Messy Taylor polynomial question

In summary, the author is trying to find a taylor polynomial approximation about the point ε = 1/2 for the following function: (x^1/2)(e^-x) and is having trouble finding the second derivative. They eventually find that if x= \frac{1}{2} then t=0, and so they can use the MacLaurin expression: \frac{\sqrt{2}}{2}\sqrt{1+2t}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} t-\frac{\sqrt{2}}{4}t^2+o(t^2). They ask
  • #1
kwal0203
69
0

Homework Statement



Find the Taylor polynomial approximation about the point ε = 1/2 for the following function:

(x^1/2)(e^-x)

The Attempt at a Solution



I'm trying to get a taylor polynomial up to the second derivate i.e.:

P2(×) = (×^1/2)(e^-x) + (x-ε) * [(e^-x)(1-2×)/2(×^1/2)] + [[(x-ε)^2]/2!] * ?

I can't find the second derivative here it just becomes a big mess am I missing something?

Thanks any help appreciated!
 
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  • #2
Let t= x-1/2 so

[itex]\sqrt{x}= \sqrt{x-\frac{1}{2}+\frac{1}{2}}= \sqrt{t+\frac{1}{2}}=[/itex]

[itex]\frac{\sqrt{2}}{2} \sqrt{1+2t}[/itex]

and

[itex]e^{-x}= e^{-x+\frac{1}{2}-\frac{1}{2}}= e^{-t-\frac{1}{2}}= \frac{1}{\sqrt{e}}e^{-t}[/itex]

Observe that if [itex]x= \frac{1}{2}[/itex] then [itex]t=0[/itex], so you can use the MacLaurin expression:

[itex]\frac{\sqrt{2}}{2}\sqrt{1+2t}=\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2} t-\frac{\sqrt{2}}{4}t^2+o(t^2) [/itex]

and

[itex]\frac{1}{\sqrt{e}}e^{-t}= \frac{1}{\sqrt{e}}-\frac{t}{\sqrt{e}}+\frac{t^2}{2\sqrt{e}}+o(t^2)[/itex]


(You have to remember the Maclaurin series of the functions:

[itex]e^{s}=1+s+\frac{s^2}{2}+... [/itex]

[itex]\sqrt{1+s}=1+\frac{s}{2}-\frac{s^2}{8}+... [/itex]


)

Now multiply:

[itex]\frac{\sqrt{2}}{2}\sqrt{1+2t}\frac{1}{\sqrt{e}}e^{-t}=\frac{1}{\sqrt{2 e}}-\frac{t^2}{\sqrt{2 e}}+o(t^2)[/itex]

but [itex]t= x-\frac{1}{2}[/itex] so

[itex]\frac{\sqrt{2}}{2}\sqrt{1+2\left(x-\frac{1}{2}\right)}\frac{1}{\sqrt{e}}e^{-(x-\frac{1}{2})}=\frac{1}{\sqrt{2 e}}-\frac{(x-\frac{1}{2})^2}{\sqrt{2 e}}+o((x-\frac{1}{2})^2)[/itex]
 
  • #3
Wow thanks, this is one question in a group of six and seems to be significantly harder than the others for some reason.

Can you provide me with any insight as to how you came up with the solution? Is it just a lot of practice and seeing similar problems in the past?

I want to develop my problem solving skills but I would never have come up with this method!
 
  • #4
It is a classic exercise on Taylor series :) It is very important to know the macLaurin series of common functions and all the properties of "elementary functions". Practice will help you! :)

[Sorry, my English is awful :|]
 

What is a Taylor polynomial?

A Taylor polynomial is a mathematical function that approximates a more complex function by using a series of simpler polynomial functions. It is used to estimate the value of a function at a specific point, by using information about the function at nearby points.

How is a Taylor polynomial calculated?

A Taylor polynomial is calculated using a process called Taylor series expansion. This involves finding the coefficients of a series of polynomial functions that can be used to approximate the original function.

What is a "Messy" Taylor polynomial?

A "Messy" Taylor polynomial refers to a Taylor polynomial that is not a perfect representation of the original function. This can occur when the function is highly complex or has a large number of terms, making it difficult to accurately approximate.

Why use a Taylor polynomial instead of the original function?

Taylor polynomials are useful for approximating functions because they can be simpler and easier to work with, while still providing a good estimate of the original function. They can also be used to find derivatives and integrals of functions that may otherwise be difficult to solve.

What are some real-world applications of Taylor polynomials?

Taylor polynomials are used in a variety of scientific fields, such as physics, engineering, and economics. They can be used to model and predict the behavior of complex systems, such as the trajectory of a projectile or the growth of a population. They are also used in computer graphics to create smooth curves and surfaces in 3D models.

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