Tin=xit+h and =z h Knowing the value of a function f at each node in the mesh, your objective is to calculate the derivative of f at node . o derive the two formulas you'l be using, we start with the definition of the derivative: ) Sle+h) h If we applied this formula to our grid values, we would get the forward difference expression = Limyy ~ flEi)fla) [ () = == and the backward difference expression o~ fE) S (@) = T2 Note that these are approximations to the value of the derivative, since we're not taking the limit as h goes to zero; but we can improve the approximation by taking the average of these two difference formulas: 1 =2 [ (=i which simplifies to the centered difference expression (o) = L) Sec) With this background, here's your assignment: - Assume the function f is defined as f(x) = 5x* - 9x% + 2 Use the power rule to find the derivative f'(x) and evaluate that derivative at x = 1.7. Note: To avoid round-off error, retain at least six decimal places in your calculations. Use the "forward difference\" and "centered difference\" formulas to estimate f'(x) at x = 1.7 for three different values of the mesh sizes o h=0.1 o h=0.01 h=0.001 Note: To avoid round-off error, retain at least six decimal places in your functional evaluations, and retain the maximum possible number of decimal places in calculations of the forward and centered difference approximations. + Use your calculated values to fill in this table: Calculate derivatives using two finite difference formulas: forward difference | centered difference o h o i exact derivative approximation approximation 0.1 001 0.001 + Answer the following two questions: = Which formula yields a better approximation: The forward difference or the centered difference? & What effect does reducing the mesh size h have upon the accuracy of these approximations? Upload your results using the blue "Submit Assignment" button at the top of the page. Be sure to show all of your work 9_these calculations