Calculating Taylor polynomials , Multiple Questions

[TsVeryOwn]

Whats up guys ! currently studying for calculas exam and could use someone going over my answers !

Homework Statement

Q1. Calculate the taylor polynomial of degree 5 centred 0 for f(x) = e^-x. Simply coeffcients and use the error formula to estimate the error when p5(0.1)

Q.2 Q1. Calculate the taylor polynomial of degree 6 centred 0 for f(x) = cos(x). Simply coeffcients and use the error formula to estimate the error when p6(-2)

Q3. Calculate the taylor polynomial of degree 3 centred x=1 for g(x) = x^3/4. Simply coeffcients and use to approximate (0.9)^3/4

Homework Equations

The Attempt at a Solution

Q1. 1 + x + -x²/2 - x³/6 + x⁴/24 + x⁵/120, , , 1.104 for next bit ?

Q2. 1- x²/2 + x⁴/24 + x⁶/720 ,, 2.422 for next part

Q3. Unsure on how to do

Could someone look over my answers , i think there right I am just unsure about the polaritys of some of the figures and also I am unsure if i answered it correctly when it asked me to "Estimate the errors"

Thanks in advance !


P.S
I`m new to this forum , I am wondering if it would be possible to post up a PDF with around 8 diffrent questions on it ? i need them all answred for studying purposes as i have a repeat college exam coming up soon

 

[Mogarrr]

Homework Equations

Below is an equation for a Taylor polynomial:

P_n(x) = f(a) + f'(a)(x-a) + \frac {f''(a)}{2!}(x-a)^2 + ... + \frac {f^{(n)}(a)}{n!}(x-a)^{n}.

The "error function" is also called the "remainder function", which is the following:

R(x) = f(x) - P_{n,a}(x).

It's the difference between the value of the function at a certain point and the value of the Taylor polynomial centered at a.

Post if you have any troubles with the questions.

 

[Ray Vickson]

Below is an equation for a Taylor polynomial:

P_n(x) = f(a) + f'(a)(x-a) + \frac {f''(a)}{2!}(x-a)^2 + ... + \frac {f^{(n)}(a)}{n!}(x-a)^{n}.

The "error function" is also called the "remainder function", which is the following:

R(x) = f(x) - P_{n,a}(x).

It's the difference between the value of the function at a certain point and the value of the Taylor polynomial centered at a.

Post if you have any troubles with the questions.

That is NOT the error formula. The typical error formula would be
\text{error} = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x-a)^{n+1},
where $\xi$ is a number in the interval $[a,x]$ (if $x > a$) or in $[x,a]$ (if $x < a$). See, eg., http://en.wikipedia.org/wiki/Taylor's_theorem .

Note: I mean that what you have done is define the error, not given a useful formula for it.

 

[HallsofIvy]

Whats up guys ! currently studying for calculas exam and could use someone going over my answers !

Homework Statement

Q1. Calculate the taylor polynomial of degree 5 centred 0 for f(x) = e^-x. Simply coeffcients and use the error formula to estimate the error when p5(0.1)

Q.2 Q1. Calculate the taylor polynomial of degree 6 centred 0 for f(x) = cos(x). Simply coeffcients and use the error formula to estimate the error when p6(-2)

Q3. Calculate the taylor polynomial of degree 3 centred x=1 for g(x) = x^3/4. Simply coeffcients and use to approximate (0.9)^3/4

Homework Equations

The Attempt at a Solution

Q1. 1 + x + -x²/2 - x³/6 + x⁴/24 + x⁵/120, , , 1.104 for next bit ?

You have your signs really confused here. Is "+ -x²" a typo? The Taylor polynomial for e^x is 1+ x+ x²/2+ x³/6+ x⁴/24+ x⁵/120. To get the Taylor polynomial for e^-x, replace x with -x which means every odd power will become negative.

Q2. 1- x²/2 + x⁴/24 + x⁶/720 ,, 2.422 for next part

Q3. Unsure on how to do

Could someone look over my answers , i think there right I am just unsure about the polaritys of some of the figures and also I am unsure if i answered it correctly when it asked me to "Estimate the errors"

Thanks in advance !


P.S
I`m new to this forum , I am wondering if it would be possible to post up a PDF with around 8 diffrent questions on it ? i need them all answred for studying purposes as i have a repeat college exam coming up soon

Please don't do that!
If you want people to help you, you do the work of posting problems. Don't make the people you are asking for help do the work of opening and reading pdfs for you.

 

[TsVeryOwn]

Cool thanks for the help guys but could you`s maybe do out the question so i could see it more clearly ?

Q3. Calculate the taylor polynomial of degree 3 centred x=1 for g(x) = x3/4. Simply coeffcients and use to approximate (0.9)3/4^^^This on in particular if someone does it step by step would be cool !

 

[Ray Vickson]

Cool thanks for the help guys but could you`s maybe do out the question so i could see it more clearly ?

Q3. Calculate the taylor polynomial of degree 3 centred x=1 for g(x) = x3/4. Simply coeffcients and use to approximate (0.9)3/4


^^^This on in particular if someone does it step by step would be cool !

It may be cool for you but it is very much against the Forum rules. You should read them; in particular, we are NOT allowed to do solutions step-by-step. We are allowed to help and give hints, etc., but YOU must do the work. If you ask more specific questions you might get better answers. Just saying you don't know how to do it is not good enough.