00:01
Let's say we're given the following equation for the height of a ball, negative 16 t squared plus 25t plus 170, where t is your time.
00:11
And of course, this whole equation is equal to your height.
00:14
And we want to find out a little bit about the trajectory of this ball.
00:18
One thing that we can do is we can make a table.
00:21
So we could say put our time as our x value and our height as our y value.
00:31
And we can just plug in, say, every 0 .25, right? and what we're basically doing here is we're just going to take these times, and we're just going to plug them directly into our equation.
00:45
Right.
00:46
So if we do it for zero, then these first two terms actually cancel out because anything times zero is zero, and we get 170.
00:56
So this means we know we have a coordinate of 0 -170, and we can basically do that all the way down the line where if we're plugging in 0 .25, then we can't cancel those out.
01:10
We actually have to do this math here, 170.
01:15
And if we do that math in our calculator, that's going to give us 175 .25.
01:24
So we'll have 0 .25 -175 -25.
01:28
And if we keep on doing this all the way down, we'll get the following point.
01:36
0 .5178 .5, 0 .75, 179 .75, 179.
01:45
And 1179, right? and we could do this for as long as we want.
01:51
I mean, we could do it down to four, down to eight, right? but what we can see here is if we look at these y values, which remember represent the height, our height starts at 170, it goes up, but then it looks like it goes down somewhere here.
02:05
Right somewhere between 0 .75 and 1 our height has reached some kind of maximum right because because like from from for the first four is going up from 170 175 178 and now it's going down right so we know that our maximum height has to be somewhere between 0 .75 and and and and 1.
02:29
We also get a bit of a hint of this because this equation here is a quadratic because it has a t squared in it.
02:40
And because of this negative sign here, that means it has to open up this way, right? that means that it has to open up downward, right? so we know that our height has to be somewhere here.
02:52
And to find that exact value, we actually want to do some calculus.
02:56
So we want to do the first derivative test.
03:00
So we're going to take the derivative of our height equation.
03:05
Well, actually, let's just rewrite our height equation here, just so everything can be in the same place.
03:19
All right.
03:20
So now, if we're taking the derivative of this, we're going to get negative 32t.
03:26
Because remember, we take our exponent, we multiply it by the coefficient, and then we reduce the exponent by one.
03:32
And we do the same thing here with the 25t, right? we multiply it by our exponent 1, and we decrease that exponent down to 0.
03:40
So our t actually disappears.
03:44
The derivative of any constant is just zero.
03:47
So we take our first derivative, we set it equal to zero to find out what our critical points are...