numbers, bases, algebra, functions, equations and other calculus concepts 1 enm 503 fundamentals rd

Numbers, Bases, Algebra, Functions, Equations and other Calculus Concepts 1 ENM 503 Fundamentals rd

n(n+1)/2 rd">

Primitives & Axioms 2 Is 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 Number An 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)/2 rd

Numbers rd 3 Cardinal: zero, one, two, used for counting Ordinal: first, second, denote position in sequence Integers: negative, zero and positive whole numbers -3 -2 -1 0 1 2 3 Fractions: parts of whole, etc. Numerals: symbols describing numbers Digits: specific symbols to denote numbers Arabic Numerals: 0 1 2 3 4 5 6 7 8 9 Roman Numerals: I II III IV V VI VII VIII IX X 1 googol = 10 100 ; 1 googolplex = 10 googol =

Types of Numbers rd 4 Rational Prime Perfect Algebraic roots of equations with integer coefficients Irrational 2 is algebraic since x 2 2 = 0 Imaginary and Complex, i = Transcendental Liouville; e, ; most frequent, not algebraic, not roots of integer polynomials Transfinite Numbers - Cantor Figurate Numbers Omega ; aleph null 0 ; aleph-one 1

CASTING OUT NINES rd 5 + 281 393 426 10910 1 1 Checks Same procedure for subtraction and multiplication 25 * 25 = 625 ~ 4 after casting out 9's 7 * 7 = 49 ~ 4 after casting out 9's

The Real Number System 6 natural numbers N = {1, 2, 3, } Integers I = { -3, -2, -1, 0, 1, 2, } Rational Numbers R = {a/b | a, b I and b 0} Algebraic Numbers Irrational 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, / rd

Binary Arithmetic rd 7 SumDifference 1011 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=2 rd 8 Let x = y xy = y 2 xy x 2 = y 2 x 2 x(y x) = (y x)(y + x) x = y + x 1 = 2 qed. Quad erat demonstrandum meaning which was to be demonstrated.

Three Classical Insolvable Problems rd 9 Using 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

Multinomials rd 10 Find the coefficient of x 3 yz 2 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)

Poly^n rd 11 (x + y + z) 3 = x 3 + 3x 2 y + 3x 2 z + 3xy 2 + 6xyz + 3xy 2 + y 3 + 3y 2 z + 3yz 2 + z 3 (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) x 3 + 3x 2 y + 3x 2 z + 3xy 2 + 6xyz + 3xy 2 + y 3 + 3y 2 z + 3yz 2 + z 3

Joseph Liouville 12 Born: 24 March 1809 in Saint-Omer, France Died: 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. rd

More Real Numbers 13 Real Numbers Rational (-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. rd

Numbers in sets 14 transcendental numbers Did you know? That irrational numbers are far more numerous than rational numbers? Consider where a and b are integers rd

Identity Property 15 The 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 = 3 1 is called the identity for multiplication. rd

Algebraic Operations 16 Basic Operations addition (+) and the inverse operation (-) multiplication (x) and the inverse operation ( ) Commutative Law a + b = b + a a x b = b x a* Vectors, Matrices non-commutative Associative Law a + (b + c) = (a + b) + c a(bc) = (ab)c Distributive Law a(b + c) = ab + ac rd

Functions 17 Functions 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 graph rd

More on Functions 18 A 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) dependent variable independent variable examples: function of n variables rd

On Domains 19 Suppose 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). rd

A more interesting function 20 Sometimes we need more than a single formula to specify a function algebraically, as in the following piecemeal example The 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). rd

Functions and Graphs 21 The graph of a function f(x) consists of the totality of points (x,y) whose coordinates satisfy the relationship y = f(x). x y |||||||||||| ______________ a linear function the zero of the function or roots of the equation y = f(x) = 0 y intercept where x = 0 rd

Graph of a nonlinear function 22 Sources: Bureau of Justice Statistics, New York State Dept. of Correctional Services/The New York Times, January 9, 2000, p. WK3. rd

Polynomials in one variable 23 Polynomials are functions having the following form: n th degree polynomial linear function quadratic function Did you know: an nth degree polynomial has exactly n roots; i.e. solutions to the equation f(x) = 0 Karl Gauss rd

