Knowing the value of a twodimensional 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 yaxis; 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 handwritten work, typed calculations, or excel formulas. To avoid roundoff 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:


· Answer the following questions:
· For which partial derivative ( or is the finite difference approximation consistently more accurate? Why do you think the finite difference approximation for the other partial derivative is consistently less accurate? (Hint: What happens to the tan x function near x = 1.56?). Under what realworld conditions might we see such poor approximations?