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Calculus 3 Essay

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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 ).

11

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

12

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).

13

Homework 14: Multivariable Functions...

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Calculus 3. Anti Essays. Retrieved December 13, 2018, from the World Wide Web: http://snehaedu.com/free-essays/Calculus-3-357634.html