# 1. Find and sketch the domains of the following: (a) g(x, y) = p ...

## Transcribed Text

1. Find and sketch the domains of the following: (a) g(x, y) = p 9 − x 2 − y 2 (b) f(x, y) = ln(16 − x 2 − y 2 ) 2. Evaluate lim(x,y)→(1, π 2 ) q x+cos y x2+3 sin y . Show work. 3. Show that lim(x,y)→(0,0) xy2 x2+y 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 2 y + x √y. Show work. 6. Find dw dt if w = x 2 + zey + sin(xz), and x = t, y = t 2 , z = t 3 . 7. If g(s, t) = f(s 2 − t 2 , t2 − s 2 ) show that g satises t ∂g ∂s + s ∂g ∂t = 0. 8. Suppose y is a function of x, F(x, y) = 0 and Fy 6= 0. Show that dy dx = − Fx Fy . 9. Find the critical points of f(x, y) if any exist, for f(x, y) = e x sin y. 10. Calculate the iterated integral: R π 0 R 2 1 y sin(xy) dx dy. 11. Evaluate the double integral R D R xy2dA, where D is the region in the first quadrant bounded by y = √ x and y = x 3 . 12. Evaluate the double integral R R D (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 D 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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