 38}{a - 3} \] 3. Simplify the numerator: \[ 5a^2 - 7 - 38 = 5a^2 - 45 \] So, the average rate of change becomes: \[ \text{Average Rate of Change} = \frac{5a^2 - 45}{a - 3} \] ### Final Expression Thus, the average rate of change of \( f(x) = 5x^2 - 7 \) on the interval \([3, a]\) is: \[ \frac{5a^2 - 45}{a - 3} \]