Facts on Polynomial Equations 24 Used 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 roots A polynomial equation has 0 as a root if and only if the constant term a 0 = 0. rd

Make-polynomial with roots rd 25 (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 Function 26 Graphs as a parabola vertex: x = -b/2a if a > 0, then convex (opens upward) if a < 0, then concave (opens downward) rd

The Quadratic Formula 27 rd (quadratic 1 4 3) (-1 -3)

A Diversion ~ convexity versus concavity 28 Concave: Convex: rd

29 Concave vs. convex

More on quadratics 30 If a, b, and c are real, then: if b 2 4ac > 0, then the roots are real and unequal if b 2 4ac = 0, then the roots are real and equal if b 2 4ac < 0, then the roots are imaginary and unequal discriminant rd

Interesting Facts about Quadratics 31 If x 1 and x 2 are the roots of a quadratic equation, then Derived from the quadratic formula rd

Equations Quadratic in form 32 quadratic in x 2 factoring Imaginary roots A 4 th degree polynomial has 4 roots rd

The General Cubic Equation 33 rd Polynomials of odd degree must have at least one real root because complex roots occur in pairs.

The easy cubics to solve: 34 rd

The 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) rd

Learning Curves Cost & Time 36 (sim-LC 1000 10 90) T n = T 0 n b 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*5 b => b = -152. rd

Exponential Functions (growth curves, probability functions) 37 often the base is e=2.718281828459045235360287471352662497757... For c 0 > 0, f(x) > 0 For c 0 > 0, c 1 > 0, f(x) is increasing For c 0 > 0, c 1 < 0, f(x) is decreasing y intercept = c 0 e x > 0 rd

y = e x rd 38 y = e x y e b = e b (x b) y' = e x (b, e b ); intercept is x - 1

y = ln x rd 39

Law of Exponents 40 rd 2 3 2 4 = 8 * 16 = 128 = 2 7 2 5 /2 3 =32/8 = 4 = 2 2 (2 3 ) 4 = 8 4 = 4096 = 2 12 2 1/2 = 1.41421356237

Calculation Rules for Roots 41 Radical is Radicand is N, n is the root index. rd

Properties of radicals 42 but note: rd Not a linear operator

Radar Beam rd 43 334.8F = 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 Exponents 44 rd

Multiplying a Multinomial by a Multinomial 45 Using 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) = x 2 - 3x + 4x -12 = x 2 + 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) rd

Logarithmic Functions (nonlinear regression, probability likelihood functions) 46 natural logarithms, base e base note that logarithms are exponents: If x = a y then y = log a x For c 0 > 0, f(x) is a monotonically increasing For 0 < x < 1, f(x) < 0 For x = 1, f(x) = 0 since a 0 = 1 For x 0, f(x) is undefined rd

Least Common Multiple (LCM) rd 47 the smallest positive integer that is divisible by the numbers. 8 = 2 2 28 16 24 32 40 12 = 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) rd 48 The largest positive integer that divides the numbers with zero remainder 102 and 30 102 = 3 * 30 + 12 30 = 2 * 12 + 6 12 = 2 * 6 + 0 6 is gcd (div 102) (1 2 3 6 17 34 51 102) (div 30) (1 2 3 5 6 10 15 30)

Friend Ben => B e = n 49 Log Base Number = Exponent Base Exponent = Number Log B N = E B E = N Log 2 16 = 4 2 4 = 16 Log 2 16 = ln 16 / ln 2 = 2.7725887/0.6931472 = 4 rd

Properties of Logarithms 50 The all important change of bases: log 2 16 = ln16/ln 2 = 4 rd

Properties rd 51 1. Log i (x i ) = log(x i ) 2. for all x and y, x > y implies log(x) > log(y) 3. for all x between 0 and 1, log(x) is negative

Examples of logarithms 52 1. If Ln x = 2 Ln 3 - 3 Ln 2, then x = ___ 2. Log 2 16 = 4 = Log 2 4 2 = 2Log 2 4 = 2*2 = 4 3. Log 2 2 = 1 4. X = Logarithms are exponents X is called the anti-logarithm 1 can never be a base rd

