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Knowing the value of a two-dimensional function at each node in the mesh, your objective is to calculate the partial derivatives and at node

[Note that, in this example, the mesh sizes in x and y are identical (h); strictly speaking, this need not be true. In some applications, we may need more resolution along the x- or y-axis; we could then use separate mesh sizes hx and hy.]

By definition, the partial derivative of a function with respect to x

and the partial derivative with respect to y is

If we applied these formula to our grid values, we would get the finite difference expressions

Note that these are approximations to the values of the partial derivatives since we're not taking the limit as h goes to zero; but as h becomes smaller, the approximations should improve.

With this background, here's your assignment:

*You must show sufficient detail to support your work to earn credit for your calculations. This can be hand-written work, typed calculations, or excel formulas. To avoid round-off error, retain at least six decimal places in all of your calculations.  Complete all trigonometric calculations in radians.

· Assume the function f is defined as f(x, y) = 3 tan x cos y

· Use differentiation rules to find the exact partial derivatives and , and evaluate those exact partial derivatives at (1.56, -2.1).

· Use the finite difference formulas to estimate and , at (1.56, -2.1) for three different values of the mesh size

· h = 0.01

· h = 0.001

· h = 0.0001

· Use your calculated values to fill in this table:

Estimated partial derivatives using finite difference formulas:

 h finite difference approx. to Exact finite difference approx. to Exact 0.01 0.001 0.0001