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# International Baccalaureate: Maths

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Get help from 80+ teachers and hundreds of thousands of student written documents 1. ## Infinite Surds

/ 2a An = 1 �V(-1)2-4(1)(-1)) / 2(1) An = 1 �V(5) / 2 a.) 1 +V(5) / 2 b.) 1 -V(5) / 2 < 0 c.) answer: 1 +V(5) / 2 An2 - An - 1 = 0 : As n gets very large An - An-1 = 0 ; An = An-1 2.) V(2+V2+V2+V2+...) A1 V2+V2 1.8477 A2 V(2+ A1) 1.9615 A3 V(2+ A2) 1.9903 A4 V(2+ A3)

• Word count: 504
2. ## Logans Logo

A parameter is a quantity that defines characteristics of functions. There are several parameters relevant to this portfolio: o The thickness of the lines: 0.5 mm o The length of the logo: 6.5 cm o The height of the logo: 6.1 cm o The type of function used A variable is a value that may vary. There are several variables relevant to this portfolio: o The x-value o The measurements, as others may have measured differently o The size of the card It is hard to find the coordinates of curves at a logo, so what I did was that I glued a millimetre paper to the logo.

• Word count: 1727
3. ## Type I - Logarithm Bases

This means that the next two base values for the next two terms are 243 and 729. This is because 35 = 243, and 36 = 729. The next two terms for this sequence is added below: Log3 81, Log9 81, Log27 81, Log81 81, Log243 81, ... The nth term can be simplified to be written as Log3n34. The next sequence is: Log5 25, Log25 25, Log125 25, Log625 25, ... In this sequence shown above, the base for each term is being multiplied by 5 each time, which shows that 5 can be to a power of n since 5 is being multiplied each time.

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4. ## Type I - Parallels and Parallelograms

Firstly you have A1, A2, and A3. However, A1 A2 form another parallelogram, which makes A4. Also, A2 A3 form another parallelogram, which creates A5. Lastly, A1 A2 A3 create a final parallelogram, A6. Therefore, there are six parallelograms formed when a fourth transversal is added to the pair of horizontal parallel lines. When five transversals are added to this figure, the figure appears as: As a fifth transversal is added to the same figure, this contains much more parallelograms.

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5. ## IB Math SL portfolio

The equation can be simplified even more by removing the variable from the rest of the equation, thus allowing for the use of the quadratic formula to solve the equation. a = 1 b = -1 c = -1 a = 1� (-1)2-4(1)(-1) 2(1) a = 1 � 5 2 Looking back at the data, it appears that the answer cannot equal a negative number, an obvious indicator being that the graph shows all values above zero, and there that there are no negatives in the initial equation.

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6. ## Infinite Surds

Surds have an infinite number of non-recurring decimal. Hence, surds are irrational numbers and are considered infinite surds. Following expression is the example of an infinite surd: Considering this surd as a sequence of terms where: = = = etc. Q: to find a formula for in terms of Answer: � = = 1.4142... � = or = 1.5537.... � = or = 1.5980.... As it is observed that = ; therefore, it can be understood that = as the trend has been observed as this until now. Hence, = = 1.6118... = = 1.6161... = = 1.6174...

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7. ## Type I - Logarithm Bases

and determine an expression for the nth term: 1 2 3 4 5 6 7 2 2 2 2 2 2 2 2 4 8 16 32 64 128 The value 2 was used to determine the nth term. is worked out from the table above in the form of by applying the change of base rule which states that: An application of this rule is as follows: The equations shown below will be used to determine and solve the sequences shown in the portfolio: = = Terms in the form of : The numerator and denominator both have the

• Word count: 777
8. ## Investigating Divisibility

II) Consider n=k+1: P(k+1) = (k+1) (k+1-1) =k (k+1) = k2+k =(k2- k) + 2k = 2M + 2k = 2 (M+k) By the principle of math of induction, the statement is true for all natural numbers. b) x =3, is P(n)= n(n-1)(n+1) divisible by 3? Yes because when you have the product of three consecutive integers, one of them will always be a multiple of 3. c) x = 4, is P(n)= n(n -1) (n2 + n +1) divisible by 4? No because there is no coefficient of four, and it's not the product of four consecutive numbers. d) x=5, is P(n) = n(n2+1) (n +1) (n -1)