Logarithms 53 Common logarithm ~ lg to the base 10, log x Ln ~ Natural logarithm with base e Lb x Binary logarithm with base 2 Log a x logarithm of x to the base a Log a 2 x = (log a x) 2 Log a log a x = log a (log a x) Let a = 2 and x = 16 Log 2 log 2 16 = log 2 (log 2 16) = log 2 4 = 2. (log (log 16 2) 2) 2 rd

Logarithm to any base using e rd 54 Find the log of 12 to the base 6. Repeat for 8 base 2 Take the log of 12 to the base e and divide by the natural log of 6. log 6 12 = ln 12 /ln 6 = 1.3868 log 2 8 = ln 8 / ln 2 = 2.07944 /0.693147 = 3

Logarithms rd 55 log a b * log b c = log a c for any a, b and c Let a = 12, b = 37, and c = 59 Then log 12 37 = 1.45314026 log 37 59 =1.12922463 log 12 59 = 1.64092178 (* 1.4531402 1.12922463) 1.64092178

The absolute value function (rectilinear distance problems, forecasting (MAD*), multi- criteria decision-making) 56 x a *mean absolute deviation rd

Properties of the absolute value 57 |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| < 5 therefore -2 8 rd

Trigonometric Functions (forecasting, bin packing problems) 58 Identities: squared relationships reciprocal relations rd a c b

Forecasting with Trig Functions 59 Quadratic trend with seasonal (monthly) effects t = time rd

60 sin = DB = OE, tan = BC, sec = OC, cos = OD = EB, cot = AB, csc = OA

Non-important Functions 61 Hyperbolic and inverse hyperbolic functions Gudermannian function and inverse gudermannian rd

Composite 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: rd

The multi-variable polynomial 63 rd

Inequalities 64 An 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 + c if both sides are multiplied or divided by the same positive number: if a > b, then ca > cb where c > 0 The 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 < 0 rd

More on inequality 65 An absolute inequality is one which is true for all real values: x 2 + 1 > 0 A conditional inequality is one which is true for certain values only: x + 2 > 5 Solution 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 0 For 2 < x < 3, f(x) < 0 For x > 3, f(x) > 0 Therefore X 3 are the solutions rd

An absolute inequality 66 example problem: solve |x 3| < 5 Write: -5 < (x 3) < 5 Conclude -2 < x < 8 for x > 3, (x - 3) x < 8 for x 3, -(x - 3) x -2 Therefore, -2 < x < 8 rd

An important multi-valued function ( Euclidean distance problems, constrained optimization) 67 x y Pythagorean theorem x y r rd

Implicit and Inverse Functions 68 implicit function explicit function inverse function rd

Inverse Functions 69 Inverse 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)/2 rd

The Devils Curve 70 y 4 - x 4 + ay 2 + bx 2 = 0 An implicit relationship that is not single-valued rd

Symmetry 71 f(x, y) = 4x 2 + 9y 2 = 36 (-x, y)(x, y) (-x, -y)(x, -y) rd

Multiplying Polynomials 72 Example 6 Find the product (2t -3)(5t 3 + 3t -1) (poly*poly #(t -3 2) #(t -1 3 0 5)) #(T 3 -11 6 -15 10) ~ 3 11t + 6t 2 15t 3 +10t 4 (poly^n #(x -3 2) 5) #(X -243 810 -1080 720 -240 32) = -243 + 810x -1080x 2 + 720x 3 -240x 4 +32x 5 rd

Adding/Subtracting Polynomials 73 (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) rd

Create Poly with Roots 74 (my-make-poly '(1 2 3)) (1 -6 11 -6) p(x) = x 3 - 6x 2 +11x - 6 (cubic 1 -6 11 -6) (3 2 1) Short Division 1 -6 11 -6 |3 3 -9 6 1 -3 2 0 |2 (quadratic 1 -3 2) (2 1) 2 -2 1 -1 0 |1 1 1 0 rd

Binary and Decimal Base Numbers 75 2 3 2 2 2 1 2 0 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 = 11 1 1 0 0 0 = 16 + 8 = 24 10 3 10 2 10 1 10 0 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) = 985 rd

Base Numbers rd 76 Write 246 in base 7 7 2 7 1 7 0 49 7 1 5 0 1 7 into 246 = 35 Remainder 1 7 into 35 = 5 R 0 7 into 5 = 0 R 5 Try 221 in base 3 73 R 2, 24 R 1, 8 R 0, 2 R 2, 0 R 2 22012

