[PPT] - Linear functions A. Functions in general A. Functions in general PowerPoint Presentation

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1 Linear functions

Handbook: E. Haeussler, R. Paul, R. Wood (2011). Introductory Mathematical Analysis for business, economics and life and social

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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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Example Taxidriver

Price of a 7 km ride? What does a taxi ride cost me with company A?

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Example Taxidriver

Price of an x km ride? What does a taxi ride cost me with company A?

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Definition

: VARIABLES

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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Definition

y is FUNCTION of x,

(Handbook: Section 2.1 p82)

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Three representations

First way: Most concrete form! Through a TABLE, e.g. for y = 2x + 5: But: limited number of values no overall picture x y 5 1 7 2 9 … …

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Three representations

Second way: Most concentrated form! Through the EQUATION, e.g. y = 2x + 5.

formula y = 2x + 5: EQUATION OF THE FUNCTION 11

Three representations

Third way: Most visual form!

Through the GRAPH rectangular coordinate system: x-coordinate, y-coordinate

y

7 6 5 4 3 2 1

(Handbook: Section 2.5 p99) x y 5 1 7

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Three representations

Third way: Most visual form!

Through the GRAPH e.g. for y = 2x + 5: STRAIGHT LINE!

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Exercises The demand q of a product depends on the price p. For a local pizza parlor some weekly demands and prices are given Remark: this table is called a demand schedule (a) What is the input variable? What is the output variable ? (b) Indicate the points in the table on a graph

p q 10 640 12 560 14 480 (Handbook: Section 2.1 p85 – example 5)

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Exercises Suppose a 180-pound man drinks four beers in quick succession. The graph shows the blood alcohol concentration (BAC) as a function

(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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Summary

input x, output y

table equation y=f(x) graph in rectangular coordinate system Extra: Handbook - Problems 2.1: Ex 17, 48, 50 16

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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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Example Taxidriver

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;

y = b + m x 19

Equation Caution: m and b FIXED: parameters x and y: VARIABLES!

(Section 3.1 p138)

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. 20

Applications

drive it for x km, if the costs amount to 0.8 Euro per km? y = 20 000 + 0.8x hence … y = mx + b!

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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Applications

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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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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Exersises

!! Not all functions are first degree functions

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. Note: Demand functions are not always linear !

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Example Taxidriver y = 2x + 5 The graph of a linear function with equation y=mx +b is

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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 ? 28

Significance of the parameter b

Here b = 5: the base price.

b can be considered as the VALUE OF y WHEN x = 0.

b shows where the graph cuts the Y-axis: Y-INTERCEPT

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Significance of the parameter m

km.

INCREASED BY 1 INPUT OUTPUT x y 3 11 4 13 ∆x = 1 ∆y = 2

m is the RATE OF CHANGE of the linear function 30

Significance of the parameter m

if x is increased by 1 unit, y is increased by m units

m is the SLOPE of the straight line 31

INPUT OUTPUT x y 3 11 6 17

Significance of the parameter m

km.

y will be increased by 2 × 3 = 6 (we have to pay 6 Euro more).

∆y = m∆x (INCREASE FORMULA)

∆x = 3 ∆y = 3x2=6

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Significance of the parameter m

if x is increased by ∆x units, y is increased by m ∆x units

Increase formula:

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Significance of the parameters b and m The graph of a linear function with equation y=mx +b is

The equation y=mx +b is called the slope- intercept form of the line with slope m and intercept b. It is also called an explicit equation of the line. 34

Exercises

produced of a good is given by c = 200 + 15 q.

cost changes when the production of the good is increased by 12 units.

cost changes when the production of the good is decreased by 2 units. 35

Supplementary exercises

are to be used!)

