Taylor polynomial of third degree and error estimation

[Telemachus]

Homework Statement

It seems that I'm a little bit lost about this exercise. It says: Find the taylors polynomial of third degree centered at the origin for $$z=\cos y \sin x$$. Estimate the error for: $$\Delta x=-0.15,\Delta y=0.2$$.

So, I did the first part (the easy one), the taylors polynomial for z at (0,0) looks like this:

$$f(x,y)=x+\displaystyle\frac{1}{3!}(-x^3+3xy^2)+R_4((x,y),(0,0))$$

Then I've found the expression for the error:
$$R_4=\displaystyle\frac{1}{4!}(\cos c_2 \sin c_1 x^4+4\sin c_2 \cos c_1 x^3y+6\cos c_2 \sin c_1 x^2y^2+4\sin c_2 \cos c_1 xy^3-\cos c_2 \sin c_1 y^4)$$

Now, how do I estimate the error? I have to say that the error will be something like $$R_4\leq{}k$$, I have to find "k". Should I consider $$c_1=0.15, c_2=0.2$$? How do I proceed from there?

Bye and thanks for posting!

 

[Mark44]

Homework Statement

It seems that I'm a little bit lost about this exercise. It says: Find the taylors polynomial of third degree centered at the origin for $$z=\cos y \sin x$$. Estimate the error for: $$\Delta x=-0.15,\Delta y=0.2$$.

So, I did the first part (the easy one), the taylors polynomial for z at (0,0) looks like this:

$$f(x,y)=x+\displaystyle\frac{1}{3!}(-x^3+3xy^2)+R_4((x,y),(0,0))$$

Then I've found the expression for the error:
$$R_4=\displaystyle\frac{1}{4!}(\cos c_2 \sin c_1 x^4+4\sin c_2 \cos c_1 x^3y+6\cos c_2 \sin c_1 x^2y^2+4\sin c_2 \cos c_1 xy^3-\cos c_2 \sin c_1 y^4)$$

Now, how do I estimate the error? I have to say that the error will be something like $$R_4\leq{}k$$, I have to find "k". Should I consider $$c_1=0.15, c_2=0.2$$? How do I proceed from there?

Bye and thanks for posting!

All of your sine and cosine expressions have a maximum value of 1.

You know what f(0, 0) is, right? You're interested in approximating f(0 - 0.15, 0 + 0.2)

 

[Telemachus]

With f you reefer to the taylors polynomial? its zero anyway, z its zero at that point, and f is zero too, cause the error its zero at that point.

I've considered $$c1=x=-0.5,c2=y=0.2$$ and I get an error of 0.000143, which I think its a good approximation, I've evaluated the function at that point, and the polinomial, and it gives an error of 0.00002.

Is this right?

What I did know was that $$x\leq{}c_1\leq{}0$$ and $$0\leq{}c_2\leq{}y$$, but I didn't know which value should I take for c_1 and c_2 on the interval. I've just made it equal, but I think I should looked for a maximum. It worked on this case, but I think this wouldn't work in other cases.

Bye there, thanks Mark.

 

[Mark44]

With f you reefer to the taylors polynomial?

No, I'm not. f is the function you want to approximate by its Taylor polynomial.

its zero anyway, z its zero at that point, and f is zero too, cause the error its zero at that point.

I've considered $$c1=x=-0.5,c2=y=0.2$$ and I get an error of 0.000143, which I think its a good approximation, I've evaluated the function at that point, and the polinomial, and it gives an error of 0.00002.

No, because you should not be setting c1 to -.5 or c2 to .2. c1 and c2 are unknown numbers in the intervals [-.5, 0] and [0, .2] respectively.

Is this right?

What I did know was that $$x\leq{}c_1\leq{}0$$ and $$0\leq{}c_2\leq{}y$$, but I didn't know which value should I take for c_1 and c_2 on the interval. I've just made it equal, but I think I should looked for a maximum. It worked on this case, but I think this wouldn't work in other cases.

Bye there, thanks Mark.

I would just replace the cosine and sine terms by 1, since they will never be larger than 1.

 

[Telemachus]

Thanks Mark!