Convert 324d to Base 2 rd 77 324 10 = 101000100 2 162 0 81 0 40 1 20 0 10 0 5 0 2 1 1 0 0 1

Base Arithmetic rd 78 For what bases does the number 121 b represent a square number? Check 1331 b. b 2 + 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 = 100 in base 33 121 is 1089 + 66 + 1 = 1156 = 34 2

Quadratic Equation in Base 5 rd 79 Solve x 2 + 3x + 2 = 0 in Base 5 where you have the integers {0 1 2 3 4} 3, 4 are the roots. Check.

Base ? rd 80 2 3 5 11 15 21 25 ?

rd 81 Subtraction, 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 15 11 2D 45 (- #xD9 #xAC) 2D AE 16 + 76 8 = _________ 4 236 3230

Irrational Number rd 82 Can 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 Roots rd 83

Logarithm/Exponential Equation rd 84 Solve x + 3e 2y - 8 = 0 for y in terms of x: e 2y = (8 x)/3 2y Ln e = Ln((8 x)/3) y = (1/2) Ln((8 x)/3)

Six steps to solving word problems 85 1. Picture the Problem Try to visualize the problem. Draw a diagram showing as much of the given information as possible, including the unknown. 2. Understand the Words Look up the meanings of unfamiliar words in a dictionary, handbook, or textbook. 3. Identify the Unknown(s) and the constants Be sure that you know exactly what is to be found in a particular problem 4. Summarize and write in mathematical form what is given 5. Estimate the Answer It 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. 6. Write and Solve the Equation(s) The unknown quantity must now be related to the given quantities by means of equation(s). rd

First word problem 86 In 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 shift Then 3x = number of employees on the day shift And (x + 2) number of employees on the swing shift x + 3x + (x + 2) = 102 5x = 100 x = 20 on the night shift 3x = 60 on the day shift (x + 2) = 22 on the swing shift rd

Financial Problem 87 A 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 - 32,450 Her tax is 28% of that amount, plus $4867, so tax = 4867 + 0.28(x - 32,450) = 7285 Solving for x, we get x = $41,086 rd

Mixture Problems 88 From 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 40L amount of lead removed = 0.5(20.0) =10.0 kg amount of lead added = 30.0 kg final amount of lead = 50.0 - 10.0 + 30.0 = 70.0 kg rd

More Mixture Problems 89 How 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 = 3.25 The 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 0.0525x The 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% steel 3.25 * x = 1.48 t of 2.84% steel rd

Sequences rd 90

Sequences & Series rd 91 A 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 + 27 Series are either finite or infinite; convergent or divergent

Arithmetic Sequences rd 92 Complete the sequences at the * a) 2 7 12 17 * *b) 5 13 21 * * c) 11 15 * 23 * d) * * 20 29 38e) 4 * 18 * 32 f) * 33 * 65 * g) 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)

Arithmetic Sequences rd 93 a)The 100 th term of 2 5 8 11 14 * * * is ____. ans. 299 b)b) The 20 th term of 11 15 19 23 * * * is ____. ans. 87 c)Find 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)/2

Arithmetic Series rd 94 Sum S n of a finite arithmetic series is given by S n = n(a 1 + a n )/2 Example: 2 + 4 + 6 + 8 +... + 100 = 50(2 + 100)/2 = 2550; where n = 100/2 = 50; a 1 = 2; a n = 100 1 + 5/3 + 7/3 +... + 201 =. 1 + (n 1)(2/3) = 201 => n = 301 terms => S n = 301(1 + 201)/2 = 30,401

Harmonic Series rd 95 Arithmetic sequence 1 4 7 10 13 16 19 22 Reciprocals: 1 1/4 1/7 1/10 1/13 1/16 1/19 1/22 is 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))))) 2

Harmonic Series rd 96 Find the 36 th term of the series 1 + 1/4 + 1/7 + 1/10 + 1/13 + The arithmetic series is 1 4 7 10 13 and the 36 th term is 1 + 35*3 = 106 => 1/106.

