Transcript B. Linear Functions

1
Linear functions
A. Functions in general
B. Linear functions
C. Linear (in)equalities
Handbook: E. Haeussler, R. Paul, R. Wood
(2011). Introductory Mathematical Analysis for
business, economics and life and social
sciences. Pearson education
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A. Functions in general
1. definition
3
A. Functions in general
Introduction
In every day speech we often hear economists
say things like
“ interest rates are a function of oil prices”,
“pension income is a function of years worked”
Sometimes such usage agrees with
mathematical usage, but not always.
(Handbook: Section 2.1 p80, paragraph 1-2)
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A. Functions in general
Example Taxidriver
What does a taxi ride cost me with company A?
• Base price: 5 Euro
• Per kilometer: 2 Euro
Price of a 7 km ride?
price  5  2  7  19
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A. Functions in general
Example Taxidriver
What does a taxi ride cost me with company A?
• Base price: 5 Euro
• Per kilometer: 2 Euro
Price of an x km ride?
y  5  2x
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A. Functions in general
Definition
•
x
and
(length of ride in km)
• y depends on x:
y
: VARIABLES
(price of ride in euro)
INPUT  OUTPUT
x
y
y: DEPENDENT VARIABLE
x: INDEPENDENT VARIABLE
Function: rule that assigns to
each input at most 1 output
(Section 2.1 p81, last 4 paragraphs)
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A. Functions in general
Definition
• We say:
y is FUNCTION of x,
or in short f of x
• We denote: y(x) or y=f(x)
• Outputs are also called function values
(Handbook: Section 2.1 p82)
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A. Functions in general
1. definition
2. Three representations
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A. Functions in general
Three representations
First way: Most concrete form!
Through a TABLE, e.g. for y = 2x + 5:
x
0
1
y
5
7
2 9
… …
But: limited number of values  no overall picture
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A. Functions in general
Three representations
Second way: Most concentrated form!
Through the EQUATION, e.g. y = 2x + 5.
formula y = 2x + 5:
EQUATION OF THE FUNCTION
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A. Functions in general
Three representations
y
Third way:
Most visual form!
Through the GRAPH
rectangular coordinate system:
x-coordinate, y-coordinate
7
6
5
4
3
2
1
-4 -3 -2 -1 0
-1
(Handbook: Section 2.5 p99)
1
2 3
x
y
0
5
1
7
4
x
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A. Functions in general
Three representations
y
14
Third way:
Most visual form!
Through the GRAPH
e.g. for y = 2x + 5:
STRAIGHT LINE!
2
x
1
Note: In this example, the graph is a only a part of a straight line
5
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A. Functions in general
Exercises
The demand q of a product depends on the price p.
For a local pizza parlor some weekly demands and prices
are given
p q
Remark: this table is called
10 640
a demand schedule
12
560
14
480
(a) What is the input variable? What is the output variable ?
(b) Indicate the points in the table on a graph
(Handbook: Section 2.1 p85 – example 5)
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A. Functions in general
Exercises
Suppose a 180-pound man drinks
four beers in quick succession.
The graph shows the blood alcohol
concentration (BAC) as a function
of the time.
(a) Input ? Output ?
(b) How much BAC is in the blood after 5 hours ?
(c) What will be the maximal BAC ?
After how much time, will this maximum be attained ?
(d) What’s the behavior of the BAC as a function of time ?
(Section 2.1 p79)
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A. Functions in general
Summary
- Definition
input x, output y
- 3 representations :
table
equation y=f(x)
graph in rectangular coordinate system
Extra: Handbook - Problems 2.1: Ex 17, 48, 50
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B. Linear Functions
1. equation
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B. Linear functions
Example Taxidriver
y = 5 + 2x
FIXED PART + VARIABLE PART
FIXED PART + MULTIPLE OF
INDEPENDENT VARIABLE
FIXED PART + PART PROPORTIONAL
TO THE INDEPENDENT VARIABLE
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B. Linear functions
Example Taxidriver
• Examples: cost of a ride with company B, C?
B  base price: 4.5 euro, price per km: 2.1 euro
C  base price 8 euro, price per km: 0.5 euro
y = 4.50 + 2.10x; y = 8 + 0.5x;
• In general: y = base price + price per km  x
y=
b
+
m x
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B. Linear functions
Equation
A function f is a linear function if and only if
f(x) can be written in the form
f(x)=y=mx + b
where m, b are constants.
Caution: m and b FIXED: parameters
x and y: VARIABLES!
(Section 3.1 p138)
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B. Linear functions
Applications
• Cost y to purchase a car of 20 000 Euro and
drive it for x km, if the costs amount to 0.8
Euro per km?
y = 20 000 + 0.8x hence … y = mx + b!
