First name:   Last name:

Consider the function \(f(x,y) = xy\) on the region \(R = \{ (x,y) : x^2 + y^2 \leq 1\}\).

1.  The region \(R\) is

a.  an open circle.

b.  a closed circle.

c.  an open square.

d.  a closed square.

e.  a closed triangle.

2.  How many critical points does \(f\) have?

a.  0

b.  1

c.  2

d.  3

e.  4

3.  How many of these critical points are OUTSIDE the region \(R\)?

a.  0

b.  1

c.  2

d.  3

e.  4

4.  If \(g(x,y) = x^2 + y^2 - 1\), at how many places on the boundary of \(R\) is the gradient of \(f\) parallel to the gradient of \(g\)?

a.  0

b.  1

c.  2

d.  3

e.  4

5.  At how many of the points you found in question 4 does \(f\) have an absolute maximum on the region \(R\)?

a.  0

b.  1

c.  2

d.  3

e.  4

6.  At how many of the points you found in question 4 does \(f\) have an absolute minimum on the region \(R\)?

a.  0

b.  1

c.  2

d.  3

e.  4

7.  Evaluate \(\int_1^2\int_1^4 \frac{1}{xy} dy dx\)

a.  \(\frac{1}{8}\)

b.  \(1\)

c.  \(\frac{1}{6}\)

d.  \(\ln(8)\)

e.  \(\ln(4)\ln(2)\)