00:01
For this problem, we are asked to find the values of a and b such that the line y equals negative 2x plus b is tangent to the curve y equals ax to the third at x equal to negative 3.
00:16
When we're thinking about tangent, we must automatically think.
00:20
So if you think about a tangent, we immediately must think about that the slopes are the same, right? if we're thinking about a tangent here, if i just have a random curve, let's say i have something like this, and i'm at x equal to negative 3, at this point in particular, these slopes must be the same right here.
00:44
And so if we're going to have the slopes be the same, this curve, our line has a slope of negative 2.
00:51
If i've written this in point slope form, moving that over.
00:54
I want to find when, firstly, this function has a slope of negative 2.
01:03
What value of a has a slope of negative 2 when x is equal to negative 3? okay, that's my first point.
01:14
And secondly, i also want to find this b here such that this line actually touches that point too.
01:22
So there's two points we need to make.
01:23
The first one will solve for a.
01:25
In order to solve for a we need to find when this function has a because a has a derivative that is equal to negative two at x equal to negative three so if i'm thinking about the derivative we take the derivative and we will get three a x squared we then plug in our value of negative three and we will get three a negative three squared and so this is going to give us 27a and we want this specifically to be equal to negative 2.
01:58
We want to find the a that gives us this because we want this slope to be equal to negative 2.
02:05
And so therefore, a has to be equal to negative 2 over 27.
02:13
If this happens, then we will have a slope of negative 2.
02:17
So a has to be negative 2 over 27.
02:20
Secondly, here, we need to find when this curve, when this line hits this function at negative 3...