• Production cost c to produce q units, if the
fixed cost is 3 and the production cost is 0.2
per unit?
c = 3 + 0.2q hence y = mx + b!
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B. Linear functions
Applications
•
The demand q of a product depends on the
price p and vice versa. For a local pizza
parlor the function is given by
p=26-q/40
Note: The function p(q) is called the
demand function by economists
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B. Linear functions
Exersises
Rachel has saved $7250 for college expenses.
She plans to spend $600 a month from this account.
Write an equation to represent the situation.
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B. Linear functions
Exersises
For a local pizza parlor the weekly demand function
Is given by p=26-q/40.
(a) What will be the revenue for the pizza parlor
if 400 pizza’s are ordered ?
(b) Express the revenue as a function of the
demand q.
!! Not all functions are first degree functions
Note: Demand functions are not always linear !
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B. Linear Functions
1. Equation
2. Graph
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B. Linear functions
Example Taxidriver
y = 2x + 5
y
The graph of a linear
function with equation
y=mx +b is
- a STRAIGHT LINE
14
2
x
1
5
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B. Linear Functions
1. Equation
2. Graph
3. Significance parameters b, m
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B. Linear functions
Example Taxidriver
A: y = 2x + 5
B: y = 4.5x + 2.1
C: y = 0.5x + 8
What’s the effect of
the different values
for m ? For b ?
y
14
2
x
1
5
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B. Linear functions
Significance of the parameter b
• Taxi company A: y = 2x + 5.
Here b = 5: the base price. 14
y
• Numerically:
b can be considered as the
VALUE OF y WHEN x = 0.
• graphically:
b shows where the graph cuts2
the Y-axis: Y-INTERCEPT
x
1
5
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B. Linear functions
Significance of the parameter m
• Taxi company A: y = 2x + 5, m = 2: the price per
km.
• Numerically: m is CHANGE OF y WHEN x IS
INCREASED BY 1
INPUT
OUTPUT
x
y
3
11
x = 1
y = 2
4
13
m is the RATE OF CHANGE of
the linear function
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B. Linear functions
Significance of the parameter m
• Graphically:
if x is increased by 1 unit,
y is increased by m units
m is the SLOPE of the straight line
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B. Linear functions
Significance of the parameter m
• Taxi company A: y = 2x + 5, m = 2: the price per
km.
• If x is increased by e.g. 3 (the ride is 3 km longer),
y will be increased by 2  3 = 6 (we have to pay 6
Euro more).
x = 3
• Always:
INPUT
x
3
6
OUTPUT
y
11
17
y = 3x2=6
y = mx
(INCREASE FORMULA)
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B. Linear functions
Significance of the parameter m
if x is increased by x units, y is increased by m x
units
Increase formula:
y  m  x
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B. Linear functions
Significance of the parameters b and m
The graph of a linear function with equation
y=mx +b is
- a STRAIGHT LINE
- with y-intercept b
- and slope m
The equation y=mx +b is called the slopeintercept form of the line with slope m and
intercept b. It is also called an explicit
equation of the line.
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B. Linear functions
Exercises
1. The cost c in terms of the quantity q
produced of a good is given by
c = 200 + 15 q.
• Give a formula for the change of cost Δc.
• Use this formula to determine how the
cost changes when the production of the
good is increased by 12 units.
• Use this formula to determine how the
cost changes when the production of the
good is decreased by 2 units.
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B. Linear functions
Supplementary exercises
•
•
Exercise 1
Exercise 2 A, B, D (only the indicated points
are to be used!)
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B. Linear functions
Exercises
y
B
Exercise 2
F
(3,9)
(0,7)
(6,6)
(0,3)
A
C
(2,0)
x
D
E
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B. Linear functions
Slope of the line m
Consider again supplementary Exercise 2
- Compare the slopes of lines A and D
- What is the slope of line C ?
- Compare the slopes of line A and B
- Compare the slopes of lines D and E
(Section 3.1)
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B. Linear functions
Slope of the line m
(Section 3.1 p128-129)
Sign of m determines whether the linear function is
- increasing / constant(!!) / decreasing
y
y
2
y
2
2
x
-2
2
m<0
2
-2
2
m>0
m=0
-2
x
x
-2
-2
-2
- Note: what about a vertical line ?
(Section 3.1 p131Example 6)
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B. Linear functions
Slope of the line m
Size of m determines how steep the line is
Note: the slope and thus the steepness of the line
depends on the scale of the axes
(Section 3.1 p128-129)
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B. Linear functions
Parallel lines
(Section 3.1 p128-129)
Parallel lines have the same slope
Perpendicular lines
(Section 3.1 p133-134)
Two lines with slopes m1 and m2 are perpendicular
to each other if and only if
1
m1 
m2
Note: any horizontal line and any vertical line are
perpendicular to each other
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B. Linear functions
Slope of the line m
Remember: y = mx (INCREASE FORMULA).
