Question: (a) Check the local linearity of f(x, y) = e-x sin y near x = 1, y = 2 by making a table of values of f for x = 0.9, 1.0, 1.1 and y = 1.9, 2.0, 2.1. Express values of f with 4 digits after the decimal point. Then make a table of values for x = 0.99, 1.00, 1.01 and y = 1.99, 2.00. 2.01, again showing 4 digits after the decimal point. Do both tables look nearly linear? Does the second table look more linear than the first?
(b) Give the local linearization of f(x, y) = e-x sin y at (1, 2), first using your tables, and second using the fact that fx(x, y) = -e-x sin y and fy(x, y) = e-x cos y.