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9. ## Math Portfolio higher level type 1

Therefore, is divisible by 3. To make sure this is true let's take a few examples. Let Now using GDC substitute the following values of in the expression Which is divisible by 3 And this number is divisible by 3 Again this number is divisible by 3. Therefore is divisible by 3 The 3rd case is when Now by plugging 4 in the expression it will turn out to be like this: And now by factorizing it, the expression will look like this: Now by factorizing the expression further it will look like this: I can't find anything in the expression that shows that it is divisible by 4.

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10. ## infintite surds

The values of an, are not exact because the values of an, have an infinite number of decimal points since it is a surd. When the value of n increases, the difference between two consecutive values of n are smaller than Graph 1 Graph 1 exemplifies the direct relationship between an and n. As the value of n increases, the value of an increases. In other words, as n gets larger, the value of gets smaller. When n gets larger and gets closer to infinity, an increases at a slower rate (ex:).

• Word count: 674
11. ## IB Fish

To clear the second 0 of the first column, the second row has to be multiplied by 3 and be subtracted to the third row. To clear the second column's last row to a 0, the second row has to multiplied by 1/2. To clear the second column's first row to a 0, the last row has to be multiplied by -1/2. To clear the third column's third row to a 0, the last column should be multiplied by 3 Finally to make the second column contain a 1, the second row is divided by 2.

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12. ## derivitaive of sine functions

The derivative, in respect to its calculus definition is defined as a limit and arises when finding slopes of tangents and rates of change on a given graph. Therefore the derivative of the function, , is the limit of the slope of the graph at any given points. The portfolio will begin by firstly providing a proof and deriving the function to create the derivative. Then the portfolio will continue on by investigating the graph of the function by analyzing the graph and comparing it to its derivative and also by looking at the behaviour of the gradient of the function as its approaches.

• Word count: 1727
13. ## matrix power

In mathematics, a matrix is a rectangular table of elements, or entries, which may be numbers or, any abstract quantities that can be added and multiplied. Matrices are used to describe linear equations, keep track of the coefficients of linear transformations and to record data that depend on multiple parameters. Matrices can be added, multiplied, and decomposed in various ways, which also makes them a key concept in the field of linear algebra. The purpose of this mathematical paper is to explore and investigate the powers of matrix. A matrix may be squared or even raised to an integer power.

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14. ## Parallels and Parallelograms

Figure 3. Sixth transversal crossing two parallel lines Sum of Parallelogram(s) Evidence 1 ,,,, 2 ,,, . 3 ,, 4 , 5 A seventh parallel transversal is added to the diagram as shown in Figure 4. Fifteen parallelograms are formed:,,,,,,,,,, , ,,, ,,,,, , Figure 4. Seventh parallel transversal crossing two parallel lines Sum of Parallelogram(s) Evidence 1 ,,,,, 2 ,,, ,. 3 ,, , 4 ,, , 5 ,, 6 After repeating the process of adding transversals consecutively, Figure 5 shows the conclusions that were made between the relationship between transversals and number of parallelograms formed between two parallel lines.

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15. ## sunrise over newyork

This involves predicting the times at which it rises and sets and the motion at which the sun moves throughout the day, and ultimately the entire year. This paper will take on a technological approach to analyzing the times at which sunrise occurs over New York, analyzing the data and determining the uses and advantages of applying the new found knowledge to the real world. The following chart is a data chart, which shows the times at which sunrise occurs over the horizon of New York.

