Nonlinear Control Sy.. - Free

62 CHAPTER 2. MATHEMATICAL PRELIMINARIES
(2.16) Determine whether the function
x2 2
for (x, y) # (0, 0)
0 for x=y=0
is continuous at (0, 0). (Suggestion: Notice that y 7 < x2.)
(2.17) Given the following functions, find the partial derivatives and
(i) f (x, y) = ex' cos x sin y
(ii)
f (X, x2y2
y) = x2 + y2
(2.18) Use the chain rule to obtain the indicated partial derivatives.
(i) z = x3 + y3, x = 2r + s, y = 3r - 2s, N, and.
(ii) z = xr+--y7 x = r cos s, y = r sin s, az, and a
(2.19) Show that the function f = 1/x is not uniformly continuous on E = (0, 1).
(2.20) Show that f = 1/x does not satisfy a Lipschitz condition on E = (0, 1).
(2.21) Given the following functions f : IR -- R, determine in each case whether f is (a)
continuous at x = 0, (b) continuously differentiable at x = 0, (c) locally Lipschitz at
x=0.
(i) f (x) = ex2
(ii) f (x) = Cos X.
(iii) f (x) = satx.
(iv) f (x) = sin(' ).
(2.22) For each of the following functions f : 1R2 -+ IIY2, determine whether f is (a) continuous
at x = 0, (b) continuously differentiable at x = 0, (c) locally Lipschitz at x = 0, (d)
Lipschitz on some D C R2.
(i)
(ii)
x1 = x2 - x1(xl +x2)
22 = -x1 - x2(xl + x2)
:i1 =x2-}'xl(N2-x1 -x2)
{
l
x2 = -x1 + x2 /32 - x1 -x2 2
( )