Transcript 1.1 Real Numbers, Inequalities, Lines

§ 1-1 Real Numbers, Inequalities,
Lines, and Exponents
The student will learn about:
the Cartesian plane,
straight lines,
an application,
integer exponents, and
fractional exponents.
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Introduction
• Quite simply, calculus is the study of rates of
change. We will use calculus to analyze
rates of inflation, rates of learning, rates of
population growth, and rates of natural
resource consumption, etc.
• In this first section we will study linear
relationships between two variables — that
is, relationships that can be represented by
lines.
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The section starts using basic
definitions of the real numbers,
inequalities, set and interval
notation. The student is
responsible for knowing this
material.
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Cartesian Coordinate System
Students should be familiar with the
basic terminology of the Cartesian
coordinate system.
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Lines and Slopes
The symbol ∆ (read “delta,” the Greek letter D)
for mathematicians means “the change in.”
For any two points (x1, y1) and (x2 , y2) we define
x = x2 – x1. The change in x is the difference
in the x-coordinates.
y = y2 – y1. The change in y is the difference
in the y-coordinates.
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Slope of a Line
1. Def: If P1 (x1, y1) and P2 (x2, y2) are two
points on a line then the slope of that line is
given by the formulas:
( y2  y1 )
Vertical Change
Rise y
m



