1.3
Algebraic Expressions
Variable
A variable is:
A letter that can represent any number from
a given set of numbers.
An algebraic expression is:
Variables (e.g. x, y, z) and some real numbers,
combined using addition, subtraction,
multiplication, divis
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Topic 2.1-C Velocity in function notation
Our goal in this lesson is to follow up on the velocity in function notation lecture with
some experience.
x(t + t) - x(t)
v =
t
The above formula for average velocity is basically the result of the ab
Topic 2.1 Extension 07
Acceleration
To follow Worksheet "Acceleration"
1)
Recall that the speed or velocity of an object is the time rate of
change of the position of the object, or in formula form <v> = x/t
and v = dx/dt. What is the difference between <
Topic 2.1 Extension 02
Instantaneous velocity v
To follow the workseet "Instantaneous velocity v"
1)
In the lesson "Average velocity <v>" you learned how to find the
average speed of a particle given a graph of its height vs. the time.
What happened to t
The energy-momentum four-vector
REFERENCE: Hartle
Recall the definition of the four-momentum
given by
p = m0u.
of a particle of mass m0
[Eq. 1]
We will call m0 the rest mass for reasons that will follow.
definition:
Here is a
"The rest mass m0 of a partic
The scalar product and the metric of flat spacetime
PROBLEM SET
PROBLEM 1. Show that the following relationships hold between the basis
four-vectors. Hint: Look at Eq. 5 of the lecture.
et et = -1,
ex ex = 1,
ey ey = 1,
ez ez = 1,
e e = 0 for .
PROBLEM 2.
Topic 2.1 Extension 03
The derivative dy/dt
1)
In the lesson "Instantaneous velocity v" you learned how to find
the instantaneous speed of a particle given a graph of its height vs.
the time. To find the instantaneous speed of the particle we took the
lim
Relativistic addition of velocities
PROBLEM SET
PROBLEM 1. Show that the relativistic addition of velocities formulas
reduce to the Newtonian ones whenever v < c.
PROBLEM 2. Show that the equations
dy'
dz'
(dx - vdt)
dx'
2
dt' = (dt - dxv/c2) , dt' = dy/(
The scalar product and the metric of flat spacetime
REFERENCE: Hartle
Suppose we have two four-vectors a and b. We can express them in terms
of the basis four-vectors as a = ae and b = be. Their scalar product
(or dot product) can be written
a b = (ae)(be
Zero rest mass particles
PROBLEM SET
PROBLEM 1. What is the
frequency is f = 2000kHz?
energy
Hint:
carried by
= 2f.
a
photon
whose
linear
PROBLEM 2. Explain why
we can't use the energy-momentum four vector in
the form p = ( m0c, m0V ) for photons or grav
Topic 2.1 Extension 07
Acceleration
1)
Recall that the speed or velocity of an object is the time rate of
change of the position of the object, or in formula form <v> = x/t
and v = dx/dt. What is the difference between <v> and v?
FYI:
We define the averag
Topic 2.1 Extension 06
More differentiation
1)
Do the work for the following neatly in the space provided.
Place the UNSIMPLIFIED answers in the blanks.
a)
c)
e)
g)
i)
k)
(d/dx)(e3cos x)
(d/dx)(sin 3x)
b)
_
_
(d/dx)(4x 3)4
d)
_
_
(d/dx)(2x sin 3x)
f)
_
_
Topic 2.1 Extension 04
Derivative shortcuts
FYI:
In the "The derivative dy/dt" you learned how to find the
FYI
derivative of a function without using a graph. You discovered that it
is quite a bit of work to find the derivative using this definition:
dy
l
Topic 1.3 - Project
You must okay this project with me before beginning it. This project must be typed up and
it must include the assessment criteria from this topic (shown below). Your electronic
copy must be uploaded to TurnItIn.com, and your hard copy
Proper time revisited
REFERENCE: None
(ct)
dct = cdt
Consider a particle moving from A to B in (S),
as shown in Fig. 1.
Only three of its four
dimensions
is
shown,
the
z-direction
suppressed because of our limitations (we can
only adequately show three di
The "big three" of relativity
REFERENCE: None
Remember the "big three" of Newtonian physics? They were definitions,
not laws. The big three consisted of a three-position vector
R = Xi + Yj + Zk,
[Eq. 1]
where X = X(t), Y = Y(t), and Z = Z(t) were all para
Topic 2.1 Extension 01
Average velocity <v>
To follow the worksheet Average velocity <v>
x(m)
1) How many points do you need2 to
find the slope of a line? _.
