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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

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

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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:


finite difference approx. to


finite difference approx. to





· 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 real-world conditions might we see such poor approximations?




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