Question

a car is traveling at a speed of 40 miles per hour (mph). the cars speed is recorded every second until it comes to a full stop.
time (in seconds) | speed (in mph)
--- | ---
0 | 40
1 | 30
2 | 23
3 | 16
4 | 9
5 | 0
what is the average rate of change for the car coming to a full stop over the 5 - second interval, in mph per second?

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function \( y = f(x) \) over the interval \([x_1, x_2]\) is given by \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\). Here, the function is the speed of the car as a function of time, \( x \) is time (in seconds) and \( y \) is speed (in mph). We need to find the average rate of change over the interval from \( x_1 = 0 \) seconds to \( x_2=5 \) seconds. At \( x_1 = 0 \), the speed \( f(x_1)=40 \) mph, and at \( x_2 = 5 \), the speed \( f(x_2)=0 \) mph.

Step2: Substitute the values into the formula

Substitute \( x_1 = 0 \), \( f(x_1)=40 \), \( x_2 = 5 \), and \( f(x_2)=0 \) into the formula for average rate of change:

$$ \frac{f(5)-f(0)}{5 - 0}=\frac{0 - 40}{5-0} $$

Step3: Simplify the expression

First, calculate the numerator: \( 0 - 40=- 40 \). Then, calculate the denominator: \( 5 - 0 = 5 \). Now, divide the numerator by the denominator: \(\frac{-40}{5}=-8\).

Answer:

-8