1. Find and sketch the domains of the following:
(a) g(x, y) = p
9 − x
2 − y
(b) f(x, y) = ln(16 − x
2 − y
2. Evaluate lim(x,y)→(1,
q x+cos y
x2+3 sin y
. Show work.
3. Show that lim(x,y)→(0,0)
4 does not exist.
4. Find the equation of the tangent plane to the graph z = xexy at the point (1,0,1).
5. Find all the second partial derivatives of z = x
y + x
√y. Show work.
6. Find dw
dt if w = x
2 + zey + sin(xz), and x = t, y = t
, z = t
7. If g(s, t) = f(s
2 − t
, t2 − s
) show that g satises t
∂s + s
∂t = 0.
8. Suppose y is a function of x, F(x, y) = 0 and Fy 6= 0. Show that dy
dx = −
9. Find the critical points of f(x, y) if any exist, for f(x, y) = e
10. Calculate the iterated integral:
y sin(xy) dx dy.
11. Evaluate the double integral R
xy2dA, where D is the region in the first quadrant bounded by y =
x and y = x
12. Evaluate the double integral R R
(3x + y)dA
where A is the region in the first quadrant that lies inside the circle x
2 + y
2 = 4 find
outside the circle x
2 + y
2 = 1.
Hint: Use polar coordinates
13. Evaluate the triple integral R R
xy sin z dV where D is the region in R3 described
by 0 ≤ x ≤ 1, 1 ≤ y ≤ 3, 0 ≤ z ≤ π.
14. The pressure p (in kilopascals), volume v (in litres) and temperature T (in Kelvin) of
a mole of an ideal gas are related by the equation pv = 8.31T. Use the chain rule to and
the rate at which the pressure is changing when T = 300◦K and increasing at a rate of
0.1K/s and the volume is 100L and increasing at a rate of 0.2L/S.
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