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International Baccalaureate: Maths
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While the general population may be 15% left handed, MENSA membership is populated to 20% left handed people (Left Handed Facts). Intrigued by the statistics, I decided it would be interesting to do some independent exploratory research in
Surveying solely students that are enrolled in the International Baccalaureate program will ensure that the coursework is consistent in difficulty level and that the GPAs are weighted the same, with 5.0 being the highest attainable grade point average due to the weighted grade scale of the rigorous classes. I plan to collect data via an online survey. The most efficient way that I can reach the greatest number of students is though the social networking website Facebook.com. In creating a Facebook poll, every one of my friends will be able to see the poll and answer quickly and conveniently from their household.
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The linear function does not give us very accurate results about the long run as well, and it is most suitable just for portraying present data, because with this trend, it will show that the population will grow continuously and constantly up to infinity. Points Value of the table (a) Value of the linear curve (b) Difference between the values (b-a) Systematic error percentage ((b-a)/b)*100 50 554.8 531.3 -23.5 -4.4 55 609.0 609.3 0.3 0.05 60 657.5 687.3 29.8 4.3 65 729.2 765.3 36.1 4.7 70 830.7 843.3 12.6 1.5 75 927.8 921.3 6.5 -0.7 80 998.9 999.3 0.4 0.04
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The purpose of this paper is to investigate an infinite summation patter where Ln(a) is a constant and the coefficient of x is an increasing factor to Ln(a).
Sn approaches a horizontal asymptote when y=4. There is a y-intercept at (0,1). Checks: tn = n t(n) S(n) 0.000000 1.000000 1.000000 1.000000 1.609438 2.609438 2.000000 1.295145 3.904583 3.000000 0.694819 4.599402 4.000000 0.279567 4.878969 5.000000 0.089989 4.968958 6.000000 0.024139 4.993096 7.000000 0.005550 4.998646 8.000000 0.001117 4.999763 9.000000 0.000200 4.999962 10.000000 0.000032 4.999995 As n � +?, Sn � +5 There is a horizontal asymptote as n approaches positive infinite (?). As n approaches positive infinite then Sn will approach positive five. N t(n) S(n) 0.000000 1.000000 1.000000 1.000000 1.791759 2.791759 2.000000 1.605201 4.396960 3.000000 0.958711 5.355672 4.000000 0.429445 5.785117 5.000000 0.153892 5.939009 6.000000 0.045956 5.984966 7.000000 0.011763 5.996729 8.000000 0.002635 5.999364 9.000000 0.000525 5.999888 10.000000 0.000094 5.999982 As n � +?, Sn � +6 Sn approaches a horizontal asymptote when y=6.
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How many pieces? In this study, the maximum number of parts obtained by n cuts of a four dimensional object will be analyzed by looking at the patterns of the maximum number of segments made by n cuts on a one dimensional line, the maximum number of regio
it can be seen that the common difference, d, is 1 since the values of S increase by 1 as n increases by 1. If the numbers for the variables are plugged into the equation, an = 2 + (n - 1)1, the equation for an would be an = n + 1, so the rule for S to obtain the maximum amount of cuts per segment would be S = n + 1. This result also shows a linear regression line, so the equation to find the maximum amount of cuts for a one dimensional object would use a linear regression.
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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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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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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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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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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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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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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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Stellar Numbers. Aim: To deduce the relationship found between the stellar numbers and the number of vertices in the stellar shapes that dictate their value.
Let us look at the table again: Rows Dots 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 In this table, however, instead of referring to simply 'term' we consider the number of rows, since that is a varying pattern that logically coincides with the concept of 'term'. From this we can get the general expression: 1. At first, the number of rows and dots do not seem to exhibit any discernible relation.
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Stellar Numbers. After establishing the general formula for the triangular numbers, stellar (star) shapes with p vertices leading to p-stellar numbers were to be considered.
