ENM 503 Lesson 2 Fundamentals

Numbers, Bases, Algebra, Functions, Equations and other Calculus Concepts1ENM 503 FundamentalsrdPrimitives & Axioms2Is the number 6 larger than the number 3?Has anyone ever seen a number?Is the word "cheese" on the blackboard "chalk" ?Has anyone ever seen a point?Axioms are assumptions made about primitives.Bird Testing of NumberAn Example of doing Mathematics Sum of first 100 integers : 1 + 2 + 3 + 4 + 5 + 6 1 + 6 = 2 + 5 = 3 + 4 = 7 => n(n+1)/2rdNumbersrd3Cardinal: zero, one, two, used for countingOrdinal: first, second, denote position in sequenceIntegers: negative, zero and positive whole numbers -3 -2 -1 0 1 2 3 Fractions: parts of whole, etc.Numerals: symbols describing numbersDigits: specific symbols to denote numbersArabic Numerals: 0 1 2 3 4 5 6 7 8 9Roman Numerals: I II III IV V VI VII VIII IX X 1 googol = 10100 ; 1 googolplex = 10googol =

Types of Numbersrd4RationalPrimePerfectAlgebraic roots of equations with integer coefficientsIrrational 2 is algebraic since x2 2 = 0 Imaginary and Complex, i = Transcendental Liouville; e, ; most frequent, not algebraic, not roots of integer polynomialsTransfinite Numbers - CantorFigurate NumbersOmega ; aleph null 0; aleph-one 1

CASTING OUT NINESrd5 + 281 393 42610910 1 1 ChecksSame procedure for subtraction and multiplication25 * 25 = 625 ~ 4 after casting out 9's 7 * 7 = 49 ~ 4 after casting out 9'sThe Real Number System6natural numbersN = {1, 2, 3, }IntegersI = { -3, -2, -1, 0, 1, 2, }Rational NumbersR = {a/b | a, b I and b 0}Algebraic NumbersIrrational Numbers{non-terminating, non-repeating decimals} e.g. Transcendental numbers ~ irrational numbers that cannot be a solution to a polynomial equation having integer coefficients transcends the algebraic operations of +, -, x, /

rdBinary Arithmeticrd7 SumDifference1011 111011 11 + 101 5- 101 - 5 10000 16 110 6

ProductQuotient 1011 11 10.00110 X 101 5 101 1011,0000000 1011 55-1010 0000 01 000 1011- 101 110111 110- 101#b11011 = 27; #o27 = 23; #xAB = 171; #7r54 = 39

1=2rd8Let x = yxy = y2 xy x2 = y2 x2x(y x) = (y x)(y + x)x = y + x1 = 2qed. Quad erat demonstrandum meaning which was to be demonstrated.

Three Classical Insolvable Problemsrd9Using only straight edge and compass

1. Construct a square whose area equals a circle.

2. Double the volume of a given cube.

3. Trisect an angle

Multinomialsrd10Find the coefficient of x3yz2 in the expansion of(x + y + z)6.

(poly^n #(x #(y #(z 0 1) 1) 1) 6) #(X #(Y #(Z 0 0 0 0 0 0 1) #(Z 0 0 0 0 0 6) #(Z 0 0 0 0 15) #(Z 0 0 0 20) #(Z 0 0 15) #(Z 0 6) 1) #(Y #(Z 0 0 0 0 0 6) #(Z 0 0 0 0 30) #(Z 0 0 0 60) #(Z 0 0 60) #(Z 0 30) 6) #(Y #(Z 0 0 0 0 15) #(Z 0 0 0 60) #(Z 0 0 90) #(Z 0 60) 15) #(Y #(Z 0 0 0 20) #(Z 0 0 60) #(Z 0 60) 20) #(Y #(Z 0 0 15) #(Z 0 30) 15) #(Y #(Z 0 6) 6) 1)

#(Y #(Z 0 0 15) #(Z 0 30) 15) #(Y #(Z 0 6) 6) 1)(x + y)^3 = #(X #(Y 0 0 0 1) #(Y 0 0 3) #(Y 0 3) 1)

10Poly^nrd11(x + y + z)3 = x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xy2 + y3 + 3y2z + 3yz2 + z3

