Linearization Error: Find Approximate Error of f(x) |x| < 1

[santais]

Homework Statement

I've been given the assignment to make a linearization of the function f(x) = Sin(x) e^(x) by using Taylors polynominal of 3. degree.

Next is to find the approximately error of f(x) when |x| < 1 (expressed as a function of x).

Homework Equations

So finding the linearization is rather easy to get through, however finding the estimated error is where I just can't figure it out.

The formula that I've been given from my book is as follows:

$(f''(s)/2)(x-a)^2$

for some s lying between a og x. But finding the s value is just something, I have no idea how to figure out. I tried to search some and found out that you could just find the maximum y- value of your function, within that interval, and put it instead of f''(s). Tried that and got a very unexact approximation.

Hope some of you have an idea, how to solve this problem :)

 

[Ray Vickson]

Homework Statement

I've been given the assignment to make a linearization of the function f(x) = Sin(x) e^(x) by using Taylors polynominal of 3. degree.

Next is to find the approximately error of f(x) when |x| < 1 (expressed as a function of x).

Homework Equations

So finding the linearization is rather easy to get through, however finding the estimated error is where I just can't figure it out.

The formula that I've been given from my book is as follows:

$(f''(s)/2)(x-a)^2$

for some s lying between a og x. But finding the s value is just something, I have no idea how to figure out. I tried to search some and found out that you could just find the maximum y- value of your function, within that interval, and put it instead of f''(s). Tried that and got a very unexact approximation.

Hope some of you have an idea, how to solve this problem :)

You are using the wrong error formula. The formula above is for the error in a _linear_ approximation (i.e., using a polynomial of degree 1). You have been asked to use a polynomial of degree 3 (a cubic approximation), so the error will involve the 4th derivative.

Anyway, you can easily estimate the "maximum possible absolute error" as a function of (x-a), just by estimating max_s{|f''''(s)|, a <= s <= x} and using that instead of |f''''(s)|. However, to do this we need to be able to compute exactly the values of exp(x)*sin(x) at various x. This would be OK if we wanted to do this once and then use the approximate formula thereafter (in other words, do a lot of work off-line in order to validate a formula to be used on-line). An alternative might be to use a cruder estimate that does not involve already knowing how to compute exp(x)*sin(x). Again, though, you would be estimating the maximum possible error, not the actual error itself.

RGV

 

[santais]

You are using the wrong error formula. The formula above is for the error in a _linear_ approximation (i.e., using a polynomial of degree 1). You have been asked to use a polynomial of degree 3 (a cubic approximation), so the error will involve the 4th derivative.

Anyway, you can easily estimate the "maximum possible absolute error" as a function of (x-a), just by estimating max_s{|f''''(s)|, a <= s <= x} and using that instead of |f''''(s)|. However, to do this we need to be able to compute exactly the values of exp(x)*sin(x) at various x. This would be OK if we wanted to do this once and then use the approximate formula thereafter (in other words, do a lot of work off-line in order to validate a formula to be used on-line). An alternative might be to use a cruder estimate that does not involve already knowing how to compute exp(x)*sin(x). Again, though, you would be estimating the maximum possible error, not the actual error itself.

RGV

Thanks for the answer. And sorry for the late reply.

I'm still quite unclear of what you mean, to be exact. In words, the assigment says that I have to estimate the error for |x| < 1. So I guess that has to be the maximum error, in form of a function.

But then you say, then when I have a 3rd degree Taylor's polynominal, I have to use f^(n+1) = f''''(s). But how exactly to find the maximum value of s within that interval numerical? One thing is that I can see it quite clearly on the graph, but if it happens to be a graph, where you almost have to identical points, with just a slightly difference, then there must be some way to solve it numerical.