= 10 T5 5 + (1 + 2 + 3 + 4) = 15 T6 6 + (1 + 2 + 3 + 4 + 5) = 21 T7 7 + (1 + 2 + 3 + 4 + 5 + 6) = 28 T8 8 + (1 + 2 + 3 + 4 + 5 + 6 + 7) = 36 As seen in the diagram above, the second difference is the same between the terms, and the sequence is therefore quadratic. This means that the equation Tn = an2 + bn + c will be used when representing the data in a general formula.
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In this essay, I am going to investigate the maximum number of pieces obtained when n-dimensional object is cut, and then prove it is true.
To begin the solution, consider the results for n=1, 2,3,4,5. Let R represent the maximum number of pieces in n-cuts of a line segment. The value for R is shown in the table. n 1 2 3 4 5 R 2 4 7 11 16 1Recursive rule: R1=2 R2=4=2+2 R3=7=4+3 R4=11=7+4 R5=16=11+5 ... Rn=Rn-1+n[U1] Assume that R0=1, then Rn=1+1+2+3+......+n Rn=1+(1+2+3+......+n) =1+2+3+......+n=n/2[2a+(n-1)*[U2]d][U3], a=1, d=1 1+2+3+......+n=n/2[2+(n-1)]=n/2[n+1]=(n^[U4]2+n)/2 Therefore, the recursive rule to generate the maximum number of regions is 1+ (n^2+n)/2= (n^2+n+2)/2[U5]. When n=5, R5=(5^2+5+2)/2=16, which corresponds to the tabulated value for n=5 above.
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This is important because if this sequence diverges, the general statement would be . A convergent series is a series in which the terms decrease in magnitude rapidly and for which the sum of the first several terms is not too different from the sum of all of the terms of the series. The following is an example of a divergent series. = Thus, there is no general statement to represent the infinite sum of this sequence. In order to detect whether the given sequence is a convergent sequence or not, there are 3 tests that can prove it.
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Infinite Summation Internal Assessment The idea of this internal assessment is to investigate the effect changing the value of x and a have on the graph of the general sequence given.
The next sequence will be a similar concept of the previous one, but the change in values of x=1 and a=3 which is 1, , , ... n Sn 0 1.000000 1 2.098612 2 2.702087 3 2.923082 4 2.983779 5 2.997115 6 2.999557 7 2.999940 8 2.999993 9 2.999999 10 3.000000 = This table show the value of , which is the sum of the sequence and never exceeds 3. This may be similar to the previous sequence but it varies as the increase faster throughout the same value of as reaches to 3.
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Y = 2sin(x) Y = sin (x) Now, as we can see from comparing these graphs, y= 2sin(x) is a vertically stretched version of the original basic function y=sin (x), because the A (amplitude) was higher. This contention is supported even further as we continue to examine graphs with higher amplitudes. If we graph again with FooPlot, we get the following: Y= 5 sin(x) Y = 2 sin(x) As we compare the new graph with a higher Amplitude (y=5 sin(x)), we can see a tremendous, more pronounced difference from the previous graph than we did above because this time, there is a greater increase in the amplitude "A".
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� y = p (1+r) n � y = ae (kt) These can be used as they include in the equation population factors, where the year can be included in them as well as interpreting the graph with them. By using the model of y = ar (n-1) which could be used in other terms as: y = ar t (t for time) I may develop this model with my data. Again, I am referring to the years in terms of being from year 0 to 45 instead of from 1950 to 1995.
