page 428 #12. Length of the graph of f on [a, b] is the length of the path c(t) = (t, f (t),
t [a, b].
(a) Since f is piecewise continuously dierentiable, the path c is piecewise C 1 , so we
compute c (t) = (1, f (t).
page 375 #10. A is a 2 2 matrix, nonzero determinant. Dene T (x) = Ax. Then T
takes parallelograms onto parallelograms.
Consider the parallelogram G with vertex p, and adjacent edges v, w. Then G is the
set of points
page 224 #36. Function u is C 2 on the unit disk D, and strictly subharmonic 2 u > 0.
Show u cannot have its maximum in the interior of D.
Suppose u has its maximum at an interior point (x, y ). Then the gradient u is 0 there.
Consider the Hess
April 10, 2009
page 127 #26. Show that f : Rn Rm is continuous if and only if the inverse image of
every open set is open.
We use the denition of continuous on page 118. We are to prove an if and only if
Exam 2 Answers
Use the Riemann sum denition to compute the integral
rectangle [0, 1] [1, 1] and f is dened by f (x, y ) = x + y .
f (x, y ) dA, where D is the
Fix n N. Write x = 1/n, y = 2/n, so that xi = i/n, yj = 1 + (2j/n) for
Exam 1 solutions
State the following:
(a) Lagrange Multiplier Theorem
(b) Second-Order Taylors Theorem
(c) Second derivative test (for multivariable extrema)
Discuss the solvability of the system
0 = 3x + 2y + z 2 + u + v