(poly^n #(x #(y #(Z 0 1) 1) 1) 3)#(X #(Y #(Z 0 0 0 1) #(Z 0 0 3) #(Z 0 3) 1) #(Y #(Z 0 0 3) #(Z 0 6) 3) #(Y #(Z 0 3) 3) 1)

x3 + 3x2y + 3x2z + 3xy2 + 6xyz + 3xy2 + y3 + 3y2z + 3yz2 + z3

Joseph Liouville12Born: 24 March 1809 in Saint-Omer, FranceDied: 8 Sept 1882 in Paris, France

An important area which Liouville is remembered for today is that of transcendental numbers. Liouville's interest in this stemmed from reading a correspondence between Goldbach and Daniel Bernoulli. Liouville certainly aimed to prove that e is transcendental but he did not succeed. However his contributions were great and led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number: 0.1100010000000000000000010000... where there is a 1 in place n! (n = 1,2,3, and 0 elsewhere.

rdEvery even number greater than 2 can be written as a sum of 2 primes (Goldbach's conjecture, still unproven and thus no counterexample)12More Real Numbers13Real NumbersRational (-4/5) = -0.8Irrational Transcendental (e=2.718281828459045 )(=3.141592653589793 )Integers (-4)Natural Numbers (5)

Did you know? The totality of real numbers can be placed in a one-to-one correspondence with the totality of the points on a straight line. Dense.rdNumbers in sets14

transcendental numbersDid you know? That irrational numbers are far more numerous thanrational numbers? Consider where a and b are integers

rdIdentity Property 15The numbers 0 and 1 play an important role in math since they do absolutely nothing. Any number plus 0 equals itself.a + 0 = 0 + a = a.One example of this is: 3 + 0 = 0 + 3 = 3.0 is called the identity for addition. Any number multiplied by 1 is equal to itself.a x 1 = 1 x a = a.One example of this is: 3 x 1 = 1 x 3 = 31 is called the identity for multiplication.

rdAlgebraic Operations16Basic Operationsaddition (+) and the inverse operation (-)multiplication (x) and the inverse operation ( )Commutative Lawa + b = b + aa x b = b x a* Vectors, Matrices non-commutative Associative Lawa + (b + c) = (a + b) + ca(bc) = (ab)cDistributive Lawa(b + c) = ab + acrdFunctions17Functions and Domains: A real-valued function f of a real variable is a rule that assigns to each real number x in a specified set of numbers, called the domain of f, a real number y = f(x) in the range. The variable x is called the independent variable. If y = f(x), we call y the dependent variable. A function can be specified: numerically: by means of a table or ordered pairs algebraically: by means of a formula graphically: by means of a graphrdMore on Functions18A function f(x) of a variable x is a rule that assigns to each number x in the function's domain a value (single-valued) or values (multi-valued)

dependentvariableindependentvariableexamples:function ofn variablesrdOn Domains19Suppose that the function f is specified algebraically by the formula

with domain (-1, 10]

The domain restriction means that we require -1 < x 10 in order for f(x) to be defined (the round bracket indicates that -1 is not included in the domain, and the square bracket after the 10 indicates that 10 is included).

rdA more interesting function20Sometimes we need more than a single formula to specify a function algebraically, as in the following piecemeal exampleThe percentage p(t) of buyers of new cars who used the Internet for research or purchase since 1997 is given by the following function. (t = 0 represents 1997).

rdFunctions and Graphs21The graph of a function f(x) consists of the totality of points (x,y) whose coordinates satisfy the relationship y = f(x).xy||||||_______

a linear functionthe zero of the functionor roots of the equation y = f(x) = 0y intercept where x = 0rdGraph of a nonlinear function22

Sources: Bureau of Justice Statistics, New York State Dept. of Correctional Services/The New York Times, January 9, 2000, p. WK3. rdPolynomials in one variable23Polynomials are functions having the following form:

nth degree polynomiallinear function

quadratic functionDid you know: an nth degree polynomial has exactly n roots; i.e. solutions to the equation f(x) = 0 Karl GaussrdFacts on Polynomial Equations24Used in optimization, statistics (variance), forecasting, regression analysis, production & inventory, etc.The principle problem when dealing with polynomial equations is to find its roots.r is a root of f(x) = 0, if and only if f(r) = 0.Every polynomial equation has at least one root, real or complex (Fundamental theorem of algebra)A polynomial equation of degree n, has exactly n rootsA polynomial equation has 0 as a root if and only if the constant term a0 = 0.

rdMake-polynomial with rootsrd25(my-make-poly '(1 2 3)) (1 -6 11 -6)(cubic 1 -6 11 -1) (3 2 1)

(my-make-poly '(2 -3 7 12)) (1 -18 59 198 -504)

(quartic 1 -18 59 198 -504) (12 7 2 -3)

(my-make-poly '(1 2 -3 7 12)) (1 -19 77 139 -702 504) but neither quintic nor higher degree polynomials can be solved by formula.

