Calculus III Midterm March 4 2009
The examination is 75 minutes. No notes, books or calculators are permitted. Remember to
show all your working and read questions carefully.
(1) (10 points) Show by either geometric arguments or by the properties of the d
Math 123 - Fall 2015 - Homework - 10.4
Give complete, well written solutions to the following exercises:
1. Give an example of three nonzero vectors u, v and w in R3 such that uv =
u w but v = w. What geometric relationship must the three vectors have
for
Math 123 - Fall 2015 - Homework - 10.8
Give complete, well written solutions to the following exercises:
1. Reparameterize the general line
r(t) = x0 + at, y0 + bt, z0 + ct
by arc length starting from the point t = 0 in the direction of increasing t.
2. S
Math 123 - Fall 2015 - Homework - 10.7
Give complete, well written solutions to the following exercises:
1. Show that the curve
r(t) = et cos t, et sin t, et
lies on the cone z 2 = x2 + y 2 and sketch its graph.
2. Consider the curves r1 (t) = t, 1 t, t2
Math 123 - Fall 2015 - Homework - 10.9
Give complete, well written solutions to the following exercises:
1. A particle moves in the plane with motion given by
r(t) = 2 cos t, 3 sin t, 1
a. Describe the path of the particle.
b. At which points on the plane
Math 123 - Fall 2015 - Homework - 11.2
Give complete, well written solutions to the following exercises:
xa y b
. Show that
x2 + y 2
does not exist if a + b 2.
1. Let f (x, y) =
lim
f (x, y) = 0 if a + b > 2 and
(x,y)(0,0)
2. Let x = x, y . For which valu
Math 123 - Fall 2015 - Homework - 11.1
Give complete, well written solutions to the following exercises:
1. We have seen that the graph of the function
f (x, y) = ex
2 y 2
can be described as a hill.
a. Manipulate this equation (or devise your own) to nd
Math 123 - Fall 2015 - Homework - 13.4
Give complete, well written solutions to the following exercises:
1. Show that the centroid of a region with simple closed boundary C is given
by
1
x2 dy
2A C
1
y 2 dx
y=
2A C
x=
where A is the area of the region .
2
Math 123 - Fall 2015 - Homework - 13.3
Give complete, well written solutions to the following exercises:
1. Let F = (x3 y 2 ) and let C be the path in the xy-plane from (1, 1) to (1, 1)
that consists of the line segment from (1, 1) to (0, 0) followed by t
Math 123 - Fall 2015 - Homework - 11.6
Give complete, well written solutions to the following exercises:
1. a. Two surfaces are called orthogonal at a point of intersection if their
normal lines are perpendicular at that point.
Show that surfaces with equ
Math 123 - Fall 2015 - Homework - 11.5
Give complete, well written solutions to the following exercises:
1. Suppose a particle moving along a metal plate has velocity i 4j at the point
(3, 2) and that the temperature of the plate is given by T (x, y) = y
Math 123 - Fall 2015 - Homework - 11.7
Give complete, well written solutions to the following exercises:
1. Consider the function f (x, y) = kx2 + y 2 4xy, where k is some xed
constant.
a. Show that for any value of k, (0, 0) is a critical point of f .
b.
Math 123 - Fall 2015 - Homework - 11.3
Give complete, well written solutions to the following exercises:
1. Use the diagram below to answer the questions:
a. At the point Q is fx positive or negative? fy ? Explain your reasoning.
b. List the points P, Q,
1
Final practice:
Lines and planes:
1. Computation:
(a) Find an equation for the 2-dimensional plane (in 3-dimensions) that passes through the points (2; 1; 1), ( 1; 1; 10) and (1; 3; 4).
(b) A second plane passes through (2; 0; 4) and has normal vector p
1
Midterm 2 practice:
Directional-derivatives, gradient-vectors and tangent-planes:
1. Given a special point x = (x1 ; x2 ), and a special direction ~ = (u1 ; u2 ) (such that j~ j = 1), can you describe the trajectory ~ (t) = x + t~ ? What does this traje
Math 123 - Fall 2015 - Homework - 10.5
Give complete, well written solutions to the following exercises:
1. Find parametric equations for the line through the point (0, 1, 2) perpendicular
to the line with parametric equations x = 4 + 2t, y = t, z = 3t 5.
