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Class Notes
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Canada
(511,260)

University of Toronto Scarborough
(31,651)

Mathematics
(870)

MATA30H3
(41)

Sophie Chrysostomou
(17)

Lecture

Department

Mathematics

Course Code

MATA30H3

Professor

Sophie Chrysostomou

Description

20140910 208 PMMore High School Material Reviewfm fm500WebAssignMATA30F2014 Fall 2014More High School Material Review LabInstructor Sophie ChrysostomouCurrent Score195DueSaturday September 20 2014 1159 PM EDT13 pointsSCalc7 QP1N001ReadingTopic 1 Definition and Representation of Functions0 When will I need this in CalculusThe concept of function is fundamental in calculus Calculus involves the study of change and investigating how one quantity changes in relation toanother For example we can model how fast a disease is spreading by looking at the relationship of new cases of the disease over time A thoroughunderstanding of function notation graphs of functions and how to work with functions is essential to understanding calculus1 Definitions of independent and dependent variableWe define a group or collection of objects as a set The set could refer to a set of people for example all the students at a particular college form a set ofcollege students a set of buildings the set of campus buildings on a particular college campus or a set of numbers the set of ninedigit social securitynumbers In calculus the most common set we will work with is the set of real numbers We often call sets by letter names like A or B or by somegenerally accepted symbol likefor the set of real numbersIn mathematics we use lowercase letters like x y and t to denote variables A variable is defined as a member of some set or collection of elementswhere the particular individual element referenced is unknownMany experiences that occur in everyday life involve a correspondence or relationship between elements of two sets For example we associate with eachperson his or her height That association can be thought of as a function The person is considered to be the independent variable and the personsheight is the dependent variable In mathematics a correspondence between elements in a set of independent variables and elements in a set ofdependent variables is sometimes called a relation If the relation has an additional property that each value of the independent variable corresponds toexactly one value of the dependent variable then the relation is a function That is the value of the dependent variable depends on the value of theindependent variable with which it is associated We use the word function in several ways we say that the correspondence is a function and we also saythat it defines the dependent variable as a function of the independent variableThe set of independent variables for which the function is defined is called the domain of the function The range of the function is the set of all possiblevalues of the dependent variable Domain and range of a function will be explored more fully in Topic 4Example 1 Determine if a relationship is a functionFor each of the following descriptions determine if the relationship described defines a function or not If it defines a function identify the domain andrange of the functiona Each college student at a particular college is assigned an ID numberb A particular level of taxable income is assigned an amount of US income taxSolutiona Since each student is given a unique ID number this description defines ID number as a function of the student The domain of this function is the setof all students at the college The range depends on how the ID numbers are defined Lets assume that the ID numbers for this particular college are sixdigit integers Then the range is the set of all sixdigit integers that have been assigned to students Note that probably not every sixdigit integer will bean ID number for some studentb The amount of income tax an individual pays depends on the level of taxable income of the individual but it depends on other factors as well Forexample the number of dependents the individual has will change the amount of income tax Thus we can find an example where individuals with thesame level of income pay different amounts of income tax So this relationship is not a function2 Function as a ruleA function is a rule that assigns to each element of the domain which is the set of values of the independent variable exactly one element in the rangeor set of dependent variables This means that each element in the domain has exactly one element in the range associated with it Sometimes theelements of the domain are called input values and the elements of the range are called output valuesThe rule that defines a function can be given in a variety of forms Specifically a function can be given in words as the function in Example 1 part awhich assigned to each college student his or her ID number It can also be given by an arrow diagram as a set of ordered pairs as an algebraicexpression and using a graph Well begin to look at these ways of expressing functions here and continue the discussion in Topic 2 Topic 3 and Topic4The rule that defines a function can be given as an arrow diagram showing an arrow from each value of the independent variable to the associated valueof the dependent variablePage 1 of 100httpwwwwebassignnetwebStudentAssignmentResponseslastdep991059820140910 208 PMMore High School Material ReviewExample 2 Determine if an arrow diagram depicts a functionDetermine if each of the arrow diagrams depicts a function from the set of independent variables to the set of dependent variablesabSolutiona Because the table shows that 66 inches corresponds to a weight of 175 and to a weight of 150 this diagram does not depict a function There aretwo different people represented here both with height 66 inchesb Each year corresponds to only one value of the dependent variable so this arrow diagram defines number of influenza deaths in the county as afunction of year3 Function as a set of ordered pairs or a tableThe rule that defines a function can be given as a set of ordered pairs where each first element in the pair represents a value of the independentvariable and the second element in the pair represents the associated dependent variable Remember the basic rule for functions applies each firstelement in the pair must be associated with only one second element So if a first element appears with two distinct second elements the set of orderedpairs cannot be a functionSince a function relates each value of the independent variable to a value of the dependent variable it is natural to think of the function as a table ofvalues in which each value of the independent variable is next to its corresponding value of the dependent variable The table consists of two columns ortwo rows The first column row contains each value of the independent variable and the second column row contains the corresponding value of thedependent variable lined up with its associated independent variable valueNot every 2column tworow table describes a function For a table to describe a function there must be only one secondcolumn value associated witheach value in the first column The following are two examples the first describes a function while the second does notA functionNot a functionxyxy1 21 20 10 11 21 21 21 0The first table gives y as a function of x Note that 1 is repeated in the first column but the corresponding value is the same in both cases So we canerase the last row and give the table belowxy1 20 11 2Example 3 Determine if a set of ordered pairs represents a functionDetermine if each set of ordered pairs represents a function Note that we use set notation with curly brackets around the elements of the set todenote the set and a capital letter to identify the setPage 2 of 100httpwwwwebassignnetwebStudentAssignmentResponseslastdep991059820140910 208 PMMore High School Material Reviewa b cSolutiona Because the two pairs 1 5 and 1 9 appear as elements of the set A we have two different second elements corresponding to the element 1 ofthe domain So A is not a functionb The set B is a function Each value in the domain that is each first element in the pairs has exactly one second element associated with it Its still afunction even though several of the dependent variable values are the samec The set C is also a functionExample 4 Analyze a table of dataA childs parents recorded the childs height once per month during the first six months since the child was born The following table gives themeasurementsAge in monthsHeight in inches1212233235 4245 52495626a Give the independent variable and the dependent variableb Give the function value when the value of the independent variable is 4c Give the height of the child when the child was 3 months oldd Give the age of the child when the child was 2495 inches tallSolutiona The independent variable is the childs age in months and the dependent variable is his or her height in inchesb The function value is 245 inchesc The height of the child which is also the function value is 235 inchesd When the function value is 2495 inches the value of the independent variable is 5 so the child was 5 months old when his or her height was 2495inches4 Graph of a functionA function can be given as a graph of the set of ordered pairs that comprise the function drawn on the Cartesian or rectangular coordinate system alsocalled the Cartesian plane This coordinate system consists of two number lines perpendicular to each other The horizontal number line is often called thexaxis and contains values of the independent variable while the vertical number line often referred to as the yaxis shows values of the dependentvariable The intersection point of the two axes is the origin Each point in the plane corresponds to an ordered pair of numbers denoted x y wherethe first coordinate of the pair gives the value of the independent variable and the second coordinate gives the value of the corresponding dependentvariable Points to the right of the origin have positive xcoordinate and points to the left of the origin have negative xcoordinate Points above the originhave positive ycoordinate while points below the origin have negative ycoordinate The graph of a function shows all points representing all orderedpairs of the functionExample 5 Graph a functiona Identify the coordinates of each of the given pointshttpwwwwebassignnetwebStudentAssignmentResponseslastdep9910598Page 3 of 100

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