The Quadratic Function26Graphs as a parabolavertex: x = -b/2aif a > 0, then convex (opens upward)if a < 0, then concave (opens downward)

rdThe Quadratic Formula27

rd (quadratic 1 4 3) (-1 -3)

A Diversion ~ convexity versus concavity28

Concave:

Convex:rdrd29Concave vs. convex

More on quadratics30 If a, b, and c are real, then:if b2 4ac > 0, then the roots are real and unequalif b2 4ac = 0, then the roots are real and equalif b2 4ac < 0, then the roots are imaginary and unequal

discriminantrdInteresting Facts about Quadratics31If x1 and x2 are the roots of a quadratic equation, then

Derived from the quadratic formulard Equations Quadratic in form32

quadratic in x2

factoring

Imaginary rootsA 4th degree polynomial has 4 rootsrd The General Cubic Equation33

rdPolynomials of odd degree must have at least one real root because complex roots occur in pairs.The easy cubics to solve:34

rdThe Power Function(learning curves, production functions)35

For b > 1, f(x) is convex (increasing slopes)

0 < b < 1, f(x) is concave (decreasing slopes)

For b = 0; f(x) = a, a constant

For b < 0, a decreasing convex function (if b = -1 then f(x) is a hyperbola)

rdLearning Curves Cost & Time36(sim-LC 1000 10 90) Tn = T0 nb Unit Hours Cumulative 1 1000.00 1000.00 2 900.00 1900.00 3 846.21 2746.21 4 810.00 3556.21 5 782.99 4339.19 6 761.59 5100.78 7 743.95 5844.73 8 729.00 6573.73 9 716.06 7289.79 10 704.69 7994.48 The slope of 90% learning curve is -0.1520; consider any unit, say 5. 783 = 1000*5b => b = -152.

rdExponential Functions(growth curves, probability functions)37

often the base ise=2.718281828459045235360287471352662497757... For c0 > 0, f(x) > 0For c0 > 0, c1 > 0, f(x) is increasingFor c0 > 0, c1 < 0, f(x) is decreasingy intercept = c0 ex > 0rd37y = ex rd38y = ex y eb = eb(x b) y' = ex (b, eb); intercept is x - 1

y = ln xrd39

Law of Exponents40

rd2324 = 8 * 16 = 128 = 27

25/23 =32/8 = 4 = 22

(23)4 = 84 = 4096 = 212

21/2 = 1.41421356237

Calculation Rules for Roots41Radical is

Radicand is N, n is the root index.

rdProperties of radicals42

but note:

rdNot a linear operatorRadar Beamrd43 334.8F = vf where v is vehicle speed and f = 2500 megacycles.sec aimed at you. F is the difference between the initial beam sent out and the reflected beam. Were you speeding if the difference was 495 cycles/sec?

v = 334.8 * 495/2500 = 66.29 mph, perhaps not speeding but driving a bit over the speed limit of 65 mph.Law of Exponents44

rdMultiplying a Multinomial by a Multinomial45Using the distributive law, we multiply one of the multinomials by each term in the other multinomial. We then use the distributive law again to remove the remaining parentheses, and simplify.

(x + 4)(x - 3) = x(x - 3) + 4(x - 3) = x2 - 3x + 4x -12 = x2 + x -12(x a)(x b)(x c)(x d) (x y) (x z) = _______(Poly* #(X 4 1) #(X -3 1)) #(X -12 1 1)rdLogarithmic Functions(nonlinear regression, probability likelihood functions)46

natural logarithms, base ebasenote that logarithms are exponents:If x = ay then y = loga xFor c0 > 0, f(x) is a monotonically increasingFor 0 < x < 1, f(x) < 0For x = 1, f(x) = 0 since a0 = 1For x 0, f(x) is undefinedrdLeast Common Multiple (LCM)rd47the smallest positive integer that is divisible by the numbers.8 = 2 2 28 16 24 32 4012 = 2 2 312 24 36 9 = 3 3 9 18 27 36 15 = 3 5 LCM = 2 * 2 * 2 * 3 * 3 * 5 = 360(lcm 8 12 9 15) 360(div 360) (1 2 3 4 5 6 8 9 10 12 15 18 20 24 30 36 40 45 60 72 90 120 180 360)

