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International Baccalaureate: Maths
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To find the exact infinite value for this sequence we would use the equation and rearrange in the correct way to find the value. Finding the exact value for the surd The exact value for the surd is 1.618033989 as the second answer does not fit in the problem. Another example of an infinite surd is: The first ten terms of this surd can be expressed in the sequence: From the tem terms of the sequence we can observe that the formula for the sequence is displayed as: The results had to be plotted in a graph as shown below: The graph above also shows us the relationship between n and.
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Thus, if we want to get the next shape, we should time the same number. Thus it's a geometric series which has a first term of 3 and a common ratio of 4, obviously. So we can get the geometric formula: Then, to verify the formula, we place n=0 n=1 n=2 n=3 And then we get: N0==3 N1==12 N2==48 N3==192 The results are already proved above. The length of each side: Because each side was break into 3 equal part, and the inner one is moved (As shown in the picture below). Therefore the length of each side is decrease and become one-third of its previous length in each stage of generation.
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What remains is: 4. Expressed in m, n and k. Use the = rule and then we can change the base. We change the base to 10. Then we apply the rule: = a We can cross away on both sides. What remains is: . Derived from this we can conclude that the general expression for the nth term of each sequence in the form thus is . Examples to justify this statement using technology: = () = = 1,5 = = * Now calculate the following, giving your answer in the form , where p, q .
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From theses first ten terms of the sequence we can derive a formula for in terms of . The formula is: By plotting the first ten terms of the sequence on a graph, we can study the relation between. Graph 1 This illustrates the relation between n and L in the case of .
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the heights used by two trapeziums and b the last height. This of course assumes as well that a,b and c are already calculated y-values (or heights) of the trapeziums. Therefore: Therefore: Inserting the values of the different heights and h gives: We are now going to increase the number of trapeziums to 5 to find the approximate area under the curve. First, we need to state the data: To state the six heights, or y-values, of the different trapeziums, we have to express them in the following way: g(0), g(0.2), g(0.4), g(0.6), g(0.8), g(1)
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Plot a graph for the BMI of different females of different ages in the US in year 2000 and analyse whether it is an accurate source of data.
we can assume it is a periodic function such as a cosine or sine graph. Even though you can use a cosine or a sine graph, I decided to use a cosine graph, as I am more familiar with this type of graph. After inputting the cosine function into autograph software, you would realise that transforming the function would be appropriate so that it can model the graph more accurately. In order for you to do this, you have to use the general formula of f(x)= acos(bx + c) + d. This is because it enables you to transform the function.
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For Negative Bounds: Bounds -1 to 0 The conjecture of n:1 is the exact opposite for conjecture of the negatively bounded regions in quadrant III. (1:n) If a set of positive bounds were to be changed to negative, then the ratio between A and B in the positive bounds would be equivalent to that of B and A in the negative bounds.
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The area of a trapezium can be found using the formula: Let a = the length of one side of the trapezium b = the length of the other side of the trapezium h = the vertical height between the two lengths In order, Although there are two trapeziums, there are three different values and in order to determine the area of the trapeziums at each value I will work out the values on the graph using the trapezium method.
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B. Calculate the ratio of the areas for other functions of the type between 0 and . Make a conjecture and test your conjecture for other subsets of the real numbers. Area A: 1- Area B: Conjecture: Given the function the area of A in ratio of the area of B can be given as n:1. Equation Area of A Area of B Ratio 3:1 4:1 5:1 6:1 2. Does your conjecture hold only for areas between x=0 and x=1? Examine for =0 and =2; =1 and =2 etc. Ex 1) The area of A for the function y=x2 [0,2] is 8/3 Area A: The area of B for the function y=x2 [0,2] is 16/3 Area B: Ex 2)
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Find the general statement that expresses , in terms of c and d. The general statement that expresses is: To test this statement substitute the values of a, x, and c for the values a= 2, b=4, and x= 256: Thus, Scopes and/or limitations of a, b, and x 1. a > 0, b > 0 The base of a logarithmic equation has to be greater than zero thus, a has to be greater than 0 and b has to be greater than 0.
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+ d where a controls the curve of the model, b controls the shifts of the model, c is the rate of fall, and d is the horizontal asymptote. My function was f(x) = (47(x-1)-1.5) + 1. I chose this function because it seems to fit closely with the data provided. I used the parameters stated above to create the equation in my Graphic Display Calculator and adjusted based on how well it fit the data provided. This is how my model appears by itself: And with the data points included: When graphed with Excel, a function of y = 46.096x-1.154 is found to be the best model of the data provided.