2) Suppose you have a particle
whose position x (in meters) is
given as a function of time t (in
Four-vectors
REFERENCE: Hartle
Perhaps you've begun to think about vectors in
(ct)
four-dimensional space - after all, vectors are a
major part of the language of physics. We will
b
call these vectors four-vectors to distinguish
a + b
them from the old th
Coordinate Geometry
The Coordinate Plane
The Distance and Midpoint Formulas
Graphs of Equations In Two Variables
1.9
Intercepts
Circles
Symmetry
Coordinate Geometry
The coordinate plane is the link between
algebra and geometry.
In the coordinate plane, w
2
Functions
Chapter Overview
In this chapter, we will learn:
How functions are used to model
real-world situations.
How to find such functions.
We will be covering sections
2.1
2.2
2.3
2.5
Focus on Modelling
2.1
What Is a Function?
Functions All A
1.10
Lines
The Slope of a Line
Equations of Lines
Parallel and Perpendicular Lines
Applications: Slope as Rate of Change
Lines
In this section, we find equations
for straight lines lying in a coordinate plane.
The equations will depend on how the line
is
Integumentary System
ANAT 1008
Integumentary System
Made up of:
Skin
Waterproof covering for the body
Protection from UV light, chemicals, etc
Appendages
sweat glands
oil glands
hair and nails
The skin is actually the largest of all the organs (~7%
Blood vessels
Blood is transported in different types of vessels:
Arteries carry blood away from the heart
Veins carry blood toward the heart
Capillaries allow for exchange between the
bloodstream and tissue cells (also at the lungs)
All vessels are
11/17/2014
The Endocrine System
ANAT 1008
Endocrine Function
Endocrine
Nervous
Chemical extension of nervous system which
monitors, regulates & integrates the internal
activities of body.
Endocrine glands (ductless) release hormones
which travel in the
Electrocardiography
The electrical impulses associated with
contractions of heart muscle also spread
throughout surrounding tissue (skin)
Detectable by an electrocardiograph
through electrical leads placed on the
skin
The resultant recording (trace) is
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Topic 2.1-B Function notation
Our goal in this lesson is to follow up on the function notation lecture with some
experience.
HINTS: (a+b)2 = a2 + 2ab + b2
(a+b)3 = a3 + 3a2b + 3ab2 + b3
1)
Complete the table below for the function x(t) = 2t2.
The relativistic train spacetime diagram
REFERENCE: None
The first thing we will determine
is how the Lorentz boost will be
drawn in each coordinate system.
(ct)
The Lorentz
We found the analytic solution of
boost
as
the train problem using the Lorentz
ob
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Topic 2.1-F Derivative shortcuts
Our goal in this lesson is to follow up on the derivative shortcuts lecture and practice
using the rules. Here they are, for your reference.
BASIC RULES:
dx
If x = Ctn then dt = nCtn-1.
dx
If x = sint then dt =
Zero rest mass particles
REFERENCE: Hartle
You may have gleaned from problems and lessons that no particle having a
non-zero rest mass can ever reach the speed of light. The equation for
the energy-momentum four-vector should convince
you. The energy and
Topic 2.2 Extension 03
Differential equations
1)
Recall that the process of finding the derivative has the symbolic
representation d/dt. Thus the derivative of y with respect to t is
represented as dy/dt, and means the limit as t->0 of the rate
y/t. We ca
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Topic 2.1-D The derivative
Our goal in this lesson is to follow up on the derivative lecture and develop a shortcut for
finding the derivative. In order to do so, you will have to go through the 4-step
procedure outlined in the lecture, which
Proper time revisited
PROBLEM SET
PROBLEM 1. Let A and B lie on the xaxis in Euclidean space. Let y = y(x)
and z = z(x) be functions that give
the y- and the z-coordinates of a
particle moving from A to B. Then for
each
x-coordinate,
we
have
the
position
Searches and Sort Worksheet
Name:
Sequential Search
Complexity: O(n)
1. What does the complexity of an algorithm determine in the analysis of algorithms?
2. Define a sequential search:
3. How many passes will it take to find the #78 in this list of data u