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16. ## matix bimonals

and b to calculate Solution First we need to find A, B: Find using different values of a and b Let's assume that a= 2 Also, let's assume that b=3 To obtain the general formula for we need to find a formula for without using different values of a and b, are: Question By considering integer

• Word count: 382
17. ## Investigating Ratios of Areas and Volumes

A = 1 - = Thus the ratio of area A: area B is 2:1. Now, I will consider the ratio of areas for other functions of the type y = xn, between x = 0 and x = 1. Function Area of B (under the curve) Area of A Ratio y = x3 = = - 0 = 1 - = 3:1 = 3 y = x4 = = - 0 = 1 - = 4:1 = 4 y = x5 = = - 0 = 1 - = 5:1 = 5 The graphs of these functions are shown below: After considering several examples of the function of the type y = xn, between x = 0 and x = 1, a clear pattern emerges.

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18. ## Designing a Freight Elevator

in parametric mode, one can better visualize the movement of the elevator down the shaft. The dot indicates where the function starts and stops as it runs from t = 0 to t = 6, zero to six minutes. The line represents the movement as it starts at ground level (y = 0) and descends down to a depth of 80 meters (y = 80) and then rising back up to ground level. The chart below shows the position of the elevator at different times: Time (t) Position in meters (y) 0 0 1 -12.5 2 -40 3 -67.5 4 -80 5 -62.5 6 0 By changing to function mode, we can take a better look at the position, velocity, and acceleration of the elevator.

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19. ## Parabola Investigation

* The values of SL and SR are defined and solved. SL= x2-x1� 2.382-1.764� 0.618 SR = x4-x3� 6.236-4.618� 1.618 * Calculate D=|SL- SR|. D=|SL- SR| � |0.618-1.618|= |-1|= 1 2. Find values of D for other parabolas of the form y= ax2+bx+c, a>0, with vertices in quadrant 1, intersected by the lines y=x and y=2x. Consider various values of a, beginning with a=1. Make a conjecture about the value of D for these parabolas. * I'm going to consider three different parabolas with vertices in quadrant 1 in the form of form y= ax2+bx+c, a>0.

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20. ## Math Portfolio Type 1

The functions are labeled and the legend is shown in the right corner. The results, I want present you on the following table: Function Area A Area B Ratio Y= x2 0.66 0.34 2 : 1 y=x3 0.75 0.25 3 : 1 y=x4 0.80 0.20 4 : 1 y=x5 0.83 0.17 5 : 1 Tab 1 From the table (Tab 1) I can see that, with change of the power of the function, the ratio of the areas changes as well.

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21. ## Log Bases Assignment

To find an expression of the nth term, I firstly converted the logarithms into base 10 so it may be solved on my graphics display calculator - a TI-84 Plus. I then converted the answers into fractions as shown below: Term 1 2 3 4 5 6 7 = = = = = = = = = = = = = = = = = = = For each sequence a pattern can be seen. In the 1st sequence, the numerator is seen to be a constant integer for each term.

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22. ## Math Project Infinite Surds

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 = Value for each term

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23. ## Matrices SL Type 1

X5 = = = X6 = = = Y2 can be proven with the same expression by slightly changed. Since a12 and a21 are negative, we much change the expression as Yn = to meet the demands of a12 and a21. Y2 = = = Y3 = = = Y4 = = = Just to be sure the expression works again, we can find higher values of Yn. Y7 = = = Y8 = = = By using GDC, the values of Xn and Yn were double checked for accuracy.

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24. ## trigonometric functions

Then I will examine different numbers for B, then C then D. After examining the Sine function I test to see if changing the values of A, B, C and D will have the same effect on the Cosine function. Sine Curve: Figure 1 Figure 2 I will use y-sin (x) as my base curve. This has a maximum value of 1, minimum value of -1 (which can be seen in figure 3 and 4) and an amplitude of 1, (which was found with the equation Amplitude= (max-min) / 2). The period of the base curve (which is the x-range, necessary to complete one oscillation)

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25. ## Maths portfolio Crows dropping nuts

Hence by creating a size range for the "large" nut will help to identify and shape the model better. One such parameter that could be seen from the graph is the asymptotes, one on the y-axis and the other on the x-axis. This clearly suggests that this is not a precise graph, where both axis have infinite possibilities. For example when the height of the drop is too low, the frequency of drops is too high, therefore, in the real world this would not be possible, therefore suggesting parameters for the graph would have to be done.

• Word count: 1259