Ch 1.1: Preliminaries
The Real Numbers
Visualized on number line
Set notation: A = {x : condition}
Example
A = {x : 0<x<5, x a whole number}
= {1,2,3,4}
Reals
Interval Notation
Open intervals (a, b)
Closed intervals [a,b]
Half open intervals (a, b], [a,b)
Unbounded intervals; infinity notation
Real numbers; interval notation
Proportionality
Two quantities x and y are proportional if y = kx for some
constant k
Ex: The rate of change r of a population is often
proportional to the population size p:
r = kp
Proportionality
Ex: 11(17) Experimental study plots are often squares of
length 1 m. If 1 ft corresponds to 0.305 m, express the area
of a 1 m by 1 m plot in square feet
Soln: Use proportionality. Let y be measured in feet, x in
meters. Then
y=kx
k = y/x = (1 ft)/ (.305 m) = 3.28
Then y = 3.28x and
(y ft) X (y ft) = (3.28)(1) X (3.28)(1)
Ans: 3.28 ft X 3.28 ft
Lines
Recall: x and y are proportional if y = kx for some constant
k
Suppose the change in y is proportional to the change in x:
y1 – y0 = m(x1 – x0)
This is the point-slope formula for a line
Equations of Lines
Slope: m = (y1 – y0)/ (x1 – x0)
Point-slope form
y – y0 = m(x – x0)
Slope-intercept form
y = mx + b
Standard form
Ax + By + C = 0
Vertical Lines: x = a
Horizontal lines: y = b
Equations of Lines
Parallel Lines: m1 = m2
Perpendicular Lines: m1 = -1/m2
Equations of Lines
Average CO2 levels in atmospheres (Mauna Loa)
Use data to find a model for CO2 level
CO2 Level
360
355
350
345
340
335
330
325
1970
Use the model to predict
CO2 levels in 1987 & 2005
CO2 Level
1975
360
355
y = 1.4967x - 2624.8
350
1980
1985
1990
1995
345
Year 340
335
330
325
1970
1975
1980
1985
1990
CO2 (ppm)
CO2 (ppm)
327.3
330
332
335.3
338.5
341
344.3
347
351.3
354
CO2 (ppm)
Year
1972
1974
1976
1978
1980
1982
1984
1986
1988
1990
Year
1995
Equations of Lines
Example: Find the equation of the line that passes through
(1,2) and (5, -3). [Standard form]
What is the slope of the line that is parallel to this line?
Perpendicular?
Example: Find the equation of the horizontal line that
passes through (2,3)
Example: Find the equation of the vertical line that passes
through (-4,1)
Trigonometry: Angles
There are two primary measures of angle
Degrees: 360 deg in a circle
Radians: 2pi radians in a circle
Conversion: y = radians, x = degrees
y = kx
2  rad  k 360 deg
2  rad
k
360 deg
y 
 rad
180 deg
x
Trigonometry: Angles
y
 rad
x
180 deg
Example: Convert 30 deg into radians
Example: Convert 60 deg into radians
Example: Convert 45 deg into radians
Example: Convert 1 rad into degrees
Note: 1 rad is the angle for which the arc length is equal to
the radius
Graph common angles
Trigonometric Functions
sin  
opp

hyp
cos  
adj
opp
adj
 y
1

hyp
tan  
y
x
1

y
x
 x
csc  
sec  
cot  
See Maple worksheet for more trig info.
hyp

1
opp
y
hyp
1

adj
x
adj
x
opp

y
Trigonometric Identities
sin   cos   1
2
2
tan   1  sec 
2
2
Other trig identities can be
derived and used in problem
solving.
Homework
Read Ch 1.1
10(7-10,15,25-29)