> # Various problems from the homeworks\
# 44 12.5\
u:=(x,y)->f(x,y);\
x:=(s,t)->exp(s)*cos(t);\
y:=(s,t)->exp(s)*sin(t);\
temp:=diff(u(x(s,t),y(s,t)),s,s); # this is the second partial u sub ss\
ugly:=exp(-2*s)*( diff(u(x(s,t),y(s,t)),s$2) + diff(u(x(s,t),y(s,t)),t$2) ); # true to its name\
answer:=simplify(ugly); # this is the desired answer expanded a bit
> #11\
z:=(x,y)->2^(x-3*y); x:=(s,t)->s^2*t;y:=(s,t)->s*t^2;\
diff(z(x(s,t),y(s,t)),s);\
diff(z(x(s,t),y(s,t)),t);
> #12.4 #21\
f:=(x,y)->sqrt(20-x^2-7*y^2);\
x0:=2;y0:=1;dx:=1.95 - x0;dy:=1.08-y0;\
df:=D[1](f)(x0,y0)*dx+D[2](f)(x0,y0)*dy;\
answer:=f(x0,y0)+df;
> #5\
f:=(x,y)->sin(x+y);\
x0:=1;y0:=-1;# reminber you will not be given z0 on the test\
z0:=f(x0,y0);# cause it is easy to compute\
eqn:=D[1](f)(x0,y0) *(x-x0)+D[2](f)(x0,y0)*(y-y0)-1*(z-z0)=0;
> #12.3 #71\
f:=(x,y)->x*sin(y);\
diff ( f(x,y), x,y,y);
> #73\
u:=(x,y,z)->ln(x+2*y^2+3*z^3);\
diff(u(x,y,z),z,y,x);
> #48\
f:=(x,y)->x^2+y^2+x^2*y;\
plot3d({f(x,y),diff(f(x,y),x),diff(f(x,y),y)},x=-1..1,y=-1..1);
> 
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