1. (10 marks)
Given that a, b, c > 0 find the maximum of f(x,y, = raybe subject to the condition that x + y + 2 = 1
where x, y, 2 > 0.
2. (10 marks)
Determine whether the vector field
F = - -
is conservative or not. If so or otherwise evaluate
where r is the curve of intersection of the surfaces z = In(1 + x) and y = x, directed from (0,0,0) to
(1, 1, ln 2).
3. (10 marks)
The composite function 2 = f(x, y) has the following structure:
f(x,y =F(hi(x,y))+G(h2(x,y)), =
where F and G are arbitrary, single variable functions.
(a) (7 marks)
Ignoring any arbitrary constants, determine the functions h1 (x, y) and h2(x,y) if f satisfies the equations
of Ox y of by = 2G'(h2(x,y)),
where the primes denote differentiation with respect to the respective single variable arguments of F and
(b) (3 marks)
Show that 2 = f(x, y) satisfies the differential equation
4. (10 marks)
Show that, for a > 0 and using a suitable change of variables,
/o / /o FOC (x² + 2 + + 2 2 da dy dz = 2(m + TT 3)a²
5. (10 marks)
The C² function
= + w = T
defines w as the implicit function, w = f(x,y,z). Determine values 20 and wo that satisfy the level set equation
F (0, 1, 20, wo) = T.
What is the maximum slope of the implicit function f at the point (0,1,20) and in what direction, u = (a,B,),
is that slope found?
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