Math 273
HW 2
Name:
Due: 22 February 2016
Show Appropriate Work
Scaled to 10 points
1. 2 Sketch the level curves corresponding to c = −2, −0, and 2 for f (x, y) = ex + y.
p
2. Consider the function f (x, y) = x2 + (y/2)2 − 1.
(a) 2 Find and sketch the domain of f . Express your solution in set notation.
√ √
√
(b) 2 Sketch the level curves corresponding to c = 0, 3, 8, and 15.
(c) 2 Sketch the surface.
3. Find the limit, if it exists, or show that the limit does not exist. Justify your solutions.
(a) 2
lim
(x,y)→(2,π)
x cos y
[Hint: Is f continuous?]
x2 + sin2 y
sin(x2 + y 2 )
[Hint: Polar and see Theorem 2 in §2.6.]
x2 + y 2
(x,y)→(0,0)
x arcsin y
lim
(c) 2
[Hint: Consider the paths x = 0 and y = sin x.]
(x,y)→(0,0) x2 + y 2
 3
3
 x y − xy
if (x, y) 6= (0, 0)
4. Consider the function f (x, y) =
x2 + y 2

0
if (x, y) = (0, 0)
(b) 2
lim
(a) In class we showed that f was continuous.
(b) A routine use of the quotient rule shows that for (x, y) 6= (0, 0), fx (x, y) =
y(x4 + 4x2 y 2 − y 4 )
.
(x2 + y 2 )2
(c) 2 For (x, y) 6= (0, 0) compute fy . Simplify. It should look similar to the above.
(d) We can use the definition of a partial derivative at a point to compute fx (0, 0) as follows.
f (h, 0) − f (0, 0)
=0
h→0
h
fx (0, 0) = lim
(e) 2 Use the definition of a partial derivative at a point to compute and fy (0, 0). Specifically,
f (0, h) − f (0, 0)
.
h→0
h
fy (0, 0) = lim
(f) 2 Use the definition of a partial derivative at a point to compute fxy (0, 0) and fyx (0, 0).
fy (h, 0) − fy (0, 0)
fx (0, h) − fx (0, 0)
Specifically, fxy (0, 0) = lim
and fyx (0, 0) = lim
.
h→0
h→0
h
h
(g) 1 What can be said about fxy and fyx in light of Clairaut’s Theorem (Theorm 1 in §14.3).
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5. Let f (x, y) = 36 − 9x2 − 4y 2
(a) 2 Sketch the surface.
(b) 2 Find the differential df .
(c) 2 If (x, y) changes from (1, 2) to (1.01, 1.97) compare the values of ∆f and df .
√
(d) 2 Find an equation for the tangent plane to the surface at the point (1, 2, 11).
(e) 1 Find an equation for the tangent plane to the surface at the point (0, 0, 6)