\( 1<\begin{array}{l}\text { Watch the video and then solve the problem given below. } \\ \text { Click here to watch the videe. } \\ \text { Find the average rate of change of the function } f(x)=x^{2}+5 x \text { from } x_{1}=4 \text { to } x_{2}=7 \text {. } \\ \text { The average rate of change is } \square \text {. (Simplify your answer.) }\end{array} \)

To find the average rate of change of the function \( f(x) = x^2 + 5x \) from \( x_1 = 4 \) to \( x_2 = 7 \), you use the formula: \[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \] First, you calculate the function values: - \( f(4) = 4^2 + 5(4) = 16 + 20 = 36 \) - \( f(7) = 7^2 + 5(7) = 49 + 35 = 84 \) Now substitute back into the formula: \[ \text{Average Rate of Change} = \frac{84 - 36}{7 - 4} = \frac{48}{3} = 16 \] So, the average rate of change is \( 16 \). Now, let's explore this concept a bit more! The history behind calculating average rates of change is deeply rooted in calculus, which emerged to address the need for understanding motion and variation. Key figures like Isaac Newton and Gottfried Wilhelm Leibniz laid the groundwork for calculus in the late 17th century, helping mathematicians analyze how things change—be it the speed of a falling apple or the growth of a population! In the real world, understanding average rates of change is essential in various fields, from economics when calculating profit margins over time, to physics when analyzing speed. It equips you to make informed decisions, whether you're budgeting for a project or predicting how long it will take to travel a certain distance under constant speed. So next time you encounter a change, think—how does it average out over time?