Greatest Common Divisor (GCD)rd48The largest positive integer that divides the numbers with zero remainder

102 and 30102 = 3 * 30 + 1230 = 2 * 12 + 612 = 2 * 6 + 06 is gcd(div 102) (1 2 3 6 17 34 51 102)(div 30) (1 2 3 5 6 10 15 30)

Friend Ben => Be = n49Log Base Number = Exponent

BaseExponent = Number

LogB N = E BE = N

Log2 16 = 4 24 = 16

Log2 16 = ln 16 / ln 2 = 2.7725887/0.6931472 = 4

rdProperties of Logarithms50

The all important change of bases:log216 = ln16/ln 2 = 4

rdPropertiesrd511. Log i(xi) = log(xi)2. for all x and y, x > y implies log(x) > log(y)3. for all x between 0 and 1, log(x) is negativeExamples of logarithms 521. If Ln x = 2 Ln 3 - 3 Ln 2, then x = ___

Log2 16 = 4 = Log2 42 = 2Log2 4 = 2*2 = 4

Log2 2 = 1

X = Logarithms are exponents X is called the anti-logarithm 1 can never be a base

rdLogarithms 53Common logarithm ~ lg to the base 10, log xLn ~ Natural logarithm with base eLb x Binary logarithm with base 2Loga x logarithm of x to the base a

Loga 2 x = (loga x)2

Loga loga x = loga (loga x)Let a = 2 and x = 16Log2 log2 16 = log2 (log2 16) = log2 4 = 2.(log (log 16 2) 2) 2

rd53Logarithm to any base using erd54Find the log of 12 to the base 6. Repeat for 8 base 2Take the log of 12 to the base e and divide by the natural log of 6.log612 = ln 12 /ln 6 = 1.3868

log28 = ln 8 / ln 2 = 2.07944 /0.693147 = 3

Logarithmsrd55logab * logbc = logac for any a, b and cLet a = 12, b = 37, and c = 59

Then log1237 = 1.45314026 log 3759 =1.12922463 log 1259 = 1.64092178(* 1.4531402 1.12922463) 1.64092178

The absolute value function(rectilinear distance problems, forecasting (MAD*), multi-criteria decision-making) 56

xa*mean absolute deviationrdProperties of the absolute value57|ab| = |a| |b||a + b| |a| + |b||a + b| |a| - |b||a - b| |a| + |b||a - b| |a| - |b|

solve |x 3| < 5-5 < |x 3| < 5therefore -2 < x > 8rdTrigonometric Functions(forecasting, bin packing problems)58

Identities:squared relationships

reciprocal relationsrdacbForecasting with Trig Functions59Quadratic trend with seasonal (monthly) effects

t = timerdrd60

sin = DB = OE,tan = BC,sec = OC, cos = OD = EB, cot = AB,csc = OA???60Non-important Functions61Hyperbolic and inverse hyperbolic functionsGudermannian function and inverse gudermannian

rdComposite and multivariate functions(multiple regression, optimal system design)62

A common everyday composite function:A multivariate function that may be found lying around the house:

rdThe multi-variable polynomial63

rdInequalities64An inequality is statement that one expression or number is greater than or less than another.The sense of the inequality is the direction, greater than (>) or less than ( b, then a + c > b + cif both sides are multiplied or divided by the same positive number: if a > b, then ca > cb where c > 0The sense of the inequality is reversed if both side sides are multiplied or divided by the same negative number. if a > b, then ca < cb when c < 0rdMore on inequality65An absolute inequality is one which is true for all real values: x2 + 1 > 0A conditional inequality is one which is true for certain values only: x + 2 > 5Solution of conditional inequalities consists of all values for which the inequality is true.

