Is linear approximation is to do with differentiation?

joeperry

Hi can somebody please tell me if linear approximation is to do with differentiation?

I have an exam soon and apparently the topics coming up in exam are Differentiation and taylor series.

Fremen

Yep. You can see this by doing an experiment.

Draw any differentiable function. Let's say we draw it on [-2,2]. Pick a point on the function - say (x,y) = (0, f(0)).

Now "zoom in" by re-drawing the function near your point over a smaller interval. We'll re-draw the function on the interval [-1,1], but we'll rescale out axes so our new graph is the same size as our old graph.

We can do this over and over again. As we zoom closer and closer in on this function, it will start to look more and more like a straight line. The line that it looks like is the best linear approximation to the function at the point 0, and the slope of that line is the derivative at 0.

In essence, differentiable functions are 'locally linear'. When you zoom in on them, they look like straight lines.

Try plotting functions at wolframalpha to see this graphically.

joeperry

Hi again, is this taylor series by another name? Thanks.

Brussels Sprout

No, the Taylor Series is basically a way of representing a function using a series. You need to use calculus in order to calculate each term of the series. The more terms you include in your series, the more accurately it will approximate the function

e.g. :



The red line is based on a series that consists of n terms. The blue line is the original function

Well, it is Taylor Series by another name. If you restrict the Taylor series to the terms up until the term involving x, you get a linear approximation.