Numerical Methods: Numerical Differentiation

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

1.) The partial derivative, of with respect to *x *is obtained by holding *y* fixed and differentiating with respect to *x. *Similarly, is found by holding *x* fixed and differentiating with respect to *y.* The equation below is adapted to partial derivatives:

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

(a). Let . Calculate the approximations to and using the formulas in (i) with *h* = 0.1, 0.01, and 0.001. Compare with the values obtained by differentiating

**MY ANSWERS:**

First, I solved for the derivative of with respect to *x*. I got:

Solving for using equation (i)

h_____ _ _____2h______

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

Derivative with respect to y:

Solving for using equation (i)

h_____ _ _____2h______

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 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 by numerical differentiation

(b) Compare your answer with .

**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:

(b). .