(x 2)(x 3) > 0; x > 2 and x > 3 x < 2 and x < 3

For x < 2; f(x) > 0For 2 < x < 3, f(x) < 0For x > 3, f(x) > 0Therefore X < 2 and X > 3 are the solutionsrdAn absolute inequality66example problem: solve |x 3| < 5Write: -5 < (x 3) < 5Conclude -2 < x < 8for x > 3, (x - 3) < 5 => x < 8for x 3, -(x - 3) < 5 => x < 5 - 3 or x > -2Therefore, -2 < x < 8rdAn important multi-valued function(Euclidean distance problems, constrained optimization)67

xyPythagorean theorem

xyrrdImplicit and Inverse Functions68

implicit function

explicit function

inverse functionrdInverse Functions69Inverse functions are symmetric around the line y = x.Example: Let y = 2x + 3 implying the inverse function is y = (x - 3)/2.

y = 2x + 3 y = x

y = (x 3)/2rdThe Devils Curve70y4 - x4 + ay2 + bx2 = 0

An implicit relationship thatis not single-valuedrdSymmetry71f(x, y) = 4x2 + 9y2 = 36

(-x, y)(x, y)

(-x, -y)(x, -y)rdMultiplying Polynomials72Example 6 Find the product (2t -3)(5t3 + 3t -1)(poly*poly #(t -3 2) #(t -1 3 0 5)) #(T 3 -11 6 -15 10) ~ 3 11t + 6t2 15t3 +10t4(poly^n #(x -3 2) 5) #(X -243 810 -1080 720 -240 32)= -243 + 810x -1080x2 + 720x3 -240x4 +32x5

rdAdding/Subtracting Polynomials73(poly+ #(x 1 2 3) #(x -3 -2 -1)) #(X -2 0 2)

(poly- #(x 1 2 3) #(x -3 -2 -1)) #(X 4 4 4)

rdCreate Poly with Roots74(my-make-poly '(1 2 3)) (1 -6 11 -6)p(x) = x3 - 6x2 +11x - 6(cubic 1 -6 11 -6) (3 2 1)Short Division 1 -6 11 -6 |3 3 -9 61 -3 2 0 |2 (quadratic 1 -3 2) (2 1) 2 -2 1 -1 0 |1 1 1 0rdBinary and Decimal Base Numbers 75 23 22 21 20 8 4 2 1Using only the digits 0 and 1, 1 1 0 1 = 8 + 4 + 1 = 13 1 0 1 1 = 8 + 2 + 1 = 111 1 0 0 0 = 16 + 8 = 24103 102 101 100 1000 100 10 1 1 2 3 9 = 1(1000) + 2(100) + 3(10) + 9(1) 0 9 8 5 = 0(1000)+ 9(100) + 8(10) + 5(1) = 985rdBase Numbersrd76Write 246 in base 7

72 71 70 49 7 1 5 0 1 7 into 246 = 35 Remainder 1 7 into 35 = 5 R 0 7 into 5 = 0 R 5Try 221 in base 3 73 R 2, 24 R 1, 8 R 0, 2 R 2, 0 R 2 22012

Convert 324d to Base 2rd7732410 = 1010001002 162 081 040 120 010 05 02 11 00 1Base Arithmeticrd78For what bases does the number 121b represent a square number? Check 1331b.

b2 + 2b + 1 = (b + 1)2

Touch all the bases to score a home run.For example, in base 7 121 is 49 + 14 + 1 = 64 in base 9 121 is 81 + 18 + 1 = 100in base 33 121 is 1089 + 66 + 1 = 1156 = 342Quadratic Equation in Base 5rd79Solve x2 + 3x + 2 = 0 in Base 5 where you have the integers {0 1 2 3 4}

3, 4 are the roots. Check.79Base ?rd802 3 5 11 15 21 25 ?

680rd81Subtraction, Base 16, with Hex Digits 0 1 2 3 4 5 6 7 8 9 A B C D E F Base 6 Base 16 54 34 D9 217 -35 23-AC 172 1511 2D 45 (- #xD9 #xAC) 2D

AE16 + 768 = _________4 2363230

81Irrational Numberrd82Can you have an irrational number raised to an irrational power and have the result be rational?

Yes, Proof: is irrational and is an irrational number raised to an irrational power and is either rational or irrational. If rational, then done. If not rational then the number below is.

