Calculating Taylor polynomials , Multiple Questions

In summary: The taylor polynomial for g(x) = x3/4 is:P(x) = 3x + 9For x = 0, the equation has a single solution, which is 3x = 9.For x = 1, the equation has two solutions, which are 3x = 5 and 3x = -6.For x = 2, the equation has three solutions, which are 3x = 2, 3x = -1, and 3x = 5.
  • #1
TsVeryOwn
2
0
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) = x3/4. 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 :rolleyes:
 
Physics news on Phys.org
  • #2
TsVeryOwn said:

Homework Equations


Below is an equation for a Taylor polynomial:

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

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

[itex] R(x) = f(x) - P_{n,a}(x) [/itex].

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.
 
  • #3
Mogarrr said:
Below is an equation for a Taylor polynomial:

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

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

[itex] R(x) = f(x) - P_{n,a}(x) [/itex].

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
[tex] \text{error} = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x-a)^{n+1},[/tex]
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.
 
Last edited:
  • #4
TsVeryOwn said:
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) = x3/4. 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 ex 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 :rolleyes:
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.

 
  • #5
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 !
 
  • #6
TsVeryOwn said:
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.
 

FAQ: Calculating Taylor polynomials , Multiple Questions

What is the purpose of calculating Taylor polynomials?

The purpose of calculating Taylor polynomials is to approximate a function with a polynomial in order to simplify complex calculations or to estimate values of the function at specific points.

What are the steps involved in calculating Taylor polynomials?

The steps involved in calculating Taylor polynomials are:

  1. Determine the point of expansion, also known as the center, for the polynomial.
  2. Find the derivatives of the function at the point of expansion.
  3. Substitute the point of expansion and its derivatives into the Taylor polynomial formula.
  4. Simplify the resulting polynomial to the desired degree.

How is the degree of a Taylor polynomial determined?

The degree of a Taylor polynomial is determined by the number of derivatives used in the polynomial. For example, a degree 3 polynomial would use the first three derivatives of the function at the point of expansion.

What is the difference between a Taylor polynomial and a Taylor series?

A Taylor polynomial is a finite sum of terms, while a Taylor series is an infinite sum of terms. The Taylor series is a more accurate representation of the function, as it includes all possible derivatives at the point of expansion.

How is the accuracy of a Taylor polynomial determined?

The accuracy of a Taylor polynomial is determined by the number of terms used in the polynomial. As the number of terms increases, the polynomial becomes a better approximation of the function. Additionally, the distance between the point of expansion and the point at which the polynomial is being evaluated also affects the accuracy.

Similar threads

Taylor polynomial, approximative solution of this equation
Replies
3
Views
1K
Finding Error on Taylor Polynomials using Taylor's Theorem
Replies
1
Views
2K
Basic Taylor Polynomial Question involving e^(-x)^2
Replies
9
Views
2K
Taylor Polynomial. Understanding.
Replies
2
Views
1K
Taylor Polynomials and Numerical Analysis
Replies
2
Views
3K
Taylor/Polynomial Approximation Question
Replies
2
Views
2K
Approximating accuracy of Taylor polynomials
Replies
4
Views
6K
Taylor Polynomial Problem: What is f´(1)?
Replies
5
Views
2K
Error estimate for Taylor polynomials
Replies
4
Views
2K
Taylor Polynomials- Lagrange remainder
Replies
3
Views
4K

Hot Threads

Recent Insights

Back
Top