Contemporary Math Preview

√ Contemporary Math I Using MAPLE or TI-89

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Otto Wilke

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Contemporary Math I Using MAPLE or TI-89

Otto Wilke Texas 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 Publishing Texas State Technical College Waco 3801 Campus Drive Waco, TX 76705 http://publishing.tstc.edu/ Publisher: Mark Long Graphics specialist: Grace Arsiaga Editor: Todd Glasscock Printing production: Bill Evridge Cover design: Sheri McGee

Manufactured in the United States of America First edition

Table of Contents Introduction to Maple...................................................................................................... 1 Getting Started....................................................................................................................1 Operation Symbols............................................................................................................1 Algebraic Operations...................................................................................................... 3 Algebraic Operations: Test Bank Problems....................................................................7 Number Systems........................................................................................................... 10 Number Systems: Test Bank Problems.........................................................................13 Radicals......................................................................................................................... 17 Radicals: Test Bank Problems.........................................................................................19 Sets and Lists................................................................................................................ 23 Sets and Lists: Test Bank Problems................................................................................26 Sets of Numbers............................................................................................................ 28 Sets of Numbers: Test Bank Problems...........................................................................30 Logic .............................................................................................................................. 33 Induction...........................................................................................................................33 Mathematical Induction..................................................................................................33 Deduction..........................................................................................................................34 Scientific Method..............................................................................................................41 Logic Problems.................................................................................................................42 Factoring........................................................................................................................ 44 Factoring: Test Bank Problems.......................................................................................46 Fractions........................................................................................................................ 49 Fractions: Test Bank Problems.......................................................................................52 Complex Numbers......................................................................................................... 54 Complex Numbers: Test Bank Problems......................................................................56 Coordinate Systems..................................................................................................... 59 Exponential Format....................................................................................................... 65 Units and Dimensions................................................................................................... 69 1. Definitions.....................................................................................................................69 2. The SI Units...................................................................................................................69 3. Algebra of Units and Dimensions.............................................................................77 4. Conversion Factors......................................................................................................78 Units and Dimensions: Test Bank Problems................................................................80

Linear Equations........................................................................................................... 83 Linear Equations: Test Bank Problems..........................................................................89 Linear Inequalities......................................................................................................... 96 Linear Inequalities: Test Bank Problems.....................................................................100 Quadratic Equations................................................................................................... 103 Quadratic Equations: Test Bank Problems.................................................................104 Radical Equations....................................................................................................... 107 Radical Equations: Test Bank Problems......................................................................107 Geometric Formulae................................................................................................... 109 Triangles..........................................................................................................................110 Quadrilaterals.................................................................................................................112 Geometric Formulae: Test Bank Problems.................................................................119 Functions..................................................................................................................... 123 Common Functions.......................................................................................................131 Transformation of Functions........................................................................................137 Functions: Test Bank Problems....................................................................................137 Exponential and Logarithmic Equations................................................................... 139 Exponential and Logarithmic Functions: Test Bank Problems................................145 Matrices........................................................................................................................ 149 Matrices: Test Bank Problems.......................................................................................155 Systems of Equations................................................................................................. 159 Matrix Equations............................................................................................................163 Diophantine Equations..................................................................................................165 Simultaneous Equations: Test Bank Problems...........................................................166 Systems of Inequalities.............................................................................................. 170 Simultaneous Inequalities: Test Bank Problems........................................................175 Sequences and Series................................................................................................ 178 Arithmetic Sequences and Series.................................................................................180 Geometric Sequences and Series..................................................................................181 Fibonacci Sequence........................................................................................................183 Sequences: Test Bank Problems...................................................................................184 Probability.................................................................................................................... 187 Probability: Test Bank Problems..................................................................................191 Descriptive Statistics and Regression...................................................................... 194 Statistics and Regression: Test Bank Problems..........................................................196

Percentage Equations................................................................................................ 201 Percentages: Test Bank Problems.................................................................................203 Proportionality............................................................................................................. 204 Proportionality: Test Bank Problems...........................................................................207 Graphs and Tables...................................................................................................... 209 Graphs and Tables: Test Bank Problems.....................................................................213 Trigonometric Functions............................................................................................ 215 Triangles...................................................................................................................... 219 Conic Sections............................................................................................................ 220 Conics: Test Bank Problems..........................................................................................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 [ShiftEnter] 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.

Contemporary Math 

Algebraic Operations An 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;

cat cat 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

 Contemporary Math

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, angle 1 ]

> > [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]; ( 1/2 )

( 1/2 )

 x  2, 2 x , 2 x + 3 y , 5  [4 y] 

 − 3.5  

An equation is an expression which contains an equal sign, =. > 3*x+2*y=6;

3x+2y=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;

3x+2y<6 Back 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.

Contemporary Math 

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_r ule:=a*(b+c)=expand(a*(b+c));[3]*[x+y]=3*x+3*y;multiplicative_id entity:=[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 1 multiplicative_inverse := [ a ]   = 1 a [1]  [ 6.1 ]   = 1  [ 6.1 ] 

Rules for exponentiation:

 Contemporary Math

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

(m + n)

> x^m/x^n=x^(m-n); (m − n) xm =x n x

If x is not equal to zero: > [x]^[0]=1; [x]

[0]

=1

> 0^0;

1 From the previous two rules: > [x]^[0]/x^n=x^([0]-n); [0]

[x] xn

=x

([0] − n)

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

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

[3 x]

[2]

= 9 x2

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

n x = x   yn y

Contemporary Math 

> [2/3]^[2]=(2/3)^2; 2   3

[2]

=

4 9

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

( xm ) = x

(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 -4 = 2+5 7

Algebraic Operations: Test Bank Problems Problem 101 8 + [ −24 ] Evaluate 6 + [ −1 ] .

> Answer=(8+(-24))/(6+(-1));evalf(%); -16 5 Answer = -3.200000000

Answer =

 Contemporary Math

> ?

Problem 102 Evaluate ( 9 + [ −22 ] ) ( 18 + [ −32 ] ) . > Answer=(9+(-22))*(18+(-32));

Answer = 182

> ?

Problem 103 Evaluate

a+

b −def c 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(%);

Contemporary Math 

-93550 13 Answer = -7196.153846 Answer =

> ?

Problem 104 f Evaluate a b + ( c d + e ) 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 255

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RVSD 02-08 ISBN 978-1-934302-07-1

9 781934 302071

9 0 0 0 0