Square Rootsrd83

Logarithm/Exponential Equationrd84Solve x + 3e2y - 8 = 0 for y in terms of x:

e2y = (8 x)/3

2y Ln e = Ln((8 x)/3)

y = (1/2) Ln((8 x)/3)

Six steps to solving word problems85Picture the Problem Try to visualize the problem. Draw a diagram showing as much of the given information as possible, including the unknown.Understand the WordsLook up the meanings of unfamiliar words in a dictionary, handbook, or textbook.Identify the Unknown(s) and the constantsBe sure that you know exactly what is to be found in a particular problemSummarize and write in mathematical form what is givenEstimate the AnswerIt is a good idea to estimate or guess the answer before solving the problem, so that you will have some idea whether the answer you finally get is reasonable.Write and Solve the Equation(s)The unknown quantity must now be related to the given quantities by means of equation(s).rdFirst word problem86In a group of 102 employees, there are three times as many employees on the day shift as on the night shift, and two more on the swing shift than on the night shift. How many are on each shift?Let x = number of employees on the night shiftThen 3x = number of employees on the day shift And (x + 2) number of employees on the swing shiftx + 3x + (x + 2) = 102

5x = 100x = 20 on the night shift3x = 60 on the day shift(x + 2) = 22 on the swing shiftrd Financial Problem87A consultant had to pay income taxes of $4867 plus 28% of the amount by which her taxable income exceeded $32,450. Her tax bill was $7285. What was her taxable income? Work to the nearest dollar.

Solution: Let x = taxable income (dollars). The amount by which her income exceeded $32,450 is then x - 32,450Her tax is 28% of that amount, plus $4867, sotax = 4867 + 0.28(x - 32,450) = 7285Solving for x, we get x = $41,086rdMixture Problems88From 100.0 kg of solder, half lead and half zinc, 20.0 kg are removed. Then 30.0 kg of lead are added. How much lead is contained in the final mixture?Solution:initial amount of lead = 0.5(100.0) = 50.0 kg 40Lamount of lead removed = 0.5(20.0) =10.0 kgamount of lead added = 30.0 kg

final amount of lead = 50.0 - 10.0 + 30.0 = 70.0 kgrdMore Mixture Problems89How much steel containing 5.25% nickel must be combined with another steel containing 2.84% nickel to make 3.25 tons of steel containing 4.15% nickel?Let x = tons of 5.25% steel needed. 5.25 a + 2.84b = 4.15 * 3.25; a + b = 3.25The amount of 2.84% steel is (3.25 x)The amount of nickel that it contains is 0.0284(3.25 x)The amount of nickel in x tons of 5.25% steel is 0.0525xThe sum of these must give the amount of nickel in the final mixture. 0.0525x + 0.0284(3.25 x) = 0.0415(3.25)x = 1.77 t of 5.25% steel3.25 * x = 1.48 t of 2.84% steelrd(solve '((5.25 2.84 13.4875 )(1 1 3.25))) (1.766598 1.483402)

89Sequencesrd90Sequences & Seriesrd91A sequence is a progression of ordered numbers: 3, 10, 19, 37, such that the preceding and following numbers are completely specified.In an arithmetic sequence the terms have a common difference: 1, 4, 7, 10, . In an harmonic sequence the terms are reciprocals of the terms in an arithmetic sequence: 1, 1/4, 1/7, 1/10, .In a geometric sequence the terms have a common ratio: 1, 3, 9, 27, .A series is the sum of the terms of a sequence 1 + 3 + 9 + 27Series are either finite or infinite; convergent or divergent

91Arithmetic Sequencesrd92Complete the sequences at the *

a) 2 7 12 17 * *b) 5 13 21 * * c) 11 15 * 23 * * * 20 29 38e) 4 * 18 * 32 f) * 33 * 65 * 10 * 70 h) 10 * * 70i) 10 * * * * 70 j) If each term of an arithmetic sequence is multiplied by a constant, is the resulting sequence arithmetic? a, a + d, a + 2d, a + 3d versus ka, k(a + d) k(a + 2d) k(a + 3d)92a) 22 b) 29 c) 19 27 d) 2 11 e) 11 25 f) 17 49 g) 40 h) 30 50 i) 22 34 46 58 j) YESArithmetic Sequencesrd93The 100th term of 2 5 8 11 14 * * * is ____. ans. 299b) The 20th term of 11 15 19 23 * * * is ____. ans. 87Find the sum of the sequence: 3 7 11 15 19 23 27 Add 3 7 11 15 19 23 27 + 27 23 19 15 11 7 3 30 30 30 30 30 30 30 => sum = 7(30)/2 = 105 d) Find the sum of the first 100 integers. n(n+1)/293d) 5050Arithmetic Seriesrd94Sum Sn of a finite arithmetic series is given bySn = n(a1 + an)/2Example: 2 + 4 + 6 + 8 + . . . + 100 = 50(2 + 100)/2 = 2550;where n = 100/2 = 50; a1 = 2; an = 1001 + 5/3 + 7/3 + . . . + 201 = .

