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International Baccalaureate: Maths
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A semitone is the smallest interval in western classical music. Each tone inclusive produces a unique sound. As a result, each semitone has a representational letter in an octave. The octave the note is located in is represented by a subscript number, for example, C4. On the piano, the scale range of the white notes are C, D, E, F, G, A, and B going from the left to the right. The black notes are named according to what key the music is in and what white note they are related to.
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Background Epidemiology Epidemiology is defined as a branch of medicine that deals with the study of the incidence, distribution and possible control of diseases and other factors relating to health. Within epidemiology, mathematics is used to produce epidemic models in order to predict the potential spread of a disease throughout a population. The use of mathematical modelling in epidemiology is particularly useful when a new infectious disease is thought to have the potential to cause an epidemic. Models in epidemiology have predicted the spread of diseases such as whooping cough, west Nile virus infections and influenza viruses.
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Table 1: The relationship between the # of guides and the distance from tip (cm) Guide number 1 2 3 4 5 6 7 8 Distance from tip 10 23 38 55 74 96 120 149 Part A: Variables and Parameter/Constraints Before a mathematical model can be made, the variables for the model must first be chosen. The Distance from tip (cm) is dependent on the guide number and therefore is the dependent variable and should have the variable of Y and be on the Y axis in a graph. This means that the Guide number (from tip)
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It is also more complex. It tells me to consider two functions, which are f(x)= xn and f(x)=aXn. then I will still need to find an answer for f1(x) for the function f(x)=aXn and then discuss the scope of my analysis. To make my life easier I will divide this investigation in parts. For every part, I will analyse a variety of cases, and come up with a conclusion at the end of each case and part. Consequently, after watching at all my results, I will give a final formula to this investigation.
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times of Olympic before year 1932 will become negative number, so I decided to simplify the values from the start year of Olympic, 1896, as my starting point, 0. Then my graph will be drawn by using the data: Year 36 40 52 56 60 64 68 72 76 80 84 Height (cm) 197 203 198 204 212 216 218 224 223 225 236 Here is a graph showing the data; Figure 1 winning gold medal height in men?s high jump from1932 to 1980 These smaller values for year will make the constant values lesser in equations I am going to investigate.
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If the triangles are in the same shape with the same three (in fact, two) angles, then they are called similar triangles in different ratio. 1. The Pythagoras’s theorem Sides which are opposite to the angles are labeled using small alphabets of the angles. When angle B is 90°, which means the triangle is a right triangle, then a2 +c2=b2. 1. Cosine rule This rule is used when we want to know the angles of a triangle. However, there is a limitation for using the rule, which is that the rule is able to be used only all three sides of a triangle are known.
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Example of a search space The problem is that the search can be very complicated. One does not know where to look for the solution and where to start. There are many methods, how to find some suitable solution (i.e. not necessarily the best solution), for example hill climbing, tabu search, simulated annealing and genetic algorithm. The solution found by these methods is often considered as a good solution, because it is not often possible to prove what the real optimum is.
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By this theory, the sixth numerator should be 21. If we try to find a relationship between the numerators, we see that the difference between each subsequent numerator is 1 in each row starting with 2. During my research of Pascal?s triangle, I also came across nCr equation, where ?n? is the row number and ?r? is the element number For example, 3C2 = (3!) / [( 2!) x (3 - 2)!] = 3 This can be used to find any number on the Pascal?s Triangle considering that the 1 on top of the triangle is represented by n =
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Let us proceed and use various values for and to generate different pairs of functions and , to observe their graphical relationship. has a vertex with coordinates and positive concavity. By definition, its shadow function will have negative concavity, but the same vertex: with vertex . Graphed: As we can see, is a reflection of by a line passing through the vertex and parallel to the x-axis, which we will call , the shadow-generating function. Choosing different values for and , with vertex coordinates ; .
