#### Zeros of cubic functions

Zeros of Cubic Functions I am going to investigate the zeros of a cubic function. Zeros of functions are in other words roots of functions. A cubic function might have a root, two roots or three roots. An easy way to find the roots of a function is by using the Remainder Theorem, which states that a is a root of if and only if. I will make a very god use of this Theorem troughout the essay. My mission is to find the equation of the tangent lines to the average of two of the three roots, by taking the roots two at a time. Then to find where the tangent line intersect the curve again in order to be able to state a conjecture concerning the roots of the cubic function and the tangent line at the average value of these roots. Let us consider the cubic function and take a look at the graph of it: * Window values: First I will state the roots of the function as follow * -3.0 * -1.5 * 1.5 And then prove this using the Remainder Theorem: is a root of Substituting x with the values of the roots: * * * Verification on the calculator : The next step will be to find the equation of the tangent line of two of the three roots. The first thing to do when finding the equation of a tangent line is to find the slope (gradient). Hence to find the derivative of the function and Second is to find the average of two of the roots using the formula, (a and b being the

#### Math IB portfolio assignment - MATRICES

Ankit Shahi Investigating Matrix Binomials Introduction to Matrix Binomials Matrix Binomials can be defined as a type of a 2 by 2 matrix. Generally speaking, matrix binomials come in the form . These matrix binomials can be defined as the sum of two component matrices. One component should be known as the positive matrix. All elements within the positive matrix have the same positive value. The other part should be called as the negative matrix. All elements within the negative matrix have the same magnitude but the top right and bottom left elements have a negative value. The overall goal of this project is to investigate the properties of these matrix binomials in relation to its positive and negative matrix components. The first step would be investigating the positive and negative matrix components separately as they are the simplest components. We shall begin by defining X and Y as the simplest positive and negative matrices respectively and finding their general expressions. Let From these experimentations with the matrix X, we notice a clear pattern. In each consecutive matrix, the values of all four elements are twice as great as the values within the previous matrix. Therefore, as the power of X is increased by one, the values of all the elements within the matrix are multiplied by two. This trend is understandable since the process of matrix multiplication

#### Math Project Infinite Surds

2008 Project 1 Regina Nieuwenburg Pages: 18 [REGINA NIEUWENBURG - MATHS SL TYPE 1] Title Page Infinitive Surds Regina Nieuwenburg Maths SL Type 1 Candidate Number: I.B. Contents Page Title Page Page 2 Contents Page Page 3 Introduction Page 4 Terms, scopes and limitations Page 5 Investigation: * The formula of an+1 in terms of an Page 6 * General statement that represents all the values of K for which the expression is an integer Page 14 Conclusion Page 17 Bibliography Page 18 Introduction In this project I will investigate the infinite surds. A surd is a number that cannot be changed into a fraction. They go on infinitely without any pattern. They are usually a square root of a number. My project will include two formulas. One is the formula of an+1 in terms of an. The second formula will represent the general statement that represents all the values of K for which the expression is an integer. Terms, scopes and limitations Un = Value for each term of 1n n = Term number a = The same meaning as Un et cetera. Un = Actual number for term (n) given. K = The same meaning as Un et cetera X = Can be any number V = Value for each term of 1 Un

#### IB Math Portfolio- Topic 1

International Baccalaureate Math Methods SL Portfolio 1: Logarithms December 02, 2009 The intent of this portfolio is to explore, investigate, and model the patterns found in logarithmic functions. In considering the first logarithm in the first sequence: It is important to know what it represents. By definition, a logarithm is the inverse of an exponential function. Therefore, when: The next expression in the series is: By definition: This expression is more difficult to solve as easily. However, if 8 is seen instead as: The expression becomes easier to solve There is a pattern in the bases of each consecutive expression. In the first set, the base is defined as: where n is the nth term in the sequence. In considering the whole sequence: Sequence 1: Term 1 Term 2 Term 3 Term 4 Term 5 So, in term 5, the base is Therefore, in terms 6 and 7 of all sequences, the value n must be equal to 6 and 7 respectively Term 6 Term 7 The graph below represents the resulting values in sequence 1 as the exponent of the base increases. Notice, the values of y never actually reach 0 because as the base increases, the smaller and closer to 1 the result must be in order to fit the expression. The same can be done with the other sequences: It is shown by this final sequence that the exponent of the base is increasing with direct correlation to the number of the

#### Derivative of Sine Functions

Derivative of Sine Functions Question 1 Investigate the derivative of the function f(x) =sinx. a) Graph the function f(x) =sinx. For-2x2 y=f(x). Let f(x)=sinx ,-2x2 By sketching the graph we get: Figure 1 : graph of function f(x) =sinx Figure 1 reveals the range of the function f(x) =sinx. is [-1,1]. b) Based on this graph, describe as carefully and fully as you can, the behabiour of the gradient of the function on the given domain. The gradient of each point on the curve is valued as the gradient of the tangent line of the point. According to the curve in figure 1, the behaviour of the gradient of the function indicates the following characteristics: ·The line of the tangent become flatter and flatter as the points move from left to right within -2 to -,- to -,0to , to . The line of the tangent become more and more precipitous as the points move from left to right within - to -,- to 0, to , to 2. ·In the domain of : [-2,-[, ]-,[,],2]the gradient is positive. The gradient is negative in the domain of::]-,-[, ], [. ·At the point when x equals to -,-,and ,the gradient is obviously 0. c) Use your Graphics Calculator (GC) to find numerical values of the gradient of the function at every /4 unit. Sketch your findings on a graph. . The numerical values (in 3 significant figures) of the gradient of the function at every /4 unit is shown in the table below: X

