00:01
In this question, an arrow is shot upwards and its height is given by the function f.
00:10
And in this question, we're asked to estimate the instantaneous velocity of the arrow at t equals 4, using h equals 0 .1, 0 .01 and 0 .001.
00:25
Recall that the instantaneous velocity at t equals 4 ,000, by definition equals to the limit of f of 4 plus h minus f of 4 divided by h as h goes to 0.
00:50
Now let's simplify the expression f of 4 plus h minus f of 4 divided by h.
01:03
So to get f of 4 plus h, let's calculate f of 4 plus h.
01:16
To get f of 4 plus h, we need to replace t by 4 plus 8.
01:20
In the formula for f.
01:22
We are going to get negative 16 times 4 plus h squared plus 160 times 4 plus h which equals to negative 16 times 16 plus 8h plus h squared plus 160 multiplied by 4 plus h this is f of 4 plus h let's calculate f of 4 2.
02:06
F of 4 equals to negative 16 times 4 squared plus 160 multiplied by 4.
02:18
Now we will plug in these expressions in the quotient.
02:26
So we will replace f of 4 plus h by negative 16 times 16 plus 8h plus h squared, right? plus 160 times 4 plus h minus f of 4, which equals to negative 16, times 4 squared plus 160 times 4.
02:57
All right, unfortunately i cannot feed it in one line, divided by h.
03:18
Now what we can do is, first of all, note that 4 squared is 16, and we can cancel negative 16 times 4 with negative 16 times 16.
03:36
We can also cancel 160 times 4 with 160 times 4 here.
03:43
And the whole expression will simplify to negative 16 times 8h plus h squared plus 160 h and that's all right.
04:07
That's all that's what is left after cancellations.
04:16
And then note that we can factor out h in the numerator.
04:20
We are going to get h times negative 16 multiplied by 8.
04:25
Plus h plus 160 divided by h so now we can cancel h and negative 16 times 8 equals to 80 plus 48 negative 16 minus negative 160 minus 128 equals to 32 so we're going to get 32 minus 19 minus 16 alright, this is a value of the quotient of this quotient here...