Check the continuity and differentiability of the function at (0, 0) where
(b) Find the domain and range of the function f, defined by . Also find two level curves of this function. Give a rough sketch of them.
First, we check the continuity and differentiability of [ f(x, y) = \begin{cases} \dfrac{2x^3y}{x^2 + y^2}, & (x, y) \neq (0,0) [2mm] 0, & (x, y) = (0,0) \end{cases} ] at ((0,0)). Continuity at ((0,0)): We compute the limit: [ \lim_{(x,y)\to(0,0)} \frac{2x^3y}{x^2 + y^2}. ] Try converting to polar coordinates: (x = r\cos\theta, y = r\sin\theta). Then [ f(r,\theta) _______ _____ ______ _________ __________ __________ ________ ___ ______ ________ ____.
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) Does the function
satisfy the requirement of Schwarz's theorem at (1, 1)? Justify your answer. (b) Locate and classify the stationary points of the following:
(i)
(ii)
Find the centre of gravity of a thin sheet with density , bounded by the curves
and
.
(b) Find the mass of the solid bounded by and
, the density function being
.
5. (a) State Green's theorem, and apply it to evaluate
Where C is the ellipse .
(b) Find the extreme values of the function
on the surface
.
Find the values of a and b, if
(b) Suppose S and C are subsets of . 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)
(ii)
(c) Identify the level curves of the following functions: (i)
(ii)
(iii) x - y
(iv) y / x
Check the continuity and differentiability of the function at (0, 0) where
(b) Find the domain and range of the function f, defined by . Also find two level curves of this function. Give a rough sketch of them.
State a necessary condition for the functional dependence of two differentiable functions f and g on an open subset D of . Verify this theorem for the functions f and g, defined by
(b) Using the Implicit Function Theorem, show that there exists a unique differentiable function g in a neighbourhood of 1 such that and
in a neighbourhood of (2, 1), where
defines the function F. Also find g'(y).
(c) Check the local invertibility of the function f defined by at (1, -1). Find a domain for the function f in which f is invertible.
Find the following limits: (i)
(ii)
(b) Find the third Taylor polynomial of the function
at (1, 2).
(c) Using only the definitions, find fxy(0, 0) and fyx(0, 0), if they exists, for the function
Evaluate , where C is the curve given by
.
(b) Use double integration of find the volume of the ellipsoid
.