1. 4.4 Average Value and Physics Connections

2. Average Value of a Function
The average value of a continuous function f(x) on some interval from a to b is given by the equation
It looks like an average, doesn’t it.

3. Example Find the average value of the function
on the interval from 1 to 5

4. Physics
The relationship between velocity v(t), acceleration a(t) and position x(t) is this:
To use this we need to understand 2 definitions.

5. Net Change vs Total Distance
If you drive to Starbucks then back (and park in the very same parking place) your net change is 0, right?
But, the odometer value has changed. That is, the total distance traveled is NOT zero. This works in physics when a particle is moving along a straight line path (typical situation)

6. Net Change
Net Change = displacement = net change in the position of a body moving along a line from a to b.
Mathematically
where some distances will be positive (“up” or “forward) while others will be negative (“down” or “backward”). The Net change is the result of these movements.

7. Total Distance
Total Distance = the sum of all the distances. Therefore, you will need to know where it changes from + to – directionally, and add the absolute values of all these.

8. WHAT? Imagine. A particle moves forward 10 m then backwards 7 m.
Net Change? 3 m
Total Distance? 17 m

9. Well, how do I do that?
Often, a sketch of the velocity is a very good idea. This will quickly identify where the distance is positive and where it is negative. Take the absolute value of anything negative and add all of these numbers.

10. Examples 1. Given that a particle travels according to the equation
Find the net change AND the total distance traveled by the particle.
Homework: Worksheet