Homework 11: Functions of Several Variables I

Name

Due on Tuesday, in class. Submit 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: ﬁrst square both sides, like in class)

(2) Find and sketch the domain of f .

(3) Find 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 ﬁnding 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 next class period. Submit your solutions (work and answers) on this page only!

(1) Compute all ﬁrst and second order partial derivatives of f (x, y ) = x3 y 4 + ln( x ).

y

(2) Find the equation of the tangent plane to the graph of the function z = f (x, y ) = exp(1 − x2 + y 2 ) at

(x, y ) = (0, 0). Convert to normal 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 . Compute fxy and fyx to determine if there is a

function f (x, y ) with these ﬁrst derivatives. If so, integrate to ﬁnd such a function.

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

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

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Homework 14: Multivariable Functions...