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) = \frac{2 (r\cos\theta)^3 (r\sin\theta)}{r^2} = \frac{2 r^4 \cos^3\theta \sin\theta}{r^2} = 2 r^2 \cos^3\theta \sin\theta. ] As (r \to 0), (2 r^2 \cos^3\theta \sin\theta \to 0) for all (\theta). Hence, ________ ________ ______ ___ ___ ____.
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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
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.
Let the function f be defined by Show that f has directional derivatives in all directions at (0, 0).
(b) Let ,
and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that
(c) Let ,
,
be three points in
.
Find |2b - a + 3c|.
swers.
(i)
(ii) A real-valued function of three variables which is continuous everywhere is differentiable.
(iii) The function , defined by
, is locally invertible at any
.
(iv) , defined by
is integrable.
(v) The function , defined by
, has an extremum at (0, 0).
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
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
.
Evaluate , where C is the curve given by
.
(b) Use double integration of find the volume of the ellipsoid
.