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 [ ____ __________ ____ _______ _____ _________ ___ ______.
_____ _____ _____ _________ _________ ______ _________ _____ _____ ________.
_________ _______ ____ ___ ________ __________ _____ _____ ______ __________.
_______ ____ ______ ____ __________ _______ _________ _________ ______ _______ _________.
______ _________ ___ _________ _____ __________ _____ ___ _________.
______ _____ ____ _______ ___ _________ ________ _______ __________ ____.
___ ________ __________ _______ _______ _________.
_______ ______ ________ ____ ____ __________ ________.
_____ _______ __________ ______ ____ ________ _________ __________ _________ _________ ____.
______ ______ ___ ______ _____ _______ _______ ______.
______ ______ _____ _________ _______ _____.
_______ ___ _____ ____ _____.
____ ___ _____ ________ ___ ___ ____ ___ _____ ____.
___ _______ ________ ___ _______ ______ __________.
____ __________ ______ _______ ___ _____ _______ _________ _________ __________ _______.
_____ __________ ______ _________ ________ _____ _______ ________ ___.
_______ _____ ______ _________ ______ __________ ______ ___ _________ ______ ____ ______.
_____ ___ ________ ________ ____ _______.
_____ ___ _______ ____ __________ ______ _______ ________ _____ ________ ________ _____.
_________ __________ ____ __________ _________ __________ __________ ___ ______ ______.
______ __________ _________ __________ _________.
Get Full Answer on WhatsApp
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.
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)
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|.
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