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.
Two differentiable functions (f) and (g) on an open set (D \subset \mathbb{R}^2) are said to be functionally dependent if there exists a differentiable function (\Phi) such that (\Phi(f(x,y),g(x,y))=0) for all ((x,y)\in D). A necessary condition for functional dependence is that the Jacobian determinant [ ______ ______ _______ ______ __________ __________ ______ _________ ____ ________ _________.
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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|.
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 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
.
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)
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.