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International Baccalaureate: Maths
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Instead of counting squares, using the trapezoidal method, or the midpoint method, we can estimate the value of the area/definite integral on the calculator. To get the value of the area (definite integral) Plug FnInt (Y1, X, A, B) into the graphing calculator: Ex. FnInt (3sin(2x), x, 0, ?/2)= 3 After determining one of the definite integrals for a value of b, we must continue this process to find several points for different values of b to try to find a pattern.
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2 To further clarify how these values were obtained a sample calculation is shown below: , a screenshot highlighting the formula used in Excel to complete the rest is shown in figure 1 below. Figure 1: Showing how Excel was used to calculate Sn If we now take these values and create a graph of n vs. Sn we obtain the result shown in Figure 2 From this graph it is clearly seen that the summation Sn converges on the value 2.
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Mathematics Portfolio. In this portfolio project, the task at hand is to investigate the relationship between G-force and time by developing models that represent the tolerance of human beings to G-forces over time.
On average humans can handle approximately 5 g before losing consciousness. With the data given, I will develop individual functions that model the relationship between the time tolerable for humans versus the various measurements of forward horizontal g-force on a subject, as well as the relationship between time and upward vertical g-force. I will compare hand generated and computer generated functions to see how well the models fit the data, and discuss any limitations to the models. The models will be based on the data: Time (minutes) +Gx (grams) 0.01 35 0.03 28 0.1 20 0.3 15 1 11 3 9 10 6 30 4.5 TIme (minutes)
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It can be seen that when you have a fraction that is smaller than 1 for your value the height of the sine curve ill decrease to the height of . On the other hand, having a bigger fraction than 1 increases the height of the sine curve to the value as can be seen from Graph 1.2. From these two graphs, it can be said that when is bigger than 1 the graph stretches outwards, whereas when is smaller than one the graph will stretch inwards.
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Stellar Numbers Portfolio. In this task I will consider geometric shapes, which lead to special numbers
The complete pattern in this 6-stellar is at follows: 0+1= 1 1+12=13 13+24=37 37+36=73 Therefore, S5 = 73+ 48=121 S6 = 121 + 60= 181 Similarly to the triangular numbers, the 6-stellar sequences uses the same method but this time the number of dots can be found by preceding term added to by the multiples of 12 (as shown in red). Other way to write the pattern is in this way: (1+0(12)), (1+1(12)), (1+3(12)) , (1+6(12)) , ... As soon as I wrote it this way I realized that there is clear relationship between the Triangular numbers and the way the 6-stellar numbers term.
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Hence, I will use Microsoft Excel in order to plot results in a suitable table. The first column will contain the different values of , which come from 1 to 10. The second column will contain the results obtained by replacing each of the values in the form of . And the third column will contain the gradual sum of each of the terms obtained in the second column. For example, the first value of the third column will be added to the second value of the second column giving the second value of the third column, and so on.
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Systems of Linear Equations. Investigate Systems of linear equations where the system constants have well known mathematical patterns.
As we can see in the graph the two equations intersect at a point and that point is the same as the solution we found when we solved the problem algebraically. So the point that the two equations intersect is (-1, 2). Graph Here are some examples using the same pattern: Example 1: x + 3y= 5 x= 5 - 3y x= 5 - 3(2) x= -1 5x - y= -7 5(5 - 3y) - y= -7 25 - 16y= -7 -16y= -32 y=2 y= 2 Example 2: 2x + 6y= 10 x= 5 - 3y x= 5 - 3(2 )
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The Sky Is the Limit Portfolio. In this assignment I will be building a model for the relationship between the winning heights in mens high jump and the years that they took place.
This makes the data slightly more difficult to analyze and could cause some inaccuracies later while plotting a function. Another constraint of the task is that there can easily be outliers in the data simply because of whoever is competing in the Olympics that year. A final constraint of this task is that there is a limited amount of data provided which could once again cause some inaccuracies. In this assignment I will be using two functions: the linear function and the square root function.
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U 0 This data is part of the main data showing just the English Language & Literature GCSE points for each student. I am going to compare the total score for each child against their alphabetical order numbering to see if we get any correlation and to prove if my hypothesis is right or wrong. Scope: National Curriculum Year Export Date : 10/06/2011 Name Entries English Language & Literature FEMALE GENDER MALE GENDER Total score Alphabetic Position Number of Results 181 Ab 9 34 m 34 181 Ab 9 40 m 40 180 An 8 28 m 28 179 Ar
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Math SL Circle Portfolio. The aim of this task is to investigate positions of points in intersecting circles.
