## Tuesday, February 26, 2013

### Calculus 3

planning 11: Functions of Several Variables I
Name
Let z = f (x, y ) =

4 ? x2 ? y 2 .

(1) sketch the graph of the function. (Hint: ?rst square both sides, like in class)

(2) go back and sketch the range of f .

(3) recover and sketch the contours f (x, y ) = c for c = ?1, 0, 2, 4, 5, if they exist.

(4) Find and sketch the domain of g (x, y ) = ln(4 ? x2 ? y 2 ).

11

Homework 12: Multivariable Functions II: Limits and Continuity Name
(1) Find lim(x,y)?(1,3)

(2) Find lim (x,y)?(1,1)
x =y

(3) Find lim (x,y)?(2,0)
2x?y =4

xy
.
x2 +y 2

x2 ?y 2
x?y

(hint: factor)

?

2x?y ?2
2x?y ?4

(4) Show that lim(x,y)?(0,0)
and C3 {y = x2 }.

(hint: conjugate)

2x4 ?3y 2
x4 +y 2

(5) Show that lim(x,y)?(0,0) cos

does not exist by ?nding the limit along the three paths: C1 {x = 0}, C2 {y = 0}

2x4 y
x4 +y 4

=1

12

Homework 13: Multivariable Functions III: Partial Derivatives
Name (1) look all ?rst and second assure partial derivatives of f (x, y ) = x3 y 4 + ln( x ).
y

(2) Find the equation of the burn plane to the graph of the function z = f (x, y ) = exp(1 ? x2 + y 2 ) at
(x, y ) = (0, 0). Convert to regulation form.

(3) Find the equation of the tangent plane to the surface r(u, v ) = u3 ? v 3 , u + v +1, u2 at (u, v ) = (2, 1). Convert
to normal form.

(4) Suppose that fx (x, y ) = 6xy + y 2 and fy (x, y ) = 3x2 + 2xy . deem fxy and fyx to determine if there is a
function f (x, y ) with these ?rst derivatives. If so, shuffle to ?nd such a function.

(5) Show that the function u(x, y ) = ln(

x2 + y 2 ) is Harmonic (i.e., it satis?es Laplaces equation uxx + uyy = 0).

13

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