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Triangular Numbers (Total number of dots in a triangle) The first shapes that shall be considered are the triangular figures: When the values of the triangles (the number of dots) are input into a table: Stage of triangle Image of Triangular shape Counted number of dots 1 1 1 2 1 2 3 3 3 1 3 6 6 4 1 4 10 10 5 5 15 15 The variables will be defined the same for tables to do with triangluar numbers: -n will be defined as the stage number of the triangle - as the nth stage of the triangle The variables will be defined the same for
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Statistics project. Comparing and analyzing the correlation of the number of novels read per week and the modal grade of T.I.S students
gender. Table of contents Page 1........... .......................................................Statement of task Page 2..................................................................Data collection procedure Page 3-6...............................................................Calculations (Statistics) Page 7-12..........................................................Calculations (Chi-square test) Page 13-16....................................Calculations (Pearson's Correlation co-efficient) Page 17..........................................Analysis and Conclusion Page 18............................................Reliability and Validity BOYS DATA Boys Number of books read mark 1 2 81-100 2 1 40-60 3 1 61-80 4 2 61-80 5 2 61-80 6 5 40-60 7 1 40-60 8 2 40-60 9 1 81-100 10 6 61-80 11 6 61-80 12 6 61-80 13 4 61-80 141 2 61-80 15 1 40-60 16 2 40-60 17 4 61-80 18 1 40-60 19 1 61-80 20 1 below40 21 1 61-80 23 1 61-80 24 2 81-100 25 1 40-60 26 6 61-80 27 1 61-80 28 6 below 40 29 1 40-60 30 6 61-80 MARK AVERAGES (BOYS)
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* The formula used to calculate the reducing balance depreciation is BVt= C ((1)-(rate/100)) ^n Note - n represents the number of years. Thus now I can use the present book value to calculate the rate at which the machine has depreciated. I will substitute the current book value in place of BVt in the equation. Then substitute C with the cost of the machine and substitute 2 in place of n as the current book value provided by the accountant is as of 2 years. I can then solve the equation in order to find the rate in a Graphic Display Calculator (GDC).
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Maths Modelling. Crows love nuts but their beaks are not strong enough to break some nuts open. To crack open the shells, they will repeatedly drop the nut on a hard surface until it opens. So, through this portfolio I will attempt to create a function t
- large nuts. Parameters: After analyzing the data given I noticed some constraints in the data given. Firstly it is important to note that the data given is an average of the number of drops taken to break open a large nut. It is also important to note that the number of drops must be a whole number because it is impossible to drop a nut times. The domain of a function that models this graph would also have to be greater than zero because it is not possible to have a value for height that is negative.
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Logarithm Bases - 3 sequences and their expression in the mth term has been given. All of these equations will be evaluated on a step-by-step basis in order to find an expression for the nth term.
Thus, to prove as the general expression for the first row of sequences: un = First term of the sequence = = log2n 8 (Taking the first term as u1 and so on,) L.H.S R.H.S u1 = log21 8 u1 = 3 log2 8 = 3 L.H.S = R.H.S. Hence, verified that is the general expression for the 1st row of the given sequence in the form . Similarly, other terms in the sequence were also verified by using the TI-83 where: U2: log48 (1.5)
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The independent variable was the genre of music played during memorization and recollection. The order in which the words were written did not affect their scores. All 26 participants were middle school band students aged 12 to 14, attending the Northwest Jackson Middle School. The results show that the null hypothesis must be retained for the tests conducted with the songs "Bad Mamma Jamma" and "Isn't She Lovely" (rap/hip-hop). Although there were slight decreases in the average number of words retained with these two songs, the decreases were not statistically significant. Introduction Research Question Does the type of background music played during memorization and recollection have a significant effect on students' ability to memorize and recall word lists?
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Proposez et testez un �nonc� g�n�ral pour ce cas. 5) Maintenant g�n�ralisez vos r�sultats pour m droites parall�les horizontales intersect�s par n droites parall�les obliques. 6) Pr�sentez vos r�sultats dans un tableur et utilisez cela pour trouver l'�nonc� g�n�ral pour l'ensemble de ces situations. 7) Testez la validit� de votre �nonc�. 8) Discutez sa port�e et/ou ses limites. 9) Expliquez comment vous �tes parvenu � cette g�n�ralisation. 1) Nous pouvons continuer � dessiner des obliques suppl�mentaires et former de nouveaux parall�logrammes. 2) Montrez que six parall�logrammes sont form�s lorsque nous ajoutions une quatri�me oblique � la figure 2.
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