1 + (n 1)(2/3) = 201 => n = 301 terms=> Sn = 301(1 + 201)/2 = 30,401

94Harmonic Seriesrd95Arithmetic sequence 1 4 7 10 13 16 19 22Reciprocals: 1 1/4 1/7 1/10 1/13 1/16 1/19 1/22is an harmonic sequence(+ 1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024 1/2048 1/4096 1/8192 1/16384 1/32768 1/65536 1/131072 1/262144 1/524288 1/1048576) 2097151 / 1048576= 1.9999999999(let ((x 0)) (dotimes (i 100 x) (incf x (recip (expt 2 i))))) 295(+ 1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024 1/2048 1/4096 1/8192 1/16384 1/32768 1/65536 1/131072 1/262144 1/524288 1/1048576) 2097151 / 104857621 terms524288(1 + 1048576)/2 = Harmonic Seriesrd96Find the 36th term of the series1 + 1/4 + 1/7 + 1/10 + 1/13 +

The arithmetic series is 1 4 7 10 13 and the 36th term is 1 + 35*3 = 106 => 1/106.96Harmonic Seriesrd97A cyclist travels from A to B at 40 mph and returns at 60 mph. The average speed for the round trip is . 48 b) 49 c) 50 d) 51 e) none of these (1 / [(1/40 + 1/60)/ 2] = 48Apply sensitivity analysis to explain why.97Geometric Seriesrd98Sum Sn of a geometric series of n terms is given by

Sn =

Find the sum of the geometric series 1 4 16 64 256 1024.Sum = (1 4 * 1024)/(1 4) = 1365Find the sum of the geometric series 3 18 108 . . . 839,808.(3 6 * 839,808)/(1 6) = 1,007,769

98Geometric Seriesrd99Find the sum of the following geometric series:(1 + i)0 + (1 + i)1 + (1 + i)2 + (1 + i) 3

Sum = (1 + i)0 - (1 + i)(1 + i)3 1 - (1 + i) = 1 - (1 + i)4 -i (F/A, 6%, 4) = (1 + i)4 1 i = F/A = [(1 + i)n 1]/i = 4.37462 at i = 6% A A A ASequencesrd1001. Write the first 5 terms of {1 1/(2n)} 4/5 5/6 7/8 9/10

2. Repeat #1 for {[(-1)n + 1]}0 1 0 1 0

3. Write the general term of 1, 1/3, 1/5, 1/7, 1/9Look at reciprocals 1 3 5 7 9 General term is 1/(2n 1)Sequences (continuing)rd1011. 102 103 105 107 111 113 ?

2. 3 15 14 7 18 1 20 21 12 1 20 9 15 14 ?

3. 2 12 36 80 150 252 392 ?

4. 3 5 6 2 9 5 1 4 1 ?

5. 1 1 2 3 5 8 13 ?Congratulations, n2 + n3101Triangle Inequalityrd102|a + b| |a| + |b|If both non-negative, |a + b| = a + b = |a| + |b|If both negative, |a| = -a; |b| = -b and a + b is negative then |a + b| = -(a + b) = -a + (-b) + |a| + |b|

If a > 0 and b < 0, then |a| = a, |b| = -b If |a| > |b|, then |a + b| = a + b < a b + |a| + |b| If |a| = |b|, then |a + b| = 0 < |a| + |b| etc.US Currencyrd103Bills: 1 2 5 10 20 50 100 500 1,000 5,000 10,000 100,000Find a geometric sequence of bills with common ratio 10.1 100 10,000 100,000103(+ 1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 1/512 1/1024 1/2048 1/4096 1/8192 1/16384 1/32768 1/65536 1/131072 1/262144 1/524288 1/1048576) 2097151 / 1048576

Pyramid Schemerd104104Fibonacci SequenceRecursive up a staircase one or two steps at a time with n stepsn # of ways 122: 1, 233: 111,12, 2145: 1111, 112, 121, 211, 22fn = fn-1 + fn-2 For n = 5 steps, take 1 step 45 ways 11111,1112,1121,1211,122thereafter or 2 steps 33 ways 2111, 212, 221thereafter yielding 5 + 3 = 8rd105Fibonacci Sequencerd106(1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946)Which terms are evenly divisible by 3, 5, 8, 13 and 55?Which term is the largest cube?1066765 is divisible by 3 and 5.144 is divisible by 3 and 8Intelligence Testingrd1071. John is twice as old as his sister Mary, who is now 5 years of age. How old will John be when Mary is 30 years of age?

