00:03
We have the function y equals 3 plus 4 x squared minus 2 x to the third.
00:10
The graph of this function will be a curve.
00:13
And the first thing that we want to do is find the slope of the line that is tangent to this curve at the point x equals a.
00:24
So the slope of a line tangent to the graph of this function to the curve at the point x equals a.
00:33
Will simply be the derivative of this function evaluated at a.
00:39
So the slope of a tangent line at a point a is equal to the derivative of the function at the point a.
00:46
Well, the derivative of 3 plus 4x squared minus 2x to the cubed, derivative of 3 constant is 0.
00:54
Derivative of 4x squared is 8 times x minus the derivative of 2x cubed is 6x squared.
01:03
And we have to evaluate this derivative.
01:09
That's y prime.
01:10
We have to evaluate it when x equals a.
01:13
So for any number a, the slope of the tangent line at x equals a will be 8 times x, 8a, minus 6 times x squared, or minus 6 times a squared.
01:27
So that is the slope of the tangent line to this function at the point x equals a.
01:43
Next, we're going to look at two different tangent lines to this function.
01:50
One tangent line is going to touch the graph at the point 1 comma 5.
01:58
So x1 is going to be 1.
02:01
Y 1 is going to be 5.
02:03
And if we want the equation of that tangent line, we're going to use the point slope form of the equation of a line from eligible 1.
02:14
Minus y1 equals m times x minus x1 so we are finding the equation of a line that is tangent to this graph to this curve at this point so specifically when the x value is one and that's important because the x one and the they are, but the slope is going to be given by the y prime of a expression that we found.
02:55
The slope of a tangent line is going to equal the derivative of the function at that particular value of x.
03:03
So the slope of this tangent line is going to equal 8 times a, or in this case, a, the value that you're plugging in for x, okay? think of this is your x coordinate that gets plugged into the derivative.
03:18
Your derivative is 8 times a minus six times a squared, depending on what value x is, depending on, you know, what is the x coordinate where the tangent line is touching the graph.
03:32
Well, the x coordinate is, in this case, one.
03:35
So we're going to plug in 1 where we see a.
03:39
So the slope of this tangent line is going to be 8 times a, 8 times 1, minus 6 times a squared or 6 times 1 squared.
03:51
And that's simple enough because 1 squared is 1 times 6 is 6, so 8 minus 6 is 2.
03:58
And so we have the equation of this particular tangent line, y minus y 1, 5 equals the slope m2 times x, minus x1, of course x1 is one.
04:14
Doing just a little bit of algebra, we would distribute this multiplication here.
04:19
We would get 2x minus 2, and then we would add the 5 to both sides.
04:25
So not writing down those basic algebra steps, we get y equals 2x.
04:31
And like i said, this would have been minus 2, and then you add the 5 to both sides, so it's plus straight.
04:41
And so the equation of a line that is tangent to our function y of x at this point is y equals 2 times x plus 3.
04:53
Now ultimately, we want to be able to graph the function, which you see here, along with the two tangent lines...