Harmonic Series rd 97 A cyclist travels from A to B at 40 mph and returns at 60 mph. The average speed for the round trip is. a) 48 b) 49 c) 50 d) 51 e) none of these (1 / [(1/40 + 1/60)/ 2] = 48 Apply sensitivity analysis to explain why.

Geometric Series rd 98 Sum S n of a geometric series of n terms is given by S n = Find the sum of the geometric series 1 4 16 64 256 1024. Sum = (1 4 * 1024)/(1 4) = 1365 Find the sum of the geometric series 3 18 108... 839,808. (3 6 * 839,808)/(1 6) = 1,007,769

Geometric Series rd 99 Find 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 A

Sequences rd 100 1. 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, 1/9 Look at reciprocals 1 3 5 7 9 General term is 1/(2n 1)

Sequences (continuing) rd 101 1. 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 ?

Triangle Inequality rd 102 |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 Currency rd 103 Bills: 1 2 5 10 20 50 100 500 1,000 5,000 10,000 100,000 Find a geometric sequence of bills with common ratio 10. 1 100 10,000 100,000

Pyramid Scheme rd 104

Fibonacci Sequence Recursive up a staircase one or two steps at a time with n steps n # of ways 1 1 22: 1, 2 33: 111,12, 21 45: 1111, 112, 121, 211, 22 f n = f n-1 + f n-2 For n = 5 steps, take 1 step 4 5 ways 11111,1112,1121,1211,122 thereafter or 2 steps 3 3 ways 2111, 212, 221 thereafter yielding 5 + 3 = 8 rd 105

Fibonacci Sequence rd 106 (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?

Intelligence Testing rd 107 1. 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)]

Numerology rd 108 How 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 + + 2 63 = (1 2 64 ) / (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.

Differential Equation for e rd 109 y y = 0 Assume y = e x = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + a 4 x 4 + a 5 x 5 +.. Then y = a 1 + 2 a 2 x + 3 a 3 x 2 + 4 a 4 x 3 + 5 a 5 x 4 +... and y(0) = 1 => a 0 = 1 y(0) = 1 => a 1 = 1 y(0) = 1 => 2a 2 = 1 => a 2 y(0) = 1 => 6a 3 = 1 => a 3 = 1/6 e x = 1 + x + x 2 /2! + x 3 /3! +

rd 110 Infinite Series Two 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.

Pythagorean Triples rd 111 mnm 2 n 2 2mnm 2 + n 2 32 512 13 61 3512 37 65 1160 61 7 6 1384 85 8 7 15112 113 6011 34791320 3721 = 61 2 8413 68872184 7225 = 85 2 10 6 64 = 4 3 120 136 6 3 27 = 3 3 36 45 Try one yourself by picking an m > than n.

Asking for a Raise rd 112 Would you rather receive a raise in salary of $300 every 6 months or $1000 every year?

1000 Lockers rd 113 There 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 Length rd 114 Express the length L of a chord of circle with radius r as a function of x being the distance from the center. L = 2(r 2 - x 2 ) 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 x

Random What 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 C Which answer is correct? Both are mathematically correct. What does "at random" mean? rd 115 (comb 2 2)

Radioactive Decay rd 116 N = N 0 e - t for t in days Given 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? N 0 = 100e -0.062 * 0 = 100 N/N 0 = = e -0.062t Solve for t to get 11.1788 days as the half life. N = 100e -0.062 * 11.1788 = 50 N 9 = 100e -0.062 * 9 = 57.235 mg Sanity check: As 9 is less than half life, expect more than 50 mg.

Rationalizing Denominator rd 117 Rationalize 1/2 1/2 Multiply numerator and denominator by 2 1/2 to get 2 1/2 /2

Spurious Roots rd 118 x 4 + (x 2) 1/2 = 0 (4 x) 2 = x 2 16 8x + x 2 = 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 Roots rd 119 (5x 2 + 10x 6) = 2x + 3 5x 2 + 10x 6 = 4x 2 +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 Equations rd 120 (ln x) 2 - 2 ln x - 3 = 0 Solve for x Let y = ln x Then y 2 2y 3= 0 or (y - 3)(y + 1) = 0 ln x = 3; ln x = -1 x = e 3 ; e -1

numbers, bases, algebra, functions, equations and other calculus concepts 1 enm 503 fundamentals rd

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