Locate and classify the stationary points of the following:

(i)  

 (ii)  

Does the function

 satisfy the requirement of Schwarz's theorem at

(1,1)? Justify your answer.

Identify the level curves of the following functions:

(i)  

(ii)  

(iii) x-y

(iv) y/x

Suppose S and C are subsets of R³. S is the unit open sphere with centre at the origin and C is the open cube =

Which of the following is true. Justify your answer.

(i) S ⊂ C

(ii) C ⊂ S

Find the values of a and b, if

Use double integration of find the volume of the ellipsoid 

Evaluate   where C is the curve given by

Find the domain and range of the function f , defined by 

find two level curves of this function. Give a rough sketch of them

Check the continuity and differentiability of the function at (0,0) where

Check the local inevitability of the function f defined by  at ,(1 − .1) Find a domain for the function f in which f is invertible.

Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that  in a neighbourhood of (,1,2) where 

defines the function F. Also find g′( y).

State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of .  R² Verify this theorem for the  functions f and g, defined by

Find the extreme values of the function

State Green’s theorem, and apply it to evaluate

Where C is the ellipse  

Find the mass of the solid bounded by   the density function being δ (z,y,x )= .|x| 

Find the centre of gravity of a thin sheet with density δ(x, y) = y, bounded by the

curves  

 )1,4 be three points in .  R³

Find |2 b − a + 3c l.

  be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

Let the function f be defined by  

Show that f has directional derivatives in all directions at.(0,0)

Using only the definitions, find   if they exists, for the function