#### infinite surds

The Sparkling Night There was once a panda named Kandee. Kandee had the best life ever. The best parents, the best friends, and most important of all the best boyfriend. Her boyfriend was Jimmy the penguin. They were both madly in love. The only problem was both Jimmy and Kandee's parents did not know about them dating. How could they? Penguins and pandas hated each other. One day Jimmy walked Kandee home from school and Kandee's dad saw them holding hands. He went crazy and ran after Jimmy. Luckily Jimmy was too fast and Kandee's dad didn't catch him. Kandee explained that the teacher told her to walk the poor blind penguin home. The funny thing is her dad actually believed her even though it looked like he knew what direction to run in. From then on Jimmy had to act blind in order to come to their house everyday. Kandee's mom and dad actually started liking Jimmy. Jimmy had a great personality and great sense of style. She was foolish to believe their relationship could actually workout. Kandee and Jimmy worked on homework and projects everyday after school. Everything was going well until one day Jimmy decided to stay after school. That day Kandee had to walk home alone. While she was walking a big, bad wolf jumped out of the forest and attacked poor Kandee. The crazy thing is that Jimmy tried to catch up with Kandee and actually saw the wolf walking away. Jimmy tried

#### Maths SL Portfolio - Parallels and Parallelograms

IB Standard Level Maths: Portfolio Piece 1 Parallels and Parallelograms Table of Results for 4 transversals: Transversals Number of Parallelograms Parallelograms Diagram 4 6 A1, A2, A3 A1 ? A2, A2 ? A3 A1 ? A2 ? A3 Table of Results for 5, 6 and 7 transversals: 5 0 A1, A2, A3, A4 A1 ? A2, A2 ? A3, A3 ? A4 A1 ? A2 ? A3 , A2 ? A3 ? A4 A1 ? A2 ? A3 ? A4 6 5 A1, A2, A3, A4, A5 A1 ? A2, A2 ? A3, A3 ? A4, A4 ? A5 A1 ? A2 ? A3 , A2 ? A3 ? A4, A3 ? A4 ? A5 A1 ? A2 ? A3 ? A4, A2 ? A3 ? A4 ? A5 A1 ? A2 ? A3 ? A4 ? A5 7 21 A1, A2, A3, A4, A5, A6 A1 ? A2, A2 ? A3, A3 ? A4, A4 ? A5, A5 ?A6 A1 ? A2 ? A3 , A2 ? A3 ? A4, A3 ? A4 ? A5, A4 ?A5 ?A6 A1 ? A2 ? A3 ? A4, A2 ? A3 ? A4 ? A5, A3 ? A4 ? A5 ? A6 A1 ? A2 ? A3 ? A4 ? A5, A2 ? A3 ? A4 ? A5 ? A6 A1 ? A2 ? A3 ? A4 ? A5 ? A6 --> Let n = number of transversals and let p = number of parallelograms Transversals (n) Parallelograms (p) 2 3 3 (1 + 2) 4 6 (1 + 2 + 3) 5 0 (1 + 2 + 3 + 4) 6 5 (1 + 2 + 3 + 4 + 5) 7 21 (1 + 2 + 3 + 4 + 5 + 6) n + 2 + ... + (n - 1) Use of Technology: Using the TI - 84 Plus, press STAT --> 1: Edit. Type in L1, L2: (2, 1) (3, 3) (4, 6) ...etc. Using Quadreg, L1, L2, rule --> y = 0.5 x 2 - 0.5 x + 0 = 1/2 x2 - 1/2 x = (x2 - x) ÷ 2 --> If n is the number of transversals and p is the number of parallelograms, then the rule is: General statement: If

#### Modelling Probabilities in Tennis. In this investigation I shall examine the possibilities for modelling the probabilities in tennis matches of varying complexity.

Modelling Probabilities in Tennis Introduction: In this investigation I shall examine the possibilities for modelling the probabilities in tennis matches of varying complexity. I will begin by assuming a very simple game of ten points, played by players of a consistent known strength. I will then expand this into games following the real-life rules of tennis more closely, to see how these affect the probabilities involved. I will examine how any results may be generalized to players of different strengths, and how these strengths affect the odds of such players over matches. A simple model: Let us assume two players Adam and Ben (who will be referred to as A and B), with fixed probability of scoring a given point against each other. Let A win of points, and B win of points. We can begin by simulating a 10-point game. As the probabilities are consistent, and there are only two possibilities for each point, the game is a series of Bernoulli trials and can be modelled with a binomial distribution. Let X denote the number of points scored by A. We can now state that: and therefore that: . We can hence easily calculate the probability distribution for all possibly values of X: x P(X=x) 0 0.000017 0.000339 2 0.003048 3 0.016258 4 0.056902 5 0.136565 6 0.227608 7 0.260123 8 0.195092 9 0.086708 0 0.017342 This can be presented as a histogram: This

#### IB Pre-Calculus Logarithm Bases General Information: Logarithms A logarithm is an exponent and can be described as the exponent needed to produce a certain number.