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Rewriting the first equation as and substituting for x in the second equation gives: . This we can solve for y: Substituting this y-value into the second equation gives: Solving for x gives: Therefore the only solution satisfying both equations is x=-1 and y=2. This means that the point (-1;2) is the only point that lies on both lines. It is the point of intersection. ________________ Let?s consider a second system, where the coefficients also follow an AS but with different first terms and different common differences: (first term: 1, common difference: +2) (first term: -2, common difference: -2)
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Circles Portfolio. The purpose of this assignment is to investigate several positions of points in three intersecting circles in order to discover a general statement
In diagram 1: Triangle OPA Triangle OP’A aâ=OP=2 aâ=OP’=? o= PA=2 oâ=P’A=1 p=p’=AO=1 p’=p=AO=1 To find OP’ with the cosine function we must know the angle measurement of â¢OAP’ (the opposite angle). I will find â¢OAP’ through different steps involving both triangles. I will begin with finding â¦ OAP: aâ²=o²+p²-2·o·p· A=cosË¹ () A= cosË¹ () A=75.52áµ Because is isosceles, â¢OAP=â¢AOP=75.52áµ. â¢AOP is shared by both triangles, which means that â¢OP’A is 75.52áµ as well. From this we can assume that â¢OAP’=180áµ-2·75.52áµ=28.96áµ=Aâ Now I can easily find the length of OP’: aâ²= (OP’)
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The fuel efficiency f1 of Arwa’s Toyota Etios is 18km/l where as the fuel efficiency f2 of Bao’s i10 is 20km/l. (Fuel Efficiency of a car is f km/l in general). The capacity of fuel tank t of both cars is equal to 35 liters. Therefore number of kilometers Arwa travels after which he has to refuel =(t-5) ×f1 =30l ×18km/l =540km (5l is for reserve) Therefore number of kilometers Bao travels after which he has to refuel =(t-5) ×f2 =600km (5l is for reserve)
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Considering Data Table 1 Gold metal heights achieved at various Olympic Games. Year 1932 1936 1948 1952 1956 1960 1964 1968 1972 1976 1980 Height (cm) 197 203 198 204 212 216 218 224 223 225 236 Table 1 Through the use of Spreadsheet, a table is produced to represent the data points in a chart In this case, time is the independent variable and the height is the dependant variable in Table 1. The height of the Gold medalist is measured in the scientific unit, centimetre (cm).
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the range is . The maximum height that can be jumped is set at 250cm because the highest achieved height by a human is 245cm by Javier Sotomayor. As the graph suggests that there is a positive relationship between the height and the year. Functions which could model this situation where it increases at a decreasing rate include reciprocal, logarithm, exponential to the power of a number between 1-0 etc. A reasonable polynomial function which can represent the data is a cubic Since a cubic has four variables, four points are necessary to solve for the co-efficient.
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5 + (-5) = 0 Non-real calculation Different irrational number 0.7153 0.7153 Same irrational number 0.4306 0.4306 Number is 0 Undefined Number is 0 0 It is important to take note of the scope and limitations of the general statement, to find out in what condition the general statement does and does not work. = The base cannot be negative, must be above zero, but cannot equal to 1.
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Looking at Pascal?s Triangle, we can then assume the numerator for the sixth row to be 21. The seventh row can be assumed to be 28 and the eighth row can be assumed to be 36 and so forth. To prove these assumptions, Table 1 below will be used to analyse the pattern of the numerators. Table 1 Row Numerator 1st Difference 2nd Difference 1 1 2 2 3 1 3 3 6 1 4 4 10 1 5 5 15 1 In Table 1, the difference between each following numerator increases by 1 during each trial, starting from row 2.
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In this portfolio, I will determine the general sequence tn with different values of variables to find the formula to count the sum of the infinite sequence.
value of n increases as well, but it does not exceed 3, thus the greatest value of this relationship will be 3 and therefore the domain of this relationship is , as n approaches . 1. From the above calculations can be concluded that the greatest value of these sequences is a, however in order to support and check this I will consider general sequence x=1 and, again, calculate for the first ten terms (). However this time I am going to use different values of such as a=5 and a=7.
