Name _
Multivariable Calculus
Date _
Problem Set #15
Please show all work to each of the following problems.
r
r
x
x
1. Determine whether or not the vector field F ( x, y ) = ( ye + sin y ) i + ( e + x cos y ) j is a
conservative vector field. If it is co
Name _
Date _
Multivariable Calculus
Problem Set #2
1. What type of conic section is described by the equation 2 y 2 3x 2 4 y + 12 x + 8 = 0 ? If it is
an ellipse, find the vertices and foci. If it is a parabola, find the vertex, focus, and directrix. If
Name _
Multivariable Calculus
Date _
Problem Set #4
For the test on Sections 12.5-12.7, you should be able to, using appropriate diagrams, explain
how the conversion formulas between cylindrical and rectangular coordinates, as well as
between spherical an
Name _
Multivariable Calculus
Date _
Problem Set #3
Please show all work to each of the following questions.
1. Consider the set of all points P such that the distance from P to A ( 1,5,3) is twice the
distance from P to B ( 6, 2, 2 ) . Show that the set
Name _
Multivariable Calculus
Date _
Problem Set #5
For the assessment on Sections 13.1-13.3, you should be able to prove the following statement:
r
r
r
If r ( t ) = c , where c is a constant, then r ' ( t ) is orthogonal to r ( t ) for all values of t .
Name _
Multivariable Calculus
Date _
Problem Set #6
1. Find the torsion of each of the following curves:
r
3
a) r ( t ) = t , 2t ,3t for arbitrary time t .
r
b) r ( t ) = sinh t , cosh t , t at the point ( 0,1, 0 ) . Hint: Look back at your first problem
Name _
Multivariable Calculus
Date _
Homework Set #8
Please show all work to each of the following questions.
1. Compute lim
( x + h ) ln ( ( x + h )
2
)
+ y 2 x ln ( x 2 + y 2 )
h
h 0
.
2. The wave heights h , in feet, in the open sea depend on the speed
Name _
Multivariable Calculus
Date _
Problem Set #7
Please show all work to each of the following problems.
2
2
2
1. Let g ( x, y , z ) = ln ( 25 x y z ) .
a) Evaluate g ( 2, 2, 4 ) .
b) Find the domain of g . Give a description of what this domain looks
Name _
Multivariable Calculus
Date _
Problem Set #10
Please show all work to each of the following problems.
3
2
2
2
1. Consider the function f ( x, y ) = 2 x + xy + 5 x + y . Find the local maximum and minimum
values of the function, as well as any saddl
Name _
Multivariable Calculus
Date _
Problem Set #9
Please show work to all of the following questions.
1. Four positive numbers, each less than or equal to 50, are rounded to the first decimal place
and then multiplied together. Use differentials to esti
Name _
Multivariable Calculus
Date _
Problem Set #11
Please show all work to each of the following problems.
1. Compute
8
2
2. Compute
1 x 2 y 2 dA , where D is the disk x 2 + y 2 1 . Hint: Before doing any
0
D
3
4
y
e x dx dy .
calculations, think about
Name _
Multivariable Calculus
Date _
Problem Set #14
Please show all work to each of the following problems.
y
1. Compute the gradient vector field for the function f ( x, y, z ) = x cos .
z
2. Create a rough sketch of the vector field F ( x, y ) =
rr
yi
Name _
Multivariable Calculus
Date _
Problem Set #13
Please show all work to each of the following problems.
1. Evaluate the triple integral
x dV , where E
E
is the bounded by the paraboloid
x = 4 y 2 + 4 z 2 and the plane x = 4 .
2. Use a triple integra
Name _
Multivariable Calculus
Date _
Problem Set #12
1. Find the volume of the solid that lies inside the sphere x 2 + y 2 + z 2 = 16 and outside the
cylinder x 2 + y 2 = 4 .
2. Use polar coordinates to combine the sum:
1
1
2
x
1 x 2
xy dy dx +
1
2
x
0
x
Name _
Date _
Multivariable Calculus
Problem Set #1
Please show all work to each of the following questions, including steps to
any integrals that may need to be calculated.
1. Find the area A of the region R that is inside the cardioid r = 2 + 2cos
and