Background

Wednesday, 5 September 2018

1:10 PM

We know that the tangent at a point is a good approximation to the function near that point Let 𝑓﷐𝑥﷯=﷐𝑒﷮𝑥﷯ Ink Drawings Ink Drawings Ink Drawings Ink Drawings    Ink Drawings Ink Drawings Ink Drawings   Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings Ink Drawings

The equation of the tangent line to the graph of  at 0 is

The tangent function  at point 0 is a polynomial of degree 1.

 

So when x is close to 0,

 

Note that the polynomial  has the property that

 

i.e.

 

 

If we want to improve our approximation, we could generalise this idea and write

And

 

 

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