00:01
All right, so we're going to take a look at the function y equals sine of theta, and we're going to find the local linear approximation at x equals zero, and the general form is l of theta equals y sub zero plus y prime of zero times theta minus zero.
00:18
And that's just the tangent line approximation.
00:21
On the side here on the top right, notice we need to find those values.
00:27
Y of zero is sine of zero, which is zero.
00:29
The derivative is cosine and cosine of zero is one.
00:34
So let's go ahead and now substitute in these values.
00:37
So l of theta then is zero plus slope of one times theta minus zero.
00:44
So this is kind of cool.
00:46
Our local linearization just gives us theta.
00:51
So therefore, sine of theta is approximately equal to theta.
00:55
So we can actually see how good approximation that is.
01:00
What we're going to do is we're going to see our difference between our local linear approximation and our true value.
01:11
So sign of point 0 .01, therefore is approximately .01.
01:18
And sign of 0 .1 is approximately .1.
01:23
This is using local linear and sign of one is approximately one.
01:31
Now normally as you get further and further away from where you did the local linearization your error really increases so we'll take a look at that.
01:39
Okay so now we're going to look at the true values for sign and then we're going to look at the absolute error which is just a difference absolute error.
01:52
So just subtracting the two.
01:54
Okay, so actual sign of 0 .01 is 0 .09998, with an absolute error of 1 .667 times 10 to the minus 7...