It is shown in the diagram below. In ?AOP', lines and have the same length, because both points, O and P' are within the circumference of the circle , which means that and are its radius. Similarly, ?AOP forms another isosceles triangle, because the lines and are both radii of the circle . = r of or = 1. Since the circles are all graphed, they can be given coordinates. For , the coordinates of the point O will be (0, 0), because it lies in the origin of the graph.
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Stopping distances portfolio. In this task, we may develop individual functions that model the relationship between speed and thinking distance, as well as speed and braking distance. We could also develop a model for the relationship between speed and ov
Initial Data Set The table below lists average times for these processes at various speeds. Speed (km/h) Thinking distance (m) Braking distance (m) 32 6 6 48 9 14 64 12 24 80 15 38 96 18 55 112 21 75 Speed vs. Thinking distance As it is visible from the table below when speed doubles, thinking speed doubles, and when speed triples, thinking speed triples. There is a clear linear relationship which can be found from any 2 points, like (64, 12) and (96, 18). Then if we consider m as a stable ratio, so the following relation is found to be true: then and = 0.1875 Speed (km/h)
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Infinite summation portfolio. A series is a sum of terms of a sequence. A finite series, has its first and the last term defined, and the infinite series, or in other words infinite summation
However, the numbers become so small, that they become insignificant, or in other words they are equal to 0. Now, we need to find the sum of Sn : Now, using Excel 2010, let's plot the relation between Sn and n : Looking at the graph, we can notice that Sn increases rapidly at first, and then it evens out when it reaches 2, which seems like an asymptote. The same happens with the terms' values. They decrease rapidly until they reach the 0, which if we plot will seem like its asymptote. Therefore, we can see that both move a maximum of 1 unit away from their first point, and then even out to the mentioned asymptote.
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Infinite Summation - 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.
1.999998 9 1.999998 10 1.999998 From this plot, I see that the values of Sn increase as values of n increase, but don't exceed 2, so the greatest value that Sn can have is 2. Therefore, it suggests about the values of Sn to be in domain Sn 2 as n approaches when x = 1 and a = 2. Now, doing similar as in first part, I am going to consider the sequence where and : Using GDC, I will calculate the sums S0, S1, S2, ..., S10: S0 = t0 = 1 S1 = S0 + t1 =
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Matrix power. The internal assessment will focused on observing patterns of matrix powers which will be the main key to find the general expression of matrix powers.
Explanation: When matrix P is powered by 3 it gives a result of , when matrix P is powered by 4 it gives a result of, and when matrix P is powered by 5 it gives a result of . The pattern shown is that results have a common factor of such as 4 shown in matrix, 8 shown in matrix, and then16 shown in matrix.
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Infinite summation SL Portfolio type I. Concerning the portfolio, the evaluation is composed on the sum of the series
Microsoft Excel Relation between and ( 0 1 2 1.000000 1 1 2 1.693147 2 1 2 1.933374 3 1 2 1.988878 4 1 2 1.998496 5 1 2 1.999829 6 1 2 1.999983 7 1 2 1.999999 8 1 2 2.000000 9 1 2 2.000000 10 1 2 2.000000 I can notice right now from this plot, that the value is increasing when the value increases, so simply they are positively correlated. however, the graph does not go beyond .
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This project was given to us, as we are currently working on and will soon be completing a unit based on the calculation of surface area and volume for 3-d and 2-d shapes. Conclusively, all those that have attempted the completion of this project, will have an end result of exactly which is the 'best' container for mass production. 4 Procedures, materials and methods Materials For the completion of this project there are some necessary materials that every student or pupil wanting to complete this project must have: - GDC (Graph Display Calculator) - "Graph" (www.padowan.dk) or "Grapher" (Apple Applications)
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value of 'y' by Un and the value of 'x' by n, thus: STELLAR NUMBERS NUMBER OF DOTS TO S6 STAGE S1 S2 S3 S4 S5 S6 1 13 37 73 121 181 Thus, using finite difference: The most obvious pattern is that the 1st row all numbers and odd and the second row all are even. Also all these numbers are some multiples of 12 + 1, for example: 12 also turns out to be the half of 6. 6 STELLAR NUMBER AT STAGE S7 1 + 1 (12) + 2 (12) + 3 (12) + 4(12) + 5(12)
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AIMS * To know if the number of accidents encountered by vehicles on the Accra Kumasi highway are related to number of times the drivers maintained their vehicles, the average speed at which they drove their vehicles and their ages. * To find out the strength of the relationship that exist between the number of accidents encountered by the commercial vehicles, and the number of times they maintained their vehicles, the average speed at which they drove their vehicles and their ages if there exists any at all.
