15
Vector Calculus
15.0 Still More Topological Notions
A region R R2 is called vertically simple if there exist continuous functions f1 and f2
such that
R = cfw_(x, y ) : a x b and f1 (x) y f2 (x).
Similarly, R is horizontally simple if there exist contin
10
Series Functions
10.1 Taylor Polynomials
Polynomial functions, as we have seen, are well behaved. They are continuous everywhere,
and have continuous derivatives of all orders everywhere. It also turns out that, given any
continuous function f that has
9
Sequences and Series
9.1 Sequences
Whats called a sequence in mathematics meshes with the everyday conception, which is
that of an ordered list of objects. However we have to be somewhat more precise than this.
Denition 9.1. A sequence is a function f f
11
Parametric and Polar Curves
11.1 Parametric Equations
A function f is called continuously dierentiable on an open interval (a, b) if f is
dierentiable on (a, b) and f is continuous on (a, b).
Suppose we have parametric equations
x = g (t), y = h(t); a
12
Coordinate Vectors
12.1 Vectors in a Plane
By a plane is meant the set
R2 = cfw_(x, y ) : x R and y R ,
and a coordinate vector in R2 is taken to be an ordered pair of real numbers v1 , v2 . We
write v = v1 , v2 and call the values v1 and v2 the compon
14
Multiple Integrals
14.0 More Topological Notions
A region is just a set of points in Rn . Regions in R2 will usually be denoted by R or S ,
and regions in R3 by D or E . In a general setting when the ambient space is not specied R
will usually be used.
13
Partial Derivatives
13.0 Notation & Topological Notions
Chapter 12 introduced the notion of vectors. The vectors were thought of as existing in
a set of points such as a plane, a space, or Rn in general. We can and will continue to adopt
this rather ph
8
Integration Techniques
8.1 Integration by Parts
For functions u and v , let J = Dom(u ) Dom(v ), so J is an open subset of both Dom(u)
and Dom(v ), and u and v are continuous on J . By the Product Rule of Dierentiation, for
each x J
(uv ) (x) = u(x)v (x
7
Transcendental Functions
7.1 Inverse Functions
Some functions are capable of assuming the same value at dierent points in their domain.
For instance the function f : R R given by f (x) = x2 takes on the value 9 when x is 3 or
3. A function that cannot d
2
Limits and Continuity
2.1 Neighborhoods and Limit Points
Given a point c R, a neighborhood of c is any open interval I R that contains c.
Thus, for any a < c and b > c, the interval (a, b) is a neighborhood of c. Other neighborhoods
of c are (, b), (a,
1
Foundations
1.1 Logic
Under Construction.
1.2 Set Theory
A set is a collection of objects. An object in a set is called an element of the set. If S is
a set and a is an element of S , we write a S . If a is not an element we write a S . Two
/
sets A and
3
Differentiation Theory
3.1 The Derivative of a Function
Motivated historically by the slope problem discussed in the previous section, there is the
following denition.
Denition 3.1. Let c be an interior point of Dom(f ). Then the derivative of f at c is
6
Applications of Integration
6.1 The Mean Value Theorem for Integrals
In 4.2 we encountered the traditional Mean Value Theorem, sometimes called the Mean
Value Theorem for Derivatives to distinguish it from the following.
Theorem 6.1 (Mean Value Theorem
5
Integration Theory
5.1 The Riemann Integral
Let f be a continuous function such that f (x) 0 for all a x b. The question arises:
what is the area of the region in R2 that is bound by the graphs of the equations x = a, x = b,
y = 0, and y = f (x)? Figure
Experiment 11.xlsx
mwater
0.1845 kg
Tiwater
2.6
o
mwater
0.152
kg
Tiwater
2
o
mwater
0.145
kg
Tiwater
1.7
o
mcup
0.047
Tfwater
5.2
o
mcup
0.047
kg
Tfwater
5.3
o
mcup
0.047
kg
Tfwater
5
o
msample
0.063
kg
Tisample
62.1
o
Tfsample
5
o
msample
0.063
kg
c
c
m
Kira Stone
Physics Lab # 8: Conservation of Energy
It is appropriate that the position (m) vs. time (s) and velocity (m/s) vs. time (s) graphs are periodic, as
the motion of the pendulum is periodic. Our situation is not frictionless and without air resis
Trial
Diameter (cm)
5.5
Mass (g)
42.8
1
Position (m)
2
Position (m)
3
Position (m)
4
Position (m)
5
Position (m)
Height of Motion Detector (m)
0.85
Velocity Before Bounce 1 (m/s)
Velocity After Bounce 1 (m/s)
Velocity Before Bounce 2 (m/s)
Velocity After
Kira Stone
Group Members: Ryan Pugliese & Lance Rosina
Mass of cart: 505 g
Mass of one paperclip: 0.4 g
Trial
1
2
3
4
5
6
7
8
Average
Standard Deviation
Mass Added (g)
10.4
20.4
30.4
40.4
50.0
60.0
70.0
80.0
g = 9.749 m/s2 0.067
Literature value g = 9.81
Lab 1 Questions
Kira Stone Physics 111 Lab
Dr. Ferrari
September 2, 2010
1. 30 points per second is an acceptable sampling rate because an objects velocity may
change just enough to measure in 1/30th of a second. If one walks 1 meter, for example,
they tr
Test 2: Chapters 3 5
Chapter 3
Communication in the Nervous System
Neurons and glia
-Neurons: individual cells in the nervous system that receive, integrate, and transmit
information
-The soma, or cell body, contains the cell nucleus and much of the chemi