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The population in millions is represented by the variable ?y?. Parameters The variables are time and population. Time (represented by x) cannot be negative and cannot decrease because it represents a unit of time. It can however increase infinitely. The variable population (represented by y) cannot be negative as population cannot go below 0. In this case the population also cannot increase infinitely as China will not be able to handle so many people. (Maximum carrying capacity of a country) I will now list the values of ?x? and ?y?.
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Calculator used: Texas Instruments-84 Silver Edition 1. Click 2. Select and press 3. I will now put in the values for the triangle term# in List 1 and the number of dots in List 2. 1. Then click again and select 2. Select and press 3. Press and then selecting and then. 4. On pressing We will see the following screen: 1. Substitute with . General statement of the Triangular number Stellar number images upto stage: Stellar number images upto stage I now put the above diagrams in numerical form: 1 1 2 13 3 37 4 73 5 121 I will now attempt to use quadratic regression to find out the general statement for the 6 stellar number at stage: Calculator used: Texas Instruments-84 Silver Edition 1)
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Math IA Type 1 Circles. The aim of this task is to investigate the positions of points in intersecting circles.
is the segment of line between the center of circle C1 and the center of circle C2. As stated above, O is the center of C1 and OP is the radius of C2; therefore C2 always intersects the center of C1. As well, C1 will always intersect the center of C3, since OA is the radius of circle C1 and point A is the center of circle C3. From the diagram, an isosceles triangle âAOP is identified. Since P is the center of C2 and Point A is a point of intersection of circles C1 and C2, AP is the radius of C2.
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Russell’s paradox shows an inconsistency in one of the axioms of the set theory. This example shows how mathematics fails the coherent truth test. The coherent truth test states that the premise for deductive reasoning must be logically consistent. Russell’s paradox gives an example of the incompleteness of mathematics. Gödel’s incompleteness theorem says that any axiomatic system cannot be complete and consistent at the same time. Using deductive reasoning, we may reach a point where a mathematical statement can neither be proved nor disproved. This is a simpler interpretation of the incompleteness theorem.
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The 1?s in this case are substituted for an equal fraction. Figure 2: The difference of the numerators between each row of Lascap?s fractions In this case the difference of the denominators is 2, 3, 4, and 5 going down the rows. Based on these numbers only, the pattern is a sum notation. Which we can calculate, the common difference increases by each row and the elements have no effect on the difference so they are not included in the equation. Table 1: The numerators of the first five rows of Lascap?s fractions n Numerator 1 1 2 3 (1+2)
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Lascap Fractions. In Lacsap's Fractions, when looking for a general pattern for the numerator, it can be noted that it does not increase linearly but exponentially
N(n+1)-Nn represents the equation for the graph that increases more evenly as the sequence advances. Using excel to graph the points and loggerpro to generate an equation, the general statement for finding the numerator N=0.5n2+0.5n, n having to be greater than 0. To check the validity of the equation sample equations were used: Sample Equation: 5th Row: N=0.5(5)2+0.5(5)=15 Patterns Recognized: The first pattern that could be recognized is that the difference between the numerators of the ensuing rows is 1 more than the change between the previous numerator of the two consecutive rows.
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LACSAP Fractions. The aim of this portfolio is to discover an equation which suits the pattern of Lacsaps fraction using technology
This means the numerator is increasing at a higher ratio than the row number. The sequences of the numerators are 3, 6, 10, and 15. When analyzed, we can see that the numerators are increasing one more than the previous term. 3 is increased by three to equal 6. 6 is increased by 4 to equal 10 and so forth. From this pattern, we can formulate an equation using the row number as a variable to find the numerator. We use the row number because we can observe that to get from row number, 2, to the numerator of the row, 3, we must multiply the row number by .
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