Therefore:
y
m
x
(Section 3.1 p128)
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B. Linear Functions
1. Equation
2. Graph
3. Significance parameters b, m
4. Determining a line based on the slope
and a point / two points
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B. Linear functions
Slope of the line m
Slope of a straight
line given by two
points:
y2  y1 y
vertical distance
m


horizontal distance x2  x1 x
(Section 3.1 p128)
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B. Linear functions
Exercises
John purchased a new car in 2001 for $32000.
In 2004, he sold it to a friend for $26000.
You may assume that the price is a linear function
of time.
Find and interpret the slope.
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B. Linear functions
Equation of lines
A straight line through a given point (x0, y0) and
with a given slope m satisfies the equation:
y  y0  m  x  x0 
This equation y  y0  m  x  x0  is called the pointslope form of the line
Remember:
The equation y=mx+b is called the slope-intercept
form of the line
(Section 3.1 p129-131)
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B. Linear functions
Exercises
John purchased a new car in 2001 for $32000. In
2004, he sold it to a friend for $26000. Find the
equation that expresses the price as a function of
time. You may assume that the price is a linear
function of time.
Supplementary exercises
•
•
Exercise 3
Exercise 4
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B. Linear Functions
1. Equation
2. Graph
3. Significance parameters b, m
4. Determining a line based on the slope
and a point / two points
5. Implicit equation
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B. Linear functions
Equation of lines
Note that e.g. the vertical line with equation x=2 can
not be written in the slope-intercept form nor in the
slope-point form
The equation of a straight line can always be written
using the general linear form Ax+By+C=0 (A and B
not both 0).
This is also called an implicit equation.
(Section 3.1 p129-131)
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B. Linear functions
Equation of lines
Remember :
Point-slope form
Slope intercept form y=m x + b
General linear from Ax + By + C = 0
note: vertical line: x=a
horizontal line: y=b
(Section 3.1 p129-131)
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B. Linear functions
Exercises
Find an equation of the line that has slope 2 and
passes through (1, -3) using the
-
Point-slope form
Slope-intercept form
-
General linear form
Supplementary exercises:
• Exercise 5
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B. Linear functions
Exercises
Make a graph of the the straight line, given by
the equation 80x+250y=10000
Tip:
- Write down the explicit equation
- Find two points that satisfy the equation
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B. Linear functions
Equation of lines
Sometimes, the general linear form arises naturally
Example: Invest a capital of 10 000 Euro in a certain
share and a certain bond
share: 80 Euro per unit
bond: 250 Euro per unit
How much of each is possible with the given capital?
Let qS be the number of units of the share and qB the
number of units of the bond.
We must have: 80qS + 250qB = 10 000
(Section 3.1 p129-131)
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B. Linear functions
Equation of lines
Note: An equation of the form Ax+By=D (A and B not
both 0) is also called an implicit equation and can
always be written using the general linear form as
Ax+By-D=0
(Section 3.1 p129-131)
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B. Linear functions
Equation of lines
Example: How many shares and bonds are possible if
80qS + 250qB = 10 000 ?
• There are infinitely many possibilities for qS en qB
e.g.: qS = 0, qB = 40;
qS = 125, qB = 0;
qS = 100, qB = 8
etc. …
• Not all combinations are possible!
(Section 3.1 p129-131)
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B. Linear functions
Equation of lines
Example: Connection, between qS and qB:
• 80qS + 250qB = 10 000: IMPLICIT equation
(general linear form: 80qS + 250qB - 10 000=0)
form: Ax + by + c = 0
• qB = 40  0.32qS: EXPLICIT equation
dependent variable isolated in left hand side, right
hand side contains only the independent variable,
form y = mx + q
• qS = 125  3.125qB: EXPLICIT equation
(Section 3.1 p129-131)
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B. Linear functions
Equation of lines
How to make a graph corresponding to an implicit
linear equation ?  straight line
- Strategy 1: make equation explicit first
- Strategy 2: find two points satisfying the equation
Example:
(Section 3.1 p129-131)
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B. Linear functions
Summary
- equation: first degree function y=mx+b (intercept-slope)
- graph : straight line
- Significance parameters
b: y-value for x=0, y-intercept
m: rate of change, slope
slope: in-/decreasing – steepness – parallel
special cases vertical/horizontal line
- setting up equations of straight line based on
- slope and point (point slope form)
- two points (slope from 2 points + point slope form)
- implicit linear function : (generalized linear equation)
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B. Linear functions
Summary
Extra: Handbook
Problems 2.1: Ex 45
Problems 3.1: Ex 1, 9, 13, 35, 38, 41, 50, 51, 61, 63, 64,
67, 69, 71, 72
Problems 3.2: Ex 1,4, 26, 28, 30
Problems 0.7: Ex 93
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C. Linear equations
1. Linear equations
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C. Linear equations
Example Taxidriver
y = 5 + 2x
When a client has to pay 10 euro, for how
many kilometres did he take a ride ?