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Exercises

Exercise 2

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Slope of the line m

Consider again supplementary Exercise 2

(Section 3.1)

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Slope of the line m

Sign of m determines whether the linear function is

(Section 3.1 p128-129) (Section 3.1 p131- Example 6)

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Slope of the line m

Note: the slope and thus the steepness of the line depends on the scale of the axes Size of m determines how steep the line is (Section 3.1 p128-129)

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Parallel lines Perpendicular lines

Two lines with slopes m1 and m2 are perpendicular to each other if and only if Note: any horizontal line and any vertical line are perpendicular to each other Parallel lines have the same slope (Section 3.1 p128-129) (Section 3.1 p133-134)

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Slope of the line m

Remember: ∆y = m∆x (INCREASE FORMULA).

Therefore:

(Section 3.1 p128)

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and a point / two points

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Slope of the line m

Slope of a straight line given by two points: (Section 3.1 p128)

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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

Find and interpret the slope.

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Equation of lines

A straight line through a given point (x0, y0) and with a given slope m satisfies the equation: This equation is called the point- slope form of the line (Section 3.1 p129-131) The equation y=mx+b is called the slope-intercept form of the line Remember:

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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

function of time.

Supplementary exercises

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and a point / two points

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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 (Section 3.1 p129-131) 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.

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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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Exercises

Find an equation of the line that has slope 2 and passes through (1, -3) using the

Supplementary exercises:

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Exercises

Make a graph of the the straight line, given by the equation 80x+250y=10000 Tip:

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Equation of lines

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) Sometimes, the general linear form arises naturally

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Equation of lines

(Section 3.1 p129-131) 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

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Equation of lines

Example: How many shares and bonds are possible if 80qS + 250qB = 10 000 ?

e.g.: qS = 0, qB = 40; qS = 125, qB = 0; qS = 100, qB = 8

(Section 3.1 p129-131)

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Equation of lines

Example: Connection, between qS and qB:

(general linear form: 80qS + 250qB - 10 000=0) form: Ax + by + c = 0

dependent variable isolated in left hand side, right hand side contains only the independent variable, form y = mx + q

(Section 3.1 p129-131)

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Equation of lines

How to make a graph corresponding to an implicit linear equation ? straight line

Example: (Section 3.1 p129-131)

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b: y-value for x=0, y-intercept m: rate of change, slope slope: in-/decreasing – steepness – parallel special cases vertical/horizontal line

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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 59

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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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Definition x is called the unknown

(Section 0.7 p28-29)

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. 62

Exercises Solve 5x – 6 = 3x

2x=6

x=3

(Section 0.7 – Example 3 p28)

Exercise 6 Exercise 7

Supplementary exercises

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Exercises

(Section 1.1 – Example 3,4,5 p45-46)

Exercise 8

Supplementary exercises

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Graphical interpretation equation: solution: 2.5 function with equation y=2x-5 2.5 is called a zero of the function

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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

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 (Section 3.4 p148-149)

4x+5y =335 9x+14y=850 67

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. 68

Solution : system 4 x + 5 y = 335 iff 36 x + 45 y = 3015 9 x + 14 y = 850 iff -36 x - 56 y = -3400

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 Example : factory

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Solution : system 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

x = 40 From 4 x + 5 y = 335 and x = 40 it follows that y=35

Example : factory This strategy to solve a linear set of equations is called the elimination-by-substitution method

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Solution : system

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 Example : factory

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Exercises : system

1.Solve the following system of equations using the elimination-by-addition method 2.Solve the following system of equations using the elimination-by-substitution method (Section 3.4 p150-151)

3x-4y=13 3y+2x=3

(Section 3.4 p152)

x+2y-8=0 2x+4y+4=0 72

Graphical interpretation of system System of equations:

Solution x=2, y=3 Two corresponding functions y=-0.5x+4 y=2x-1

2 is x-coordinate of intersection point y=-0.5x+4 2y+2=4x

Point (2,3) is intersection point

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Supplementary exercises

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

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Supplementary exercises Definition : inequality 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).

(Section 1.2 p51-54)

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Supplementary exercises

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