Hi everyone. I have a homework on numerical differentiation and I just want to know if my answers are correct.

1.) The partial derivative, $\displaystyle f_x(x,y) $ of $\displaystyle f(x,y)$ with respect toxis obtained by holdingyfixed and differentiating with respect tox.Similarly, $\displaystyle f_y(x,y) $ is found by holdingxfixed and differentiating with respect toy.The equation below is adapted to partial derivatives:

$\displaystyle f_x(x,y) =\frac{f(x+h,y) - f(x-h,y)}{2h}+O(h^2)

$

$\displaystyle f_y(x,y) =\frac{f(x,y+h) - f(x,y-h)}{2h}+O(h^2)

$

[The two equations above are denoted as equation (i)]

(a). Let $\displaystyle f(x,y)=\frac{xy}{(x+y)}$. Calculate the approximations to $\displaystyle f_x(2,3)$ and $\displaystyle f_y(2,3)$ using the formulas in (i) withh= 0.1, 0.01, and 0.001. Compare with the values obtained by differentiating $\displaystyle f(x,y).$

MY ANSWERS:

First, I solved for the derivative of $\displaystyle f(x,y)$ with respect tox. I got:

$\displaystyle f_x(x,y)=\frac{y(x+y)-xy^2}{(x+y)^2}=\frac{3(2+3)-2(3)^2}{(2+3)^2}=-0.12$

Solving for $\displaystyle f_x(x,y)$ using equation (i)

h_____$\displaystyle f(2+h,3)$_$\displaystyle f(2-h,3)$_____2h______$\displaystyle f_x(2,3)$

0.1_____1.23529____1.16327____0.20000___0.36014

0.01____1.20359____1.19639____0.20000___0.36000

0.001___1.20036____1.19964____0.20000___0.36000

Solving for $\displaystyle f_y(x,y)$

Derivative with respect to y: $\displaystyle f_y(x,y)=\frac{x(x+y)-yx^2}{(x+y)^2}=\frac{2(2+3)-3(2)^2}{(2+3)^2}=-0.08$

Solving for $\displaystyle f_y(x,y)$ using equation (i)

h_____$\displaystyle f(2,3+h)$_$\displaystyle f(2,3-h)$_____2h______$\displaystyle f_y(2,3)$

0.1_____1.21569____1.18367____0.20000___0.16006

0.01____1.20160____1.19840____0.20000___0.16000

0.001___1.20016____1.19984____0.20000___0.16000

Am I doing this right??

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2.) The distance $\displaystyle D = D(t)$ traveled by an object is given in the table below:

_t_____D(t)

8.0___17.453

9.0___21.460

10.0__25.752

11.0__30.301

12.0__35.084

(a) Find the velocity $\displaystyle V(10)$ by numerical differentiation

(b) Compare your answer with $\displaystyle D(t)=-70+7t+70e^\frac{-t}{10}$.

My Answers:

(a). Is it okay if I use central-difference to solve for the velocity? Or should I use forward of backward difference? If so, should I assume different step values (h)? Using central-difference, my answer is:

$\displaystyle V(10)=\frac{D(11.0)-D(9.0)}{11.0-9.0}=4.421$

(b). $\displaystyle D(t)=-70+7t+70e^(-t/10)=-70+7(10.0)+70e^\frac{-10.0}{10}=25.752$.

$\displaystyle V(10)=\frac{D(10.0)}{10.0}=\frac{25.752}{10.0}=2.5 752$