Homework Maple 3
Due Date: Tuesday, May 6th
Spring 2008
Partial derivatives Compute the second partial derivatives of f (x, y) = x3 + x2 y 3 2y 2 dfdx:=di(f,x); dfdy:=di(f,y); dfdx2:=di(f,x,x); dfdy2:=di(f,y,y); dfdxy:=di(f,x,y); dfdyx:=di(f,y,x); Compute fxxyz if f (x, y, z) = sin(3x + yz) Check that each of the following functions is a solution of the wave function 2 2u = c2 u t2 x2 (a) u = sin(kx) sin(ckt) (b) u = t/(c2 t2 x2 ) (c) u = (x ct)6 + (x + ct)6 (d) u = sin(x ct) + ln(x + ct) Method of Lagrange multipliers Find the extreme values of f (x, y) = x2 +2y 2 on the circle g(x, y) = x2 +y 2 = 1 In other words, solve the system f = g, g = 1 for (x, y, ).
sol:=solve({di(f,x)=L*di(g,x), di(f,y)=L*di(g,y), {x,y,L}); g=1}, subs(sol[1],f); subs(sol[2],f); subs(sol[3], f); subs(sol[4],f); A rectangular box without a lid is to be made from 12 cm2 of cardboard. Find the maximum volume of such a box. In other words, nd the maximum of f (x, y, z) = xyz (volume) subject to g(x, y, z) = 2xz +2yz +xy = 12 (surface) Find the extreme values of f (x, y, z) = 2x + 6y + 10z subject to g(x, y, z) = x2 + y 2 + z 2 = 35
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