Calculus 3 (Math-UA-123)
Fall 2016
Homework 7: Sections 11.7, 11.8
Give complete, well-written solutions to the following exercises.
1. Consider the function f (x, y) = kx2 + y 2 4xy, where k is some fixed constant.
(a) Show that for any value of k, (0, 0
Vectors 10.2
1. Geometrically construct the denition of the magnitude of a vector in 3D.
2. Geometrically describe direction angles and direction cosines.
3. A woman walks due west on the deck of a ship at 3 mph. The ship is moving north at a speed of 22
The Dot Product 10.3
1. Find the angle between a diagonal of a cube and one of its edges.
2. Suppose that all sides of a quadrilateral are equal in length and opposite sides are parallel. Use vector
methods to show that the diagonals are perpendicular.
3.
Calculus 3 (Math-UA-123)
Fall 2016
Homework 4: Sections 10.8-10.9
Give complete, well-written solutions to the following exercises.
1. At what point on the curve
x = t3 , y = 3t, z = t4
is the normal plane parallel to the plane 6x + 6y 8z = 1?
2. Find the
Calculus 3 (Math-UA-123)
Fall 2016
Homework 3: Sections 10.6-10.7
Give complete, well-written solutions to the following exercises.
1. (1) Match the equation with the surface it defines and (2) identify each surface by type
(ellipsoid, paraboloid, etc.).
Calculus 3 (Math-UA-123)
Fall 2016
Homework 2: Sections 10.3-10.5
Give complete, well-written solutions to the following exercises.
1. Let P , Q, R be three points not on the same line, and let L be the line through Q and
R. Let a = QR and b = QP . Show t
Calculus 3 (Math-UA-123)
Fall 2016
Homework 6: Sections 11.5, 11.6
Give complete, well-written solutions to the following exercises.
1. Find the values of
z
x
and
z
y
at the given point:
xey + yez + 2 ln(x) = 2 + 3 ln(2),
(1, ln(2), ln(3).
2. If f (u, v,
Calculus 3 (Math-UA-123)
Fall 2016
Homework 11: Sections 13.3-13.4
1. Suppose that F is an inverse square field:
F(r) =
cr
,
|r|3
where c is a constant and r = x i + y j + z k. Find the work done by F in moving an
object from a point P1 along a path to a
Calculus 3 (Math-UA-123)
Fall 2016
Homework 12: Sections 13.5-13.7
1. For each of the following vector fields, determine if the divergence is positive, zero, or
negative at the indicated point. Explain/justify your answer.
(b)
(c)
6
6
6
4
4
4
2
2
0
0
y
y
Calculus 3 (Math-UA-123)
Fall 2016
Homework 10: Sections 13.1-13.3
1. Figures (I)-(IV) contain level curves of functions of of two variables f (x, y). Figures
(A)-(B) are their corresponding gradient fields f (x, y).
Match the level curves in (I)-(IV) wit
Calculus 3 (Math-UA-123)
Fall 2016
Homework 8: Sections 12.1-12.3
1. Find the volume of the solid that lies between the surface z = x2xy
2 +1 and the plane
z = x + 2y and is bounded by the planes x = 0, x = 2, y = 0, and y = 4.
2. Evaluate the double inte
Calculus 3 (Math-UA-123)
Fall 2016
Homework 5: Sections 11.1, 11.3, 11.4
Give complete, well-written solutions to the following exercises.
1. Match each set of level curves with the appropriate graph of function. Briefly explain
your choices.
1.
2.
3.
4.
Calculus 3 (Math-UA-123)
Fall 2016
Homework 9: Sections 12.5-12.7
1. Sketch and describe the region of integration of the integral below. Include clear
explanation/justification.
Z 1 Z 1z2 Z 1x2 z2
f (x, y, z) dy dx dz.
0
1z 2
0
2. Evaluate the following
Arc Length and Curvature
We have a space curve, C and the following parameterizations:
Consider the following curve r(t) = et cos t, et sin t, et . Find the arc length from 0 t .
Space Curve Functions - Parameterize the curve with s.
Reparameterize r(t) =
Vector Functions and Space Curves 10.7 and the beginnings of 10.8
1. Find a vector equation for the tangent line to the curve of intersection of the cylinders x2 + y 2 = 25 and
y 2 + z 2 = 20 at the point (3, 4, 2).
2. If two objects travel through space
TANGENT PLANES AND LINEAR APPROXIMATIONS
1. Tangent planes
Problem 1. Find the equation of the tangent plane to the graph of the function z =
f (x, y ) = 1 x2 y 2 at the point (a, b). Try to simplify the equation by setting c =
1 a2 b2 .
Problem 2. Find t
Calculus 3 (Math-UA-123)
Spring 2017
Homework 0
Due: Friday, January 27
at 12pm (noon)
Please give complete, well-written solutions to the following exercises. You must scan and
upload your solutions to your account on GradeScope (https:/gradescope.com/).
Freshman Seminar: Death in Rome
FRSEM-UA 629
Spring 2017
Thursdays, 3:30 6:00
Professor Michael Peachin
Department of Classics
Silver Center 503D
mp8@nyu.edu
Course Description:
There is a famous sentiment attributed to Benjamin Franklin: In this world
no
Calculus 3 (Math-UA-123)
Spring 2017
Homework 1
Due: Monday, January 30
at the start of class
Please give complete, well-written solutions to the following exercises. Submit via Gradescope.
1. (a) Describe in words the set of points that satisfy the equat
Exercises Chapter 9
1. The acceleration due to gravity at the moons surface is only about one-sixth that at the earths surface. If you took a
pendulum clock to the moon, would it run fast, slow, or on time?
E.1
It would run slow.
2. A clothing rack hangs
Calculus 3 (Math-UA-123)
Fall 2016
Homework 13: Sections 13.8-13.9
Z
F dr, where F(x, y, z) =
1. Use the surface integral in Stokes Theorem to calculate
C
x2 y 3 i + j + zk and C is thee intersection of the cylinder x2 + y 2 = 4 and the hemisphere
x2 + y
Calculus 3 (Math-UA-123)
Fall 2016
Homework 1: Sections 10.1, 10.2
Give complete, well-written solutions to the following exercises.
(All questions below are about points in R3 .)
1. Consider the points A(0, 3, 1) and B(1, 2, 2).
(a) Find the projections