Math 123 - Fall 2015 - Homework - 10.6
Give complete, well written solutions to the following exercises:
1. Find an equation for the surface consisting of all points P R3 such that
their distance from (1, 0, 0) and the plane x = 1 are equidistant. Identif
College Pre-Calc
Section 12.4: Parallel and Perpendicular Vectors:
Dot Product
Definition of the Dot Product:
a b = ( a1 , a2 ) ( b1 , b2 ) = a1b1 + a2b2
also known as the scalar product or inner product
a b is a one "number" answer
College Textbook - "
Single Variable Optimization
(Chapter 8 of MFE book; 4.1, 4.3-4.5 in e-book)
Jankowski, Math for Economics I
April 8, 2015
Jankowski, Math for Economics I
Single Variable Optimization(Chapter 8 of MFE book; 4.1, 4.3-4.
Absolute maximum, minimum
Definition
Functions, and some applications
(4.1-4.5, 5.1 in MFE text; 1.1-1.2 in e-book)
Jankowski, Math for Econ 1
January 28, 2015
Jankowski, Math for Econ 1
Functions, and some applications(4.1-4.5, 5.1 in MFE text;
1.1-
Some prerequisites
The following topics a
The Chain Rule
(6.8 in MFE book; 2.5 in e-book)
Jankowski, Math for Economics 1
March 5, 2015
Jankowski, Math for Economics 1
The Chain Rule(6.8 in MFE book; 2.5 in e-book)
Here it is
Proposition
If f and g are differentiable, then so is the function F =
Continuity
(7.8 in MFE book; 1.5 in e-book)
Jankowski, Math for Econ 1
February 11, 2015
Jankowski, Math for Econ 1
Continuity(7.8 in MFE book; 1.5 in e-book)
Warm-up
Consider f below.
lim f (x) exists, but doesnt equal f (4).
x4
We had to lift our pen of
Transforming graphs and functions; inverses
(5.1, 5.2, 5.3 of MFE book; 1.2 of e-book)
Jankowski, Math for Econ 1
February 9, 2015
Jankowski, Math for Econ 1
Transforming graphs and functions; inverses(5.1, 5.2, 5.3 of MFE
Shifting graphs (5.1): Given y =
Mathematics for Economics II
Integration by parts
SCHOOL LOGO EXAMPLES
(MFE: 9.5; Stewart: 6.1)
Abtin Rahimian
ARTS & SCIENCE
Courant Institute of Mathematical Sciences, NYU
rahimian@cims.nyu.edu
March 25, 2016
1 / 16
The Magic Formula
Product rule: If u(
Mathematics for Economics II
Partial Fractions
SCHOOL LOGO EXAMPLES
(Stewart: 6.3)
Abtin Rahimian
ARTS & SCIENCE
Courant Institute of Mathematical Sciences, NYU
rahimian@cims.nyu.edu
March 27, 2016
1 / 15
Sometimes we have to integrate one polynomial divi
Mathematics for Economics II
Present
and Future Values
SCHOOL LOGO EXAMPLES
(Stewart: 10.3, 10.5)
Abtin Rahimian
ARTS & SCIENCE
Courant Institute of Mathematical Sciences, NYU
rahimian@cims.nyu.edu
March 27, 2016
1 / 19
Present vs. Future Values
The futur
Mathematics for Economics II
Integration
by Substitution
SCHOOL LOGO EXAMPLES
(MFE: 9.6; Stewart: 5.5)
Abtin Rahimian
ARTS & SCIENCE
Courant Institute of Mathematical Sciences, NYU
rahimian@cims.nyu.edu
March 25, 2016
1 / 13
Warm-Up
Lets find
Z
2
2xe x dx
Mathematics for Economics II
Introduction
to Differential Equations
SCHOOL LOGO EXAMPLES
(MFE: 9.8, 9.9)
Abtin Rahimian
ARTS & SCIENCE
Courant Institute of Mathematical Sciences, NYU
rahimian@cims.nyu.edu
April 9, 2016
1 / 31
Differential equations allow
Math 123 - Fall 2015 - Homework - 10.2
Give complete, well written solutions to the following exercises:
1. A plane can y at a speed of 180 kilometers per hour. The pilot takes o
from an aireld heading north. After 30 minutes the navigator notices that
th
Math 123 - Fall 2015 - Homework - 10.1
Give complete, well written solutions to the following exercises:
1. For each surface in R3 given by the following (in)equalities,
i. Describe the surface in words;
ii. Represent the surface in set notation; and
iii.