1. ## Taylor Polynomial

I need to calculate the taylor polynomial T2(x) at 1 for the function

f(x) = x/1+2x

I have the following, with some blanks can anyone help?

For this function

f(x) = x/1+2x, f(1) = 1/3
F'(x) = ? f'(1) = ?
F"(x) = ? f"(1) = ?

Hence the Taylor polynomial of degree 2 at 1 for f is

T2(x) = 1/3 ? (x-1) ? (x-1)^2

Then I need to show that T2(x) approximates f(x) with an error less than 1/30 on the interval [0.5,1.5].

I have with I=[0.5,1.5], a=1,r=1,n=2.

1.First, f^(3) (c) = ?

2. Thus |f^(3) (c)|= ? for c E [0.5,1.5]

so we take M =

3. Using the remainder estimate (M/(n+1)!r^n+1,

we obtain

|f(x)-T2(x)| = |R2(x)|

equalor more than M/(2+1)! r^2+1

=1/3! * ? * 1^3

= ? = ? for x E [0.5,1.5]

Thus T2(x) approximates f(x) with an error less than 1/30 on [0.5,1.5].

2. ## Re: Taylor Polynomial

Again, please try to remember your early lessons. There are reasons why you studied the Order of Operations.

These are NOT the same:

x/1+2x

and

x/(1+2x)

You really should know this, or you really have no business in calculus. Step up your game if you want to do well.

Anyway, try long division. $\frac{1}{2}\cdot\left(1 - \frac{1}{1+2x}\right)\;=\;\frac{1}{2}\cdot\left(1 - \frac{1}{1-(-2x)}\right)$

Now, why would I do such a silly thing?

Unique series don't care how you find them.

If you don't like that, just start in with your derivatives from the first form after long division. That should make your derivatives easier.

3. ## Re: Taylor Polynomial

Can anyone help me with the derivatives?

4. ## Re: Taylor Polynomial

We can help if you're more clear (like TkHunny already said):
If you have to calculate the first derivative of:
$f(x)=\frac{x}{1+2x}$
Use the quotientrule:
$D\left[\frac{f(x)}{g(x)}\right]=\frac{D[f(x)]\cdot g(x)-f(x)\cdot D[g(x)]}{[g(x)]^2}$
(To calculate the second derivative try to simplify the first derivative as much as you can because then the calculating of the second derivative will be easier.)

5. ## Re: Taylor Polynomial

Would this mean 1 * 1+2x - 1 * 2/(1+2x)^2

= 2x / (1+2x)^2 for f' ?

6. ## Re: Taylor Polynomial

You made a little mistake in the numerator:
$f'(x)=\frac{(1+2x)-2x}{(1+2x)^2}=\frac{1}{(1+2x)^2}=(1+2x)^{-2}$
Now determine $f'(1),f''(x),f''(1)$

7. ## Re: Taylor Polynomial

Would f" = -2(1+2x)^-3.

f'(1) = 1/9

f"(1) = not sure

8. ## Re: Taylor Polynomial

f"(1) = -2/27

9. ## Re: Taylor Polynomial

$f'(1)=\frac{1}{9}$ is correct, but if you want to calculate the second derivative you have to use the chain rule:
$f''(x)=D\left[(1+2x)^-2\right]=-2\cdot (1+2x)^{-3}\cdot D(1+2x)=\frac{-4}{(1+2x)^3}$
So $f''(1)=...$

Now you can determine the taylor polynomial.

10. ## Re: Taylor Polynomial

Would that be -4/27.

The taylor polynomial I have is;

1/3 + 1/9(x-1) - 4/27(x-1)^2

Correct!

12. ## Re: Taylor Polynomial

could you help me with f(3) the third derivative.

13. ## Re: Taylor Polynomial

$f^{3}(x)=-4\cdot D[(1+2x)^{-3}]=-4\cdot \left[-3\cdot (1+2x)^{-4}\cdot D(1+2x)\rigt]=...$
Try to complete.

14. ## Re: Taylor Polynomial

Would that be 4/(1+2x)^4

15. ## Re: Taylor Polynomial

Originally Posted by Arron
Would that be 4/(1+2x)^4
Look, if you're studying Taylor polynomials you should already know how to differentiate. We cannot hold your hand every step of the way.

You can check your derivatives here (and click on Show steps to see working): differentiate -4&#47;&#40;1 &#43; 2x&#41;&#94;3 - Wolfram|Alpha