International Baccalaureate IB Pre-Calculus Portfolio Logarithm Bases January 6, 2009 Student Number: 1208769 C. Leon King High School Andre Elliott January 6, 2009 Dr. Stone IB Pre-Calculus Logarithm Bases General Information: Logarithms A logarithm is an exponent and can be described as the exponent needed to produce a certain number. For Example: 23=8, from this you would say that 3 is the logarithm of 8 with base of 2 (log28). 2 is written as a subscript, and 3 is the exponent to which 2 must be raised to produce 8. The formula or definition that is used in logarithms is: logbx=e, so be=x. So base(b) with exponent(e) produces x. So from the example, 2 is the base(b), 8 is x (the number produced), and the exponent(e) is 3. So, 23=8. Logarithms in Sequences: Introduction Since logarithms can be solved (log28=3) to form numbers, this means logarithms are just another way to represent a number; and since numbers can be in sequences, so can logarithms. Given this, consider the following sequences: ) log2 8, log4 8, log8 8, log16 8, log32 8, ... 2) log3 81, log9 81, log27 81, log81 81, ... 3) log5 25, log25 25, log125 25, log625 25, ... 4) logm mk, logm2 mk, logm3 mk, logm4 mk Sequence 4 is the general statement that is used to reflect the previous sequences. So, consider the first sequence; it starts out with a base(m) of 2, the second term has a base(m)

#### Matrix Binomials IA

Maths Portfolio Standard Level International Baccalaureate Matrix Binomials The main aim of this portfolio is to investigate the matrix binomials and observe and determine a general expression from the patterns that we obtain through the workings. Throughout the project, I shall be using solely matrices of 2 x 2 formations, and investigate the patterns I find. . To begin with, we consider the matrices X = and Y =. The values of these matrices, each raised to the power of 2, 3 and 4 are calculated, as shown below; X2 = X = Y2 = x = X3 = x = and Y3 = x = X4 = x = Y4 = x = It can be observed that all the matrices calculated above are in the form of 2 X 2, they are all square matrices. The corresponding diagonal elements are also observed to be the same. Since the matrices of each nth power can be seen to be the value of 1 less than the nth term, the general expression for the matrix Xn in terms of n is - Xn = And the general expression for Yn is - Yn = Likewise, the values of the matrix (X + Y), raised to the power 2, 3, and 4 is calculated to find its general expression. The matrix: (X + Y) = + = So, (X + Y) 2 = = (X + Y) 3 = x = (X + Y) 4 = x = And from the above, we can infer that the general expression for (X + Y) n is as follows, (X + Y) n = Proof: Taking n as 3, the value is

#### models and graphs relating the Body Mass Index for females to their ages in the US in the year 2000

Anh Nhu Vu IB Mathemathics Standard Level 2008 Maths Coursework This coursework will explore models and graphs relating the Body Mass Index for females to their ages in the US in the year 2000. The ages and the corresponding BMI numbers are variables and as we generate the function that models the behaviour of the graph later on, the parameters are the values of a, b, c and d in the formula of the function. When the data points are plotted on a graph, it is interesting to see what kind of graph these points are forming. The graph BMI appears to resemble the graph of a trigonometric function. If we base on the function y=sinx to develop our model function, the type of function that models the behaviour of the graph is: y1=a×sin [b(x+c)] +d These are the reasons why I chose this type of function: Primarily, the shape of the graph resembles that of the graph of the function y=sinx. However, in comparison to the graph of the function y=sinx, the data points of the graph BMI indicates that the processes of transformation have been done: _Vertical and horizontal stretch (the scales are represented by a and b) _Vertical and horizontal translation (the scales are represented by c and d) Once we have indentified the type of the function that models the behaviour of the graph, it is possible to use algebraic methods to create an equation that

