Advanced Calculus fi..

90 Advanced Calculus, Fifth Edition5. For the given function z = ,f(.r, !I), ti nd A; and d; in terms of A.r and Ay at .r = I . Y = 1.Comp;lrc these two functions for sclcctcd values of Ax. A, near 0.6. A certain function 2 = f'(.r, p) is known to havc the valuc ,f( 1, 2) = 3 and derivativesf,( 1. 2) = 2. f,.( 1. 2) = 5. Make "reasonablc" estimates of ,f (I. I, 1.8). f ( I .2, 1.8), and.f( 1.3, 1.8).7. Lct ; = ,f!.r. x) = .ry/(.r2 + y2) except at (0, 0): let f'(0. 0) = 0. Show that ilz/i).r andi):ji)y exist for all (.r. J) and arc continuous except at (0, 0). Show by thc FundamentalLemma that; has a differential for (.r. y) # (0. 0) but not at (0.0). since f is discontinuousat (0. 0). IIt is instructive to graph thc level curves of f .]For a function of n variablesthe differential is obtained as in Section 2.6:Thus it is a linear function of d.rl. . . . . d.r,,, whose coefficients f',, . . . . , f,,, arethe partial derivatives of f at the point considered. This linear function is a closeapproximation to the increment Ay in the sense described in Section 2.6:tl -+ 0 ..... 6, + 0 as cl.rl + 0 ..... dx,, -+ 0.On occasion, one has to deal with several functions of n variables:If these functions have continuous partial derivatives in a domain D of En. then allhave differentials: