Contemporary Math Preview

≠∂ %

√Contemporary math I UsIng mapLe or tI-89

Otto Wilke

Contemporary math I UsIng mapLe or tI-89

Otto WilkeTexas State Technical College Waco

© 2007 Texas State Technical College Waco

ISBN 978-1-934302-07-1

All rights reserved, including the right to reproduce this book or any portion thereof in any form. Requests for such permissions should be addressed to:

TSTC PublishingTexas State Technical College Waco3801 Campus DriveWaco, TX 76705

http://publishing.tstc.edu/

Publisher: Mark LongGraphics specialist: Grace ArsiagaEditor: Todd GlasscockPrinting production: Bill EvridgeCover design: Sheri McGee

Manufactured in the United States of America

First edition

TabLe of ContentsIntroduction to Maple ..................................................................................................... 1 GettingStarted ...................................................................................................................1 OperationSymbols ...........................................................................................................1

Algebraic Operations ..................................................................................................... 3 AlgebraicOperations:TestBankProblems ...................................................................7

Number Systems .......................................................................................................... 10 NumberSystems:TestBankProblems ........................................................................13

Radicals......................................................................................................................... 17 Radicals:TestBankProblems ........................................................................................19 Sets and Lists ............................................................................................................... 23 SetsandLists:TestBankProblems ...............................................................................26

Sets of Numbers ........................................................................................................... 28 SetsofNumbers:TestBankProblems ..........................................................................30

Logic ............................................................................................................................. 33 Induction ..........................................................................................................................33 MathematicalInduction .................................................................................................33 Deduction .........................................................................................................................34 ScientificMethod .............................................................................................................41 LogicProblems ................................................................................................................42 Factoring ....................................................................................................................... 44 Factoring:TestBankProblems ......................................................................................46

Fractions ....................................................................................................................... 49 Fractions:TestBankProblems ......................................................................................52

Complex Numbers ........................................................................................................ 54 ComplexNumbers:TestBankProblems .....................................................................56

Coordinate Systems..................................................................................................... 59

Exponential Format ...................................................................................................... 65

Units and Dimensions .................................................................................................. 69 1.Definitions ....................................................................................................................69 2. The SI Units ..................................................................................................................69 3.AlgebraofUnitsandDimensions ............................................................................77 4.ConversionFactors .....................................................................................................78 UnitsandDimensions:TestBankProblems ...............................................................80

Linear Equations .......................................................................................................... 83 LinearEquations:TestBankProblems .........................................................................89

Linear Inequalities ........................................................................................................ 96 LinearInequalities:TestBankProblems ....................................................................100

Quadratic Equations .................................................................................................. 103 QuadraticEquations:TestBankProblems ................................................................104

Radical Equations ...................................................................................................... 107 RadicalEquations:TestBankProblems .....................................................................107

Geometric Formulae .................................................................................................. 109 Triangles .........................................................................................................................110 Quadrilaterals ................................................................................................................112 GeometricFormulae:TestBankProblems ................................................................119

Functions .................................................................................................................... 123 CommonFunctions.......................................................................................................131 TransformationofFunctions .......................................................................................137 Functions:TestBankProblems ...................................................................................137 Exponential and Logarithmic Equations .................................................................. 139 ExponentialandLogarithmicFunctions:TestBankProblems ...............................145

Matrices ....................................................................................................................... 149 Matrices:TestBankProblems ......................................................................................155

Systems of Equations ................................................................................................ 159 MatrixEquations ...........................................................................................................163 DiophantineEquations .................................................................................................165 SimultaneousEquations:TestBankProblems ..........................................................166

Systems of Inequalities ............................................................................................. 170 SimultaneousInequalities:TestBankProblems .......................................................175

Sequences and Series ............................................................................................... 178 ArithmeticSequencesandSeries ................................................................................180 GeometricSequencesandSeries .................................................................................181 FibonacciSequence .......................................................................................................183 Sequences:TestBankProblems ..................................................................................184

Probability ................................................................................................................... 187 Probability:TestBankProblems .................................................................................191

Descriptive Statistics and Regression ..................................................................... 194 StatisticsandRegression:TestBankProblems .........................................................196

Percentage Equations................................................................................................ 201 Percentages:TestBankProblems ................................................................................203

Proportionality ............................................................................................................ 204 Proportionality:TestBankProblems ..........................................................................207

Graphs and Tables ..................................................................................................... 209 GraphsandTables:TestBankProblems ....................................................................213

Trigonometric Functions ........................................................................................... 215 Triangles...................................................................................................................... 219

Conic Sections ........................................................................................................... 220 Conics:TestBankProblems .........................................................................................231

Apendix A: Conversion Factors ................................................................................ 235

About TSTC Publishing ............................................................................................. 255

Contemporary Math �

Introduction to Maple

Getting Started

Most of the Maple commands in this text correspond to the Classic Worksheet syntax. To open the Classic Worksheet go to Start—>Programs—>Maple 11—>Classic Worksheet. The prompt is the > symbol. The cursor is a vertical line, |. The cursor may be positioned by moving the arrow keys or by clicking the left mouse button. The cursor may not follow page up or page down commands or scroll up or down.

End all commands with a semicolon or a colon and then press [Enter] to execute the commands. When a command ends with a colon, execution causes the calculations to be done, but the results do not appear on the screen. To delay execution until the last of a group of commands, press [Shift-Enter] to move the cursor to the next line.

More than one command may be typed on a line, but each command must end in a colon or semicolon. Only help commands do not require a colon or semicolon.

Help commands begin with a question mark. For a general help screen, type the ? symbol, and then press [Enter]. For help with a particular command, type ? followed by the name of the command, for example, ?factor. A help menu also appears on the title bar.

>? >?factor

Operation Symbols

Symbol Operation + Addition - Subtraction, negation * Multiplication ^ Exponentiation &* Matrix multiplication, non-commutative multiplication = Equals := Assignment % Last answer %% Second to last answer

� Contemporary Math

EXAMPLES >3+2;

>5-(-8);#Two adjacent operators must be separated by parentheses.

>5^2;

>3*5/5^2;#Exponentiation first; multiplication & division at the same #time from left to right.

>#Anything following pound sign on a line is an unexecuted comment.


Algebraic OperationsAn operation is an action. In algebra operations are performed with numbers. Algebraic operations include addition, subtraction, multiplication, and exponentiation. There are other operations in mathematics. Some will be discussed later.

The four operations discussed here are binary operations, i.e. involving two numbers. Addition combines two numbers to yield a unique third number called the sum. Operators are symbols that indicate the operation to be performed. In Maple they are:

+ for addition

- for subtraction

* for multiplication

/ for division

^ for exponentiation

Maple requires all multiplication be explicitly indicated by the * in input. Notice the required * between the sets of parentheses.

> (3*x*y+4*z)*(2*x-1);

( ) + 3 x y 4 z ( ) − 2 x 1> expand(%);

− + − 6 x2 y 3 x y 8 z x 4 z

The Maple math output does not print the *. 3*x*y is indicated by the presence of a space between the characters.

> cat;c*a*t;

catc a t

Above, cat is the name of a single number, but c a t is the product of three numbers. The numbers involved in operations are given names:

addend+addend=sum

minuend-subtrahend=difference


factor*factor=product

dividend/divisor=quotient=numerator/denominator

base^exponent=result of exponentiation

Now may be a good time to define some words. A variable is a letter, symbol or word that represents a number. A numeral in a particular position relative to the decimal point represents a fixed number. The number represented by a variable differs (varies) from problem to problem. Below is a list of example variables, separated by commas.

> [x,y,theta,cat,dog,angle[1]];

[ ], , , , ,x y θ cat dog angle1

> > [2,x,2*x,3*sqrt(5),4*x^2*sqrt(y)];

[ ], , , ,2 x 2 x 3 5( )/1 2

4 x2 y( )/1 2

An expression is a single term or some terms connected with algebraic operators. Below is a list of expressions, separated by commas.

> [2,2*x,2*x+3*y,sqrt(5*x)/[4*y]-3.5];

, , ,2 2 x + 2 x 3 y −

5( )/1 2

x( )/1 2

[ ]4 y 3.5

An equation is an expression which contains an equal sign, =.

> 3*x+2*y=6;

= + 3 x 2 y 6

An inequality is an expression that contains an inequality symbol, < (less than), > (greater than), <> (not equal to), <= (less than or equal to), >= (greater than or equal to).

> 3*x+2*y<6;

< + 3 x 2 y 6Back to operations... Algebra, like a game, consists of a set of rules that govern what the outcome of operations must be. To play any game effectively you must know the rules. Four rules govern addition.

Commutative Rule The order of addends does not change the sum.


a+b=b+a 3+2=2+3=5

Associative Rule When adding three (or more) numbers, add any two, then add that sum to a third number.

(a+b)+c=a+(b+c) 2+3+4=(2+3)+4=5+4=2+(3+4)=2+7=9

Additive Identity The sum of zero and a number is the number.

a+0=a 6+0=6

Additive Inverse The sum of a number and its negative is zero. The negative of a number is called its additive inverse.

a+(-a)=0 3+(-3)=0

Rules for multiplication are similar, but include one additional rule, the distributive rule. In Maple, brackets can sometimes be used to prevent Maple from automatically performing the operation and showing only the result.

> commutative_rule:=a*b=b*a;[3]*[2]=[2]*[3];associative_rule:=[a*b]*c=a*[b*c];[3*x]*[y]=[3]*[x*y];[3]*[x*y]=3*x*y;;distributive_rule:=a*(b+c)=expand(a*(b+c));[3]*[x+y]=3*x+3*y;multiplicative_identity:=[a]*[1]=a;[6]*[1]=6;multiplicative_inverse:=[a]*[1/a]=1;[6.1]*[[1]/[6.1]]=1;

:= commutative_rule = a b a b = [ ]3 [ ]2 [ ]3 [ ]2

:= associative_rule = [ ]a b c a [ ]b c = [ ]3 x [ ]y [ ]3 [ ]x y = [ ]3 [ ]x y 3 x y

:= distributive_rule = a ( ) + b c + a b a c = [ ]3 [ ] + x y + 3 x 3 y

:= multiplicative_identity = [ ]a [ ]1 a = [ ]6 [ ]1 6

:= multiplicative_inverse = [ ]a

1a 1

= [ ]6.1

[ ]1[ ]6.1 1

Rules for exponentiation:


> x^m*x^n=x^(m+n);

= xm xn x( ) + m n

> x^m/x^n=x^(m-n);

= xm

xn x( ) − m n

If x is not equal to zero:

> [x]^[0]=1;

= [ ]x[ ]0

1

> 0^0;

1From the previous two rules:

> [x]^[0]/x^n=x^([0]-n);

= [ ]x

[ ]0

xn x( ) − [ ]0 n

> 1/x^n=[x]^[-n];[3]^[-1]=1/3;

= 1xn [ ]x

[ ]−n

= [ ]3[ ]-1 1

3

Also:

> (x*y)^n=x^n*y^n;[3*x]^[2]=(3*x)^2;

= ( )x y n xn yn

= [ ]3 x[ ]2

9 x2

> (x/y)^n=x^n/y^n;

=

xy

nxn

yn


> [2/3]^[2]=(2/3)^2;

=

23

[ ]249

> (x^m)^n=x^(m*n);

= ( )xm nx

( )m n

> [[2]^[2]]^[3]=[2]^[6];

= [ ][ ]2[ ]2 [ ]3

[ ]2[ ]6

When an algebraic expression involves multiple operations, the operations must be done in the correct order. First perform exponentiation operations from left to right. Then do multiplication and division at the same time from left to right. Finally do addition and subtraction at the same time from left to right. Expressions contained within grouping symbols (usually parentheses) should done according to the above order.

Maple uses parentheses for grouping symbols. (I only use brackets to prevent Maple from doing all the work for you.) A horizontal dividing line is also a grouping symbol. Force Maple to evaluate the numerator and evaluate the denominator before dividing by placing the numerator in parentheses and placing the denominator in parentheses.

> (`3`-7)/(`2`+5)=(3-7)/(2+5);

= − 3 7 + 2 5

-47

Algebraic Operations: Test Bank Problems

PrObLEM 101

Evaluate

+ 8 [ ]−24 + 6 [ ]−1 .

> Answer=(8+(-24))/(6+(-1));evalf(%);

= Answer -165

= Answer -3.200000000


> ?

PrObLEM 102

Evaluate ( ) + 9 [ ]−22 ( ) + 18 [ ]−32 .

> Answer=(9+(-22))*(18+(-32));

= Answer 182

> ?

PrObLEM 103

Evaluate + − a b

c d e f when a = 3, b = 22, c = 26, d = 25, e = 9, and f = 32.

Warning, the protected name Chi has been redefined and unprotected

> Answer=3+22/26-25*9*32;evalf(%);


= Answer -9355013

= Answer -7196.153846

> ?

PrObLEM 104

Evaluate + a b ( ) + c d e f when a = 2, b = 3, c = 3, d = 3, e = 1, and f = 5.

> Answer=2*3+(3*3+1)^5;evalf(%);

= Answer 100006 = Answer 100006.

> ?

Contemporary Math ��

Established in 2004, TSTC Publishing is a provider of high-end technical instructional materials and related information to institutions of higher education and private industry. “High end” refers simultaneously to the information delivered, the various delivery formats of that information, and the marketing of materials produced. More information about the products and services offered by TSTC Publishing may be found at its Web site: http://publishing.tstc.edu/.

TSTC Publishing

≠∂ %

9 7 8 1 9 3 4 3 0 2 0 7 1

ISBN 978-1-934302-07-19 0 0 0 0

RVSD 02-08

More information about this book as well as other books from TSTC Publishing may be found at:

http://publishing.tstc.edu/

Documents

Contemporary Math Preview