Solution:
Output y=10 is given
Input x is unknown: 5 + 2 x = 10
x=2.5  5 + 2 (2.5) = 10
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C. Linear equations
Definition
A LINEAR EQUATION in the variable x is an
equation that is equivalent to one that can
be written in the form m x + b=0 , where
m and b are constants and m ≠ 0.
x is called the unknown
(Section 0.7 p28-29)
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C. Linear equations
(Section 0.7 – Example 3 p28)
Exercises
Solve 5x – 6 = 3x
• We begin by getting the terms involving x
on one side and the constant on the other
2x=6
• Then we divide by an appropriate constant
x=3
Supplementary exercises
Exercise 6
Exercise 7
63
C. Linear equations
Exercises
(Section 1.1 – Example 3,4,5 p45-46)
Supplementary exercises
Exercise 8
64
C. Linear equations
Graphical interpretation
equation: 2 x  5  0
solution: 2.5
function with equation y=2x-5
2.5 is called a zero of the function
65
C. Linear equations
1. Linear equations
2. System of linear equations
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C. Linear equations
Example: Factory
Suppose that the manager of a factory is setting up a
production schedule for two models of a new product.
Model A requires 4 resistors and 9 transistors. Model
B requires 5 resistors and 14 transistors. From its
suppliers the factory gets 335 resistors and 850
transistors. How many of each model should the
manager plan to make each day so that all the
resistors and transistors are used ?
Solution: x=number of A; y=number of B
335 resistors: 4x+5y =335
850 transistors: 9x+14y=850
4x+5y =335
(Section 3.4 p148-149)
9x+14y=850
67
C. Linear equations
Definition: system
We call
ax+by =c
dx+ey =f
a system of linear equations.
The problem is to find values of x and y for
which both equations are true simultaneously.
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C. Linear equations
Solution : system
Example : factory
4 x + 5 y = 335 iff 36 x + 45 y = 3015
9 x + 14 y = 850 iff -36 x - 56 y = -3400
+
-11 y = -385
y = 35
From 4 x + 5 y = 335 and y = 35 it follows that x=40
This strategy to solve a linear set of equations is
called the elimination-by-addition method
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C. Linear equations
Solution : system
Example : factory
4 x + 5 y = 335 iff y = 67 – 4/5 x
Substituting y in 9 x + 14 y = 850 leads to
9 x +14 (67 - 4/5 x ) = 850
-11/5 x = -88
x = 40
From 4 x + 5 y = 335 and x = 40 it follows that y=35
This strategy to solve a linear set of equations is
called the elimination-by-substitution method
70
C. Linear equations
Solution : system
Example : factory
Note that you can quickly check your solution.
Indeed, your solution should satisfy both equations.
I.e.
4 (40) + 5 (35) = 335
9 (40) + 14 (35) = 850
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C. Linear equations
Exercises : system
1.Solve the following system of equations using the
elimination-by-addition method (Section 3.4 p150-151)
3x-4y=13
3y+2x=3
2.Solve the following system of equations using the
elimination-by-substitution method (Section 3.4 p152)
x+2y-8=0
2x+4y+4=0
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C. Linear equations
Graphical interpretation of system
System of equations:
y=-0.5x+4
2y+2=4x
Solution x=2, y=3
Two corresponding functions
y=-0.5x+4
y=2x-1
2 is x-coordinate of intersection
point
Point (2,3) is intersection point
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C. Linear equations
Supplementary exercises
• Exercise 9
• Exercise 10
Extra: Handbook
Problems 0.7: Ex 95
Problems 1.2: Ex 5, 11, 21
Problems 1.3: Ex 1
Problems 3.4: Ex 1, 5, 9, 19, 28, 33
74
C. Linear equations
1. Linear equations
2. System of linear equations
3. Linear inequalities
75
C. Linear equations
Supplementary exercises
• Exercise 11
Definition : inequality
(Section 1.2 p51-54)
A LINEAR INEQUALITY in the unknown x is
an inequality that can be written in the
form ax+b<0 or ax+b≤0 or ax+b>0 or
ax+b≥0, with a and b numbers (a ≠ 0).
76
C. Linear equations
Supplementary exercises
• Exercise 12
• Exercise 13
77
C. Linear equations
Graphical interpretation inequality
inequality:  0.5 x  4  2 x  1
inequality: 2 x  5  0
solution: x>2.5
solution: x<2
function with equation
two corresponding functions
y=2x-5
for x>2.5 graph is above
horizontal axis
for x<2 green graph is
higher than blue one