Test problems of 14.2 and 14.4

Huaizhen Qin

14.2

Question 1: Let f(x, y)=x+(ye^x). Verify  that  (Subscript[f, x])(x, y)-(Subscript[yf, y]) (x, y)=1.
[Solution]  Since (Subscript[f, x])(x, y)=1+(ye^x) and (Subscript[f, y])(x, y)=(e^x), it follows that   
	
	(Subscript[f, x]) (x, y) - (Subscript[yf, y]) (x, y)=[1+(ye^x)]-(ye^x)=1.

I suggest Question 1 to test how well the students manipulate partial derivative of a given multivariate function.

1  4.4
Question 2: Calculate the directional derivative  of  f(x, y)=(x^2)+(e^y) 
at  (1, 0)  in  the direction of the vector (Overscript[i,  ->]) +(Overscript[j,  ->]) .

[Solution] Let (Overscript[( u),  ->]) =(1/(Sqrt[2]))(Overscript[i,  ->]) +(1/(Sqrt[2]))(Overscript[j,  ->]) .   Note that (Subscript[f, x])(1, 0)=2 and (Subscript[f, y])(1, 0)=1.   Accordingly,we obtain that the  directional derivative of 
 f(x, y)  at (1, 0)   in  the direction of the vector (Overscript[( u),  ->])   is

(Subscript[f, (Overscript[u,  ->]) ])(1, 0)=(Subscript[f, x])(1, 0)(Subscript[u, 1])+(Subscript[f, y])(1, 0)(Subscript[u, 2])=2×(1/(Sqrt[2])) + 1×(1/(Sqrt[2])) = 
              3/(Sqrt[2]).

I suggest Question 2 to investigate how well the students understand the  directional derivative of a given function  with respect to a given direction.