Taylor approximations (1 Viewer)

[EinstenICEBERG]

Before we move on to more rules for the derivative, we’ll introduce a useful
application of it known as first-order Taylor approximations, according to which
we can approximate the change in the value of a function by using its derivative.2
Specifically, we have

f(x + h) = f(x) + (df(x)/dx)*h + R(x, h)h

where R(x, h) is some average approximation error that will approach zero as h
approaches zero. We won’t prove this theorem, but understand it using figure
(3). Recall that when taking a derivative, we are calculating the slope of a
function in one point, represented by the slope of the tangent line in the figure.
The Taylor approximation works by assuming that the slope of the function
is constant and approximating f(x + h) by the value of the tangent at x + h.
Since the derivative of f(x) is not generally constant, this gives rise to the
(total) approximation error R(x, h)h. However, as we let h approach zero, we
are approaching the point where the slope of the function is indeed exactly
df(x)/dx, and R(x, h) will also approach zero.

can someone simplify this for me, I trried understanding through many links but I don't get it still =(

 

[seanieg89]

Which part is confusing you? This is basically just the definition of differentiability.

Remember that we say that f(x) is differentiable at a if



exists. If it does, we denote this quantity by



which will just be a real number depending on a that tells us the slope of the line tangent to the graph of f(x) at x=a.

Rewriting the limit we get:



So for every a at which f is differentiable, the bracketed quantity tends to zero as h tends to zero.

We denote this bracketed quantity by R(a,h).

Then multiplying the expression for R through by h and rearranging, we get



where R is a function that tends to zero as h tends to zero, regardless of what a is.



The intuitive idea here is that if you zoom in on a smoothish (differentiable) curve, it looks pretty much like a straight line. So we expect that we can approximate such a function by a linear one (the tangent at a which has equaion L(a+x)=f(a)+xf'(a)), and that this approximation will be better and better closer to the point of tangency.

The above working gives a concrete fact about how good this approximation is. More precise things can be said about this approximation if you have more information about f.