Use the information in Example 7 to find the average speed...

Use the information in Example 7 to find the average speed of the car from $t_{1}=0$ to $t_{2}=9$ seconds. Explain why the result is less than the value obtained in part (b) of Example $7 .$ Physics In Exercises $67-70$ , (a) use the position equation $s=-16 t^{2}+v_{0} t+s_{0}$ to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from $t_{1}$ to $t_{2},$ (d) describe the slope of the secant line through $t_{1}$ and $t_{2},($ e) find the equation of the secant line through $t_{1}$ and $t_{2},$ and $(f)$ graph the secant line in the same viewing window as your position function.

Use the information in Example 7 to find the average speed of the car from $t_{1}=0$ to $t_{2}=9$ seconds. Explain why the result is less than the value obtained in part (b) of Example $7 .$
Physics In Exercises $67-70$ , (a) use the position equation $s=-16 t^{2}+v_{0} t+s_{0}$ to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from $t_{1}$ to $t_{2},$ (d) describe the slope of the secant line through $t_{1}$ and $t_{2},($ e) find the equation of the secant line through $t_{1}$ and $t_{2},$ and $(f)$ graph the secant line in the same viewing window as your position function.

Key Concepts

Average Rate of Change

This concept describes how a quantity, such as position, changes on average over a given time interval. In physics, the average rate of change of position with respect to time is equivalent to the average speed, which is calculated as the difference in position divided by the elapsed time. It provides an overall measure of motion that does not account for any fluctuations that occur at individual instants.

Instantaneous Rate of Change

The instantaneous rate of change is the derivative of the position function, representing the speed (or velocity) at a specific moment in time. Unlike the average rate, it captures the exact rate of motion at an instant, which can be higher than the average due to factors like acceleration that cause the speed to vary over the interval.

Secant Line

The secant line is a straight line connecting two points on the graph of a function. Its slope represents the average rate of change over the interval between these two points. In motion problems, the slope of the secant line corresponds to the average speed, providing a visual representation of how the overall change in position compares to instantaneous changes.

Quadratic Position Function

This function models position under constant acceleration and is typically expressed as a quadratic equation. It forms the basis for analyzing projectile motion or similar scenarios in physics. Understanding this equation is essential for extracting relevant information such as displacement, velocity, and acceleration, and for linking graphical interpretations to algebraic computations.

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