#### The Two Yachts Problem

The Two Yachts Problem Pg. 405 IB Math SL Y2 Yacht A has initial position (-10, 4) and has velocity vector. Yacht B has initial position (3, -13) and has velocity vector. . Explain why the position of each yacht at time t is given by rA = + t and rB = + t - For vector equations, the form is =. ( refers to an initial position and refers to a direction vector.) - Therefore, a vector equation for Yacht A can be written as + t. - A vector equation for Yacht B can be written as + t. 3. The position of B relative to A () is rB - rA = + t, which in coordinates will be. 4. The formula for finding the distance is: d = Therefore, d2 = 169 - 78t + 9t2 + 289 - 136t + 16t2 = 25t2 - 214t + 458 5. d2 is a minimum when t = 4.28 To find the minimum of d2, I set the derivative equal to 0. So, 50t - 214 = 0. Thus t = = 4.28. 6. The time when d is to be a minimum is the same time as when d2 is a minimum, so the closest approach occurs at t = 4.28. So, if I put t = 4.28 into the expression for d is: d = = = = = 0.2 miles

#### Plot a graph for the BMI of different females of different ages in the US in year 2000 and analyse whether it is an accurate source of data.

Body Mass Index Maths Coursework March 2008 By: 12M(2) This maths coursework is based on Body Mass Index. This is a measure of ones body fat; it is calculated by taking one's weight (kg) and dividing it by the square of one's height (m). For this coursework, I have to plot a graph for the BMI of different females of different ages in the US in year 2000 and analyse whether it is an accurate source of data and how it can help me to find other BMI's around the world. Below is my graph showing the BMI of different females of different ages in the US in the year 2000: As shown on the graph, the 'x' values are the age of the females and the 'y' values are their body mass index. The age is measured in Years. When modelling this data, the initial impression is to think that is was an f(x)= x2 graph. However once you notice that it is not a mere parabola but a wave due to the curve that levels off (shown on graph) we can assume it is a periodic function such as a cosine or sine graph. Even though you can use a cosine or a sine graph, I decided to use a cosine graph, as I am more familiar with this type of graph. After inputting the cosine function into autograph software, you would realise that transforming the function would be appropriate so that it can model the graph more accurately. In order for you to do this, you have to use the general formula of f(x)= acos(bx

#### Modeling Polynomial Functions

Modeling Polynomial Functions Polynomial functions are power functions, or sums of two or more power functions. Furthermore, a polynomial function must be made up entirely of nonnegative integer powers. These polynomial functions are commonly used to graph changes in a population or amount over a specified period of time. Derivative functions are functions that represent the slope of an exponential or polynomial function in relation to time. Therefore, they can assist in the calculation of a rate of change at a specific moment in time. To visually see how the derivative of a function relates to the original power function you must graph them both on the same set of axis. To graph the original power function either input the equation into a graphing calculator or calculate the zeros by hand and approximate. In this case, the equation for the polynomial function is: x3+6x2+9x. Graphing the derivative function without using power rule is more complicated in that you must use the equation f '(x)=f(x+0.001)-f(x)/0.001 and plug in the original power function. This equation finds a little bit above and below the point and calculates a close to instantaneous rate of change. This instantaneous rate of change becomes a point on the graph of the derivative at the same time interval as the power function. When the two graphs and graphed on the same set of axis it resembles the

#### Logarithm Bases Math IA

Logarithmic Sequences IA Andrew Cherny Math SL Scott Learned /26/11 In math class, I was given the assignment to evaluate multiple logarithmic sequences to see if any patterns were evident within these sequences. Using logarithmic rules that I previously learned in math class, I was able to discover multiple patterns within each logarithmic expression. To begin, I was given the general logarithmic expression: ,... In order to establish patterns within this general logarithmic expression, I will use multiple examples to help establish a common pattern between all the examples. The first sequence is as followed: ,... Next, the same logarithmic sequence will be evaluated but more in depth, to try and find a common pattern. Note: a pattern is already starting to show. While the numerator stays the same throughout the logarithmic sequence, the denominator increases linearly by one. The pattern is also evident within these next two examples of logarithmic sequences. ) 2) Cleary, the pattern is noticeable as the sequence goes on. Therefore, I was able to find the term for each sequence, writing it in the form where . For the general logarithmic sequence, the term is the following. ..., That being the case, the nth term for the three examples of logarithmic sequences are the following: 1) 2) 3) To justify my answer using technology, I used excel to