Calculus 3

planning 11: Functions of Several Variables I
Name
Due on Tuesday, in class. pick out your solutions (work and answers) on this page only!
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 ).

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Homework 12: Multivariable Functions II: Limits and Continuity Name
Due on Tuesday, in class. Submit your solutions (work and answers) on this page only!
(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

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Homework 13: Multivariable Functions III: Partial Derivatives
Name
Due at the beginning of our abutting class period. Submit your solutions (work and answers) on this page only!




(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).

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