Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.43

SELF EXAMINATION QUESTIONS

1. fx, fy for the following f(x, y) :

(i) (x – y)^2 (ii) x^3 + 3xy + y^3 (iii) x y^2 +^2 (iv) 2 2

1

x y+

(v) ( )

x y

x y

+

− (vi)

x y^33

x y

+

+ (vii) e

x y (viii) log (xy).

[Ans. (i) 2 (x – y) ; – 2 (x – y) (ii) 3x^2 + 3y : 3x + 3y^2

(iii) 2 2 2 2

x. y

x y x y+ + (iv) ( ) ( )

−x. x y ; y x y^2 +^2 −3/2 −^2 +^2 −3/2

(v) ( )^2 ( )^2

2y 2x;

x y x y

−

− − (vi) ( ) ( )

3 2 3 3 2 3

2 2

2x 3xy y 2y 3xy x;

x y x y

+ − + −

+ +

(vii) yexy, xexy (viii) 1 1x x, ]

2. For the following functions, find fxx, fyx, fxy, fyy :
   (i) x^2 + 2xy + y^2 , (ii) x^3 + 3x^2 y + 3xy^2 + y^3 , (iii) ex2+y2 (iv)x^2 y^2.
   [Ans. (i) 2 (every case) (ii) 6 (x + y) in every case
   (iii) 2ex3+y2(l + 2x^2 );4xyex2+y3;4xyex2+y2 : 2ex2+y2 (1 + 2y^2 ) (iv) 2y^2 ; 4xy ; 4xy ; 2x^2 ]


3. If f(x, y) = ex + y, then
   f f(x,y), f(x,y).f
   y x

∂ = ∂ =

∂ ∂

4. If f(x,y) =
   x y,
   x y

−

+ then

f f

x y

∂ = −∂

∂ ∂ , atx = y = l.

5. 
   

   If f(x, y, z) = x^2 + y^2 + z^2 , then fx (^) x = fy (^) y = fz (^) z.

   
   


6. 
   

   If u = x y^2 +^2 Vx^2 +y^2 , finduxx,uyy [Ans.

   
   

2 2

2 2 3/2 2 2 3/2

y ; x

(x y ) (x y )+ + ]

7. If = log (x^2 + y^2 ) show that

2 2

2 2

u u 0

x y

∂ +∂ =

∂ ∂

8. 
   

   If (x, y) = yx log show that fx (^) y = fy (^) x

   
   


9. 
   

   If u = x^2 + y” + z , find the value of xux + yuy + zuz. [Ans. 2]

   
   


10. 
    

    If u = x^2 y + y^2 z + z^2 x, find the value of ux + uy + uz. [Ans. (x + y + z)^2 ]