00:01
So in this question, a particle moves along the x -axis so that its velocity at time t from times zero to five is given by v of t, which is three times the quantity of t -minus -1 times the quantity of t -minus -3.
00:15
And we want to find the average velocity of the particle over the time interval from zero to five.
00:21
This information here about the position of the particle at time two is irrelevant for this question.
00:30
So for this question, we're going to remember our formula for the average value of a function f of x on the closed interval from a to b.
00:44
The average value of f of x on the closed interval from a to b is given by 1 over b minus a times the integral from a to b of f of x, dx.
00:58
So here i'm given my velocity, and i want to find my average velocity.
01:08
And so we're finding the average value of this velocity function.
01:15
So my average velocity will be 1 over 5 minus 0 times the integral from 0 to 5 of v of t d t, that is 1 5th, times the integral from 0 to 5 of 3 times the quantity of t minus 1 times the quantity of t minus 3 d t.
01:48
Now that 3, that's a constant multiple, so i can slide that through my interval.
01:54
I can say i have 3 5ths times the integral from 0 to 5, and i'm going to have to multiply out these two binomials.
02:04
When i multiply these out, i'm getting t squared minus 4t plus 3 dt.
02:16
So now i'm ready to get my anti -derivative.
02:21
What's my anti -derivative of t squared? that's t cubed over 3.
02:27
Minus, what's my anti -derivative of 4t? that's 2t squared.
02:35
While my anti -derivative of 3, that's just 3t...