#### Math Type I SL Matrices

Pure 20 IB Portfolio Project Matrix Binomials Dan Stoica Henry Wise Wood High School Calgary, Alberta Mr. Bedford Table of Contents Title Page...........................................................................................................Page 1 Table of Contents............................................................................................Page 2 Mathematical Investigation Sheet...........................................................Page 3 Calculator/Introduction..............................................................................Page 4 Determining X and Y.....................................................................................Page 5 Determining X and Y continued...............................................................Page 6 Determining A and B.....................................................................................Page 7 Determining A and B continued...............................................................Page 8 Determining A and B continued...............................................................Page 9 Determining A and B continued............................................................Page 10 Determining A and B continued............................................................Page 11 Determining Expressions for (A + B)..................................................Page

#### The comparison between the percentage of ethnic students and the average rent per week in British universities.

The comparison between the percentage of ethnic students and the average rent per week in British universities. Introduction An investigation to clarify whether when the average rent rate is higher at a British university the percentage of Ethnic students becomes higher. The selection of British universities occurred by taking 30 successive universities, from a random starting point (with aid from a random number generator) with the Average Rent per week and Percentage of Ethnic students recorded. This selection process has made the investigation more valid by giving a real spectrum of university standards. Due to the fact that increasingly more ethnic students apply to university and that many immigrants will usually invest more in education, including monetary investments, therefore it is logical the hypothesis adhere to this. A correlation between the average rent rate and the percentage of ethnic students is high. In order to discover the relationship between ethnic students and universities with different rates of rent all the data must show a strong link to each other, performing tests is an effective method for determining this. Plan To begin with, to find data suitable for statistical testing, Push.com is seen to have a realistic view of universities due to the involvement of more student input and a wider spectrum of university life is considered¹. The choice

#### Investigating the Koch Snowflake

Mathematics HL Portfolio Omar Nahhas. Class 12 "IB" (C). The Koch snowflake is also known as the Koch island, which was first described by Helge von Koch in 1904. Its building starts with an equilateral triangle, removing the inner third of each side, building another equilateral triangle with no base at the location where the side was removed, and then repeating the process indefinitely. The first three stages are illustrated in the figure below Each step in the process is the repeating of the previous step hence it is called iteration. If we let Nn = the number of sides, ln = the length of a single side, Pn = the length of the perimeter, and An = the area of the snowflake, all at nth stage, we shall get the following table for the 1st three iterations. Table no.1: the value of Nn, ln, Pn, and An, at the stage zero and the following three stages. n Nn ln Pn An 0 3 3 (V3)/4 2 /3 4 (V3)/3 2 48 /9 48/9 0(V3)/27 3 92 /27 92/27 94(V3)/243 Note: assume that the initial side length is 1. We can see from the above table that the number of sides isn multiplied by four at each iteration. The length of each side is divided by 3 in each step, thus it is 1/3 the length of the same side in the previous step. As for the perimeter, the perimeter equals the number of sides multiplied by the length of a single side, hence that we have the above values where that

#### Investigating a sequence of numbers

Type 1: Investigating a sequence of Numbers This is an investigation about series and sequences involving permutations. From a given series, I find the pattern of numbers that result from different values and use graphs to conjecture an expression from the series. By using mathematical induction and direct proof, I prove the general terms that I derived for the series. Part 1: The sequence of numbers is defined by , , , ... From the pattern of different values of n in above, I conclude that! Part 2: Let If n=1 = were ! ! If n=2 were !, ! + (2 If n=3 were !, !,! + (2 Part 3: From Part 2, I know that: ! To conjecture an expression of , I first organize the results that are derived in Part 2 to discover a pattern in the value of as n increases. Table 1.1: n 2 4 5 3 8 23 - - - - - - n ! ? The same results of from Table 1.1 can be represented as follows: Table 1.2 : n 2-1= (1+1)!=1 2 6-1= (2+1)! =5 3 24-1= (3+1)! =23 - - - - n From the patterns exhibited in Table 1.2, I notice that which is further illustrated in Graph 1.1. In Graph 1.1, I plotted the graph of for the first three values (represented by green dots) and I assumed that (n+1)! will lead to a conjecture forand plotted its values for n=1,2,3 (represented by red dots) . From the two graphs, I notice that (n+1)! is exactly 1 unit above for all three points

#### math portfolio type 1

IB Mathematics SL Portfolio Type I Matrix Powers Done by: Bassam Al-Nawaiseh IB II * Introduction: Matrices are rectangular tables of numbers or any algebraic quantities that can be added or multiplied in a specific arrangement. A matrix is a block of numbers that consists of columns and rows used to represent raw data, store information and to perform certain mathematical operations. The aim of this portfolio is to find general formulas for matrices in the form . of Each set of matrices will have a trend in which a general formula for each example is deduced. * Method 1: Consider the matrix M = when k = 1. Table 1: Represents the trend in matrix M = as n is changed in each trial. Power Matrix n = 1 n = 2 n = 3 n = 4 n = 5 n = 10 n = 20 Matrix M is a 2 x 2 square matrix which have an identity. As n changes the zero patterns is not affected while the 2 is affected. 2n is raised to the power of n. When n =1, 21 = 2, when n = 2, 2² = 4 and when n = 3, 2³ = 8 and so on. So as a conclusion, Mn = * Method 2 Consider the matrices P = and S = Table 2.1: Represents matrix P as the power n is increased by one for each trial. Power Matrix / Pn n =1 = 20 = n =2 2 = 21 = n = 3 3 = 22 = n = 4 4= 23 = n = 5 5 = 24 = As the power n of the matrix is increased by one the scalar is doubled.