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.
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Evaluate , where C is the curve given by
.
(b) Use double integration of find the volume of the ellipsoid
.
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.
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
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.
) 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 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
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|.