2. Mary is 24 years old. She is twice as old as Ann was when Mary was as old as Ann is now. How old is Ann? Let x = Ann's age: 24 = 2[x (24 x)]

1075 years older than Mary John = 10 now 10 + 25 = 35Numerologyrd108How much wheat can be put on a chessboard with 1 grain on the first square, 2 on the next, 4 on the third etc.?S = 1 + 2 + 4 + 8 + 16 + 32 + + 263 = (1 264) / (1 2) = 18,446,744,073,709,551,615 grains of wheat

Roughly a train reaching a thousand times around the Earth filled with wheat.108Differential Equation for erd109y y = 0Assume y = ex = a0 + a1x + a2x2 + a3x3 + a4x4 + a5x5+ . .

Then y = a1 + 2 a2 x + 3 a3 x2 + 4 a4 x3 + 5 a5 x4 + . . .and y(0) = 1 => a0 = 1 y(0) = 1 => a1 = 1 y(0) = 1 => 2a2 = 1 => a2 y(0) = 1 => 6a3 = 1 => a3 = 1/6ex = 1 + x + x2/2! + x3/3! + 109rd110Infinite SeriesTwo motorcyclists A and B, 100 miles apart, head for each other. A travels at 40 mph and B at 60 mph. A fly flies from As nose to Bs nose and back again and again at 70 mph. How far will the fly have flown when the two cyclists meet?Infinite series whose terms increase in magnitude have no attainable sum. If a sum exists, series is said to be convergent; if not, divergent. 110Pythagorean Triplesrd111mnm2 n22mnm2 + n232 512 1361 3512 3765 1160 617 6 1384 858 7 15112 1136011 34791320 3721 = 6128413 68872184 7225 = 85210 6 64 = 43120 136 6 3 27 = 33 36 45Try one yourself by picking an m > than n.Asking for a Raiserd112Would you rather receive a raise in salary of $300 every 6 months or $1000 every year? 1000 Lockersrd113There are 1000 lockers and all are opened. Then I go by each and reverse the state. If open, I close it; if closed, I open it. Then I repeat for every 2 lockers, then every 3 lockers , etc. When done, what are the states of the lockers?

What are you modeling mathematically?

Chord Lengthrd114Express the length L of a chord of circle with radius r as a function of x being the distance from the center.L = 2(r2 - x2)1/2

Find length of chord in circle of radius 13 that is 5 units from the center. L = 2(169 25)1/2 = 24.

r L/2 xRandomWhat does random mean? What is the probability that a randomly drawn chord is shorter than the leg of the equilateral triangle in the circle? 2/3 Repeat for the perpendicular at the midpoint of the radius. 1/4

A

B CWhich answer is correct? Both are mathematically correct. What does "at random" mean? rd115(comb 2 2)Radioactive Decayrd116N = N0e-t for t in daysGiven an element N = 100e-0.062t , find the initial amount, the half life, and verify that the amount at half life is half of the initial amount. How much is present after 9 days?

N0 = 100e-0.062 * 0 = 100 N/N0 = = e-0.062t Solve for t to get 11.1788 days as the half life.

N = 100e-0.062 * 11.1788 = 50 N9 = 100e-0.062 * 9 = 57.235 mg Sanity check: As 9 is less than half life, expect more than 50 mg. Rationalizing Denominatorrd117Rationalize 1/21/2 Multiply numerator and denominator by 21/2 to get 21/2 /2Spurious Rootsrd118x 4 + (x 2)1/2 = 0(4 x)2 = x 216 8x + x2 = x -2

(quadratic 1 -9 18) (3, 6)Check 6: 6 4 = (6 2) 2 = 2 3: 3 4 + (3 2 -1 1 reject 3; accept 6.Spurious Rootsrd119(5x2 + 10x 6) = 2x + 3

5x2 + 10x 6 = 4x2 +12x + 9 :Squaring both sides

Solve to get (quadratic 1 -2 -15) (5, -3)

Check to see that -3 is spurious and rejected, but that root 5 checks OK.

Logarithmic Equationsrd120(ln x)2 - 2 ln x - 3 = 0 Solve for x

Let y = ln x

Then y2 2y 3= 0 or (y - 3)(y + 1) = 0

ln x = 3; ln x = -1

x = e3; e-1120