Theory

Theory 1

Linear approximation is the technique of approximating a specific value of a function, say , at a point that is close to another point where we know the exact value . We write for , and , and . Then we write and use the fact that:

Computing a linear approximation

For example, to approximate the value of , set , set and , and set so .

Then compute: So .

Finally:

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Now recall the linearization of a function, which is itself another function:

Given a function , the linearization at the basepoint is the functional form of the tangent line, the line passing through with slope :

The graph of this linearization is the tangent line to the curve at the point .

The linearization may be used as a replacement for for values of near . The closer is to , the more accurate the approximation is for .

Computing a linearization

We set , and we let .

We compute , and so .

Plug everything in to find :

Now approximate :

Theory 2

Taylor polynomials

The Taylor polynomials of a function are the partial sums of the Taylor series of :

These polynomials are generalizations of linearization. Specifically, , and .

The Taylor series is a better approximation of than for any .

Facts about Taylor series

The series has the same derivatives as at the point . This fact can be verified by visual inspection of the series: apply the power rule and chain rule, then plug in and all factors left with will become zero.

The difference vanishes to order at :

The factor drives the whole function to zero with order as .

If we only considered orders up to , we might say that and are the same near .