.
( x2  x1 ) Horizontal Change Run x
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Slopes of Lines
Discuss slope.
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Example – FINDING SLOPES
AND GRAPHING LINES
Find the slope of the line through the following pair
of points, and graph the line. (2, 1), (3, 4)
Solution:
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Slope-Intercept form of a Line
Def: The equation y = mx + b is called the
slope-intercept form of a line. The slope is m
and the y intercept is b.
We will use this form to graph a line on
graph paper and on our calculator.
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Slope-Intercept Form of a Line
The slope-intercept form when
graphing.
y = 2x - 3
We first plot the y intercept
(0, - 3).
The slope is 2 = 2/1 so we then
plot another point 2 units up
and 1 unit over from the y –
intercept. m = 2 = Δy .
1
Δx
We finish by connecting those
points to form the line.
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Graphing: Calculator
Sketch a graph of y = 2x – 1.
By calculator
1. Turn your calculator on
and in the y = window
enter 2x – 1.
2. On the zoom screen
choose “zoom standard”
and touch “enter”.
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Point-Slope Form of a Line
Def: The equation y – y 1 = m (x – x 1) is
called the point-slope form of a line. The
slope is m and (x 1, y 1) is a point on the line.
We will use this form to solve problems that
ask for the equation of a line.
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Example 3 – USING THE
POINT-SLOPE FORM
Find an equation of the line through (6, –2) with
slope
y – y1 = m(x – x1)
Solution:
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Special Cases
A vertical line has
an equation of x = a.
x=2
A horizontal line has
an equation of y = b.
y=1
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Equations of Lines
There is one form that covers all lines, vertical
and nonvertical.
Ax + By = C
For constants A, B, C, with
A and B not both zero.
Any equation that can be written in this form is
called a linear equation, and the variables are
said to depend linearly on each other.
We will sometimes use this form to write the
final form of our line.
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X - Intercepts
The x intercept of f (x) occurs where and if
the graph of the function crosses the x-axis.
Algebraically it is the x value where f (x) is
zero, or (x, 0). That is, to find the x intercept,
let f (x) = 0 and solve for x. Remember f(x)
is y!
2x + 3y = 6
2x + 3· 0 = 6
2x = 6
x=3
The x intercept is 3
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Y - Intercept
The y intercept of f (x) occurs where and if
the graph of the function crosses the y-axis.
Algebraically it is the f (x) value where x is
zero, or (0, f (x)). That is, to find the y
intercept let x = 0 and solve for f (x).
2x + 3y = 6
2· 0 + 3y = 6
3y = 6
y=2
The y intercept
is 2
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Intercepts on a Calculator
2
1. Consider 2x + 3y = 6. First
y
x2
3
solve the equation for y.
2. Graph this equation
Sometimes you may
on your calculator.
also use Table to find
3. To find the x-intercept both the x-intercepts
use the “zero” function
and the y-intercept at
under the “Calc” menu. the same time!
4. To find the y-intercept use the “value” function
under the “Calc” menu with an x value of 0.
Note: x-intercept is 3 or (3,0) and the y-intercept is 218or
(0,2).
Review
Equations of a Line
General
Ax + By = C
Not of much use. Test answers.
Slope-Intercept Form
y = mx + b
Graphing on a calculator.
Point-slope form
y – y1 = m (x – x1)
Finding the equation of a line.
Horizontal line
Vertical line
y=b
x=a
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Application Example
Linear Depreciation. Office equipment was
purchased for $20,000 and is assumed to have
a scrap value of $2,000 after 10 years. If its
value is depreciated linearly (for tax purposes)
from $20,000 to $2,000:
1. Find the linear equation that relates
value (V) in dollars to time (t) in years.
We know two points. What are they?
t = 0 and V = $20,000 (0, $20,000) and
t = 10 and V = $2,000 (10, $2,000)
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Application Example Continued
Linear Depreciation. Office equipment was purchased for
$20,000 and is assumed to have a scrap value of $2,000 after
10 years. If its value is depreciated linearly for (tax purposes)
from $20,000 to $2,000:
y – y 1 = m (x – x 1)
Points are (10, 2000) and (0, 20000).
The slope is V/ t or
m = (2000 – 20000) / (10 – 0) = - 1800
You may use either point. I will choose (10, 2000).
Substituting into V – V1 = m (t – t1), yields
V – 2000 = - 1800 (t – 10) = - 1800 t + 18000
So
V = -1800t + 20000.
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Application Example Continued
Linear Depreciation. Office equipment was purchased for
$20,000 and is assumed to have a scrap value of $2,000 after
10 years. If its value is depreciated linearly for (tax purposes)
from $20,000 to $2,000:
V = -1800t + 20000.
2. What would be the value of the
equipment after 6 years?
V(6) = -1800 • 6 + 20000 = - 10800 + 20000
= $9,200
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Application Example Continued
3. Graph the equation V = -1800t + 20000.
4. Use “value” to find V (6) on
the calculator.
5. Write a verbal interpretation
of the slope of the line found in
the first part of this problem.
0 ≤ t ≤ 10
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Positive Integer Exponents
Definition: The 3 4 is an exponential expression.
The base is 3 and the exponent or power is 4. It
is an abbreviation for repeated multiplication.
I.e. 3 4 = 3 · 3 · 3 · 3 = 81
n
More generally x n = x · x · · · x
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Exponential Properties
1. Exponential laws:
xm xn = xm+n
xm / xn = xm–n
(xy) n = x n y n
(x/y) n = x n / y n
(x m) n = x m n
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Zero and Negative Exponents
1. By definition
2. x
1
x0 = 1
1
1
2
 and x  2
x
x
and in general x
 x
H int :  
y
1
n
1
 n
x
 x
y

and  
x
y
n
y
  
 x
n
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Rational Exponents
1. By definition  9 = 3 means the principal
square root or the positive root if there are two.
2. x
3. x
1
2
m
n
1
3

x and x 
3
x
1
n
and in general x 

n
x
n
x
m
4. On a calculator
x
m
n
is
x^(m/n)
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Summary.
• We had a brief introduction to the Cartesian plane
and graphing.
• We learned about straight lines, slope, and the
different forms for straight lines.
• We did an applied problem involving a straight line
graph and saw the meaning of the y-intercept and
the slope of the graph.
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Summary.
• We learned about integer exponents positive, zero,
and negative.
• We learned fractional exponents.
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ASSIGNMENT
§1.1 on my website
15, 16, 17, 18, 38, 39, 40, 41, 42.
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