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Mathematics SL Portfolio Type II. This portfolio considers commercial fishing in a country in two different environments, namely the sea and fish farms. The statistics are obtained from UN Statistics Division Common Database from the year 1980 to 2006. T
Part I Defining Variables, Parameters and Constraints There are two variables in the data given. These two variables affect each other's value. The independent variable is the year and the dependent value is the total mass of fish caught in the sea shown in thousands of tonnes. The change in year affect the total mass of fish caught. The year increases by one. Parameters in the context of mathematical model are constants involved in the relation between the independent and dependent variable (Bard, 1974). Possible parameters that may affect the fish production are weather, government policy, technological advancement, environmental factors and demands.
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I chose the exponential trend line because after looking at the other available trend lines in the Excel program, the line that will provide me with the r2 value closest to 1 is the exponential decay trend line with the r2 value of .9942. A trend line is most reliable and reasonable when the r2 value is at one or very close to one. Also, the exponential line looks like it is the most reasonable and best fitting. The graph I made is very similar to the model given.
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Sequence and n2 1 2.5 4 5.5 7 8.5 10 11.5 Second difference This second difference illustrates the value for 'b' which is equal to. However the value of 'c' has not yet been determined. It was calculated using an example: Using n=2: n2 + n + c = 6 (2)2 + (2) + c = 6 5 + c=6 c=1 From the example we can verify that 'c' must be equal to 1 to reach the desired figure. To check that these are the correct values two more examples were used: --> Using n=5 n2 + n + 1 = 21 (5)2 + (5)
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Infinite Surds Investigation. This graph illustrates the same relationship as was demonstrated in the infinite surd of 1. The largest jump between values occurs between 1 and 2.
9 10 -0.000012055 10 11 -0.000003752 The exact value of this infinite surd could be found by using the idea that: Now substitute the value for an+1 into the equation and solve for an by setting it equal to zero. The surd is canceled once the whole equation is squared and this is left: Now this is solved using the quadratic formula: The negative answer, , must be disregarded because a negative answer for a surd is not possible. Therefore the exact value of the infinite surd is: For the infinite surd expression: This would be the sequence of terms for an: etc.
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Fishing Rods Portfolio. Leo has a fishing rod with an overall length of 230 cm. The diagram shows a fishing rod with eight guides, plus a guide at the tip of the rod.
The graph of the data would look like this: Quadratic Function: (ax2+bx+c) To find a quadratic function for this data, I used the first three guide numbers and distances from the tip: (1, 10), (2, 23), (3, 38) and used these numbers in the matrix method To find [X], we could use this formula: Quadratic function is: Cubic Function: ax3+bx2+cx+d The same method was used to find the cubic formula, except that the first 4 guide numbers and distances were used in the matrix method.
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Discover how to solve inverse functions both graphically and algebraically, whilst investigating their relations, properties and patterns.
2. a) The linear function, f(x) = 2x + 4 is reflected. So the points on the graph, (0,4) become (4,0). b) (3x - 1)/(x + 2) is reflected and becomes, (2x + 1)/(3 - x). This results in a mirror image of the original function. c) f(x) = x� is reflected resulting in the inverse function, g(x) = �Vx. 3. Using the linear function, f(x) = 4x + 8, it is clear that my results in Q2 are indeed correct, as they are confirmed by the inverse function of the above linear function. It is flipped resulting in g(x) =.
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To try and detect a correlation, this can be attempted with other values of n, using the same limits. For n=3, between x=0 and x=1: For n=4, between x=0 and x=1: Further examples: Thus far, we have a correlation that for power n, the ratio of areas is also equal to n. Before making a conjecture, we should examine whether this occurs with other limits as well. For n=2, between x=0 and x=2 (note that the limits on y change for the calculation of A, but the process remains otherwise the same): For n=3, between x=0 and x=2: For n=4, between x=0 and x=2: I then tested the results for when.
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