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Because the in the change of base rule we can simply cancel out the expressions with each other and that will leave us with. Then the sequence will look like. This illustrate that k in the nominator is constant and the denominator will increase by 1. Now both n and k is an element of any real number. This gives us an general statement for the nth term of the sequence. If we convert the first sequence to the form of we then get; We can see that the 2's in the expression match the m in the expression.
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2. What type of function models the behavior of the graph? Explain why you chose this function. Create an equation (a model) that fits the graph. I chose the sine function because the curve would fit the data provided. The curve I see shows a great resemblance to a sine graph and after stretching and shifting a sine function, it will match the graph. I did not choose cosine because there is no regression for cosine. Also there are lots of similarities between sine and cosine simple because they have the same function translation.
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This is because the body would only start response when the viral particles go up to 1 million. We could then put: Solve t. Increasing rate = 200% per 4 hrs Increasing rate per hr = 1.18920711500272 hour viral particles Fever? hour viral particles Fever? 0 10,000 not yet 21 380,546 not yet 1 11,892 not yet 22 452,548 not yet 2 14,142 not yet 23 538,174 not yet 3 16,818 not yet 24 640,000 not yet 4 20,000 not yet 25 761,093 not yet 5 23,784 not yet 26 905,097 not yet ... ... ... 27 1,076,347 begin 20 320,000 not yet 28 1,280,000 begin 29 1,522,185 begin 30 1,810,193 begin From this table we can see that after about 26 to 27 hours, the immune system starts to react to the viral particles.
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� un This is rather obvious because population size of the next year is equal to population size this year multiplied by the growth rate. Now, let us explore what would happen if we left this population of fish to grow for 20 years, assuming that our models for total population and growth rate are correct. It is best to use Microsoft Excel 2007 to calculate this. Initially there is a population of 10000: Year Population of Fish Rate of Growth t un+1 = (-0.00001 � un + 1.6)
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A5 ? A6 --> Let n = number of transversals and let p = number of parallelograms Transversals (n) Parallelograms (p) 2 1 3 3 (1 + 2) 4 6 (1 + 2 + 3) 5 10 (1 + 2 + 3 + 4) 6 15 (1 + 2 + 3 + 4 + 5) 7 21 (1 + 2 + 3 + 4 + 5 + 6) n 1 + 2 + ... + (n - 1) Use of Technology: Using the TI - 84 Plus, press STAT --> 1: Edit. Type in L1, L2: (2, 1) (3, 3)
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View the diagrams in figure 2 and figure 3. Figure 2. Figure 3. Logarithms follow exponential rules being exponents themselves. A few general statements include that one can observe in any base, the logarithm of 1 is 0. View the diagram in figure 4. Up until this point the examples used, contained positive numbers. Can we find the logarithm of a negative number? View the diagram in figure 5. In addition, in any base, the logarithm of the base itself is 1. View the diagram in figure 6. Logarithms follow other rules such as inverse properties.
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Pythagoreans Theorem: a2 + b2 = c2 where c is the hypotenuse while a and b are the sides of the triangle. Sine, Cosine, Tangent: For any unknown angle � within a right-angled triangle, sin � = opposite/hypotenuse, cos � = adjacent/hypotenuse and tan � = opposite/adjacent sides. Sine Law: For any triangle, sin(a)/a = sin(b)/b = sin(g)/c Cosine Law: For a triangle with sides of length a, b and c, and angle ? opposite the side of length c, the cosine law says that, c2 = a2 + b2 - 2ab cos(?).
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-->As 'n' keeps increasing, the value of an - an+1 keeps decreasing , and gradually at that. The exact value of this infinite surd is 1.62. Q3) Consider another infinite surd, where the first term is Repeat the entire process above and find the exact value for this surd. Answer) Basically repeat the same process for the first infinite surd that had 1, instead of 2. So, in this case we take the surd as a sequence of terms, where is; a1 = so, a2 = a3 = also, a2 = a3 = Squaring on both sides (a3)2 = ()2
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So we can conclude that more the amplitude is low and more the graph is wide. I put -2 sin x to show that when the value of the amplitude is negative, the graph will flip up sat down. Part 3: Investigation of the graphs y = sin (x+C) I choose sin (x+1) and sin x to show that when the number is positive, the period will move to the left by one because it will start at 1 for the amplitude. Here you can see that when the value in the parenthesis is negative, the period will shift to the right by one and the amplitude will start at -1.
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- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - pg26 Introduction Through this coursework I will investigate the case of a standard adult patient who is infected by a virus and produce models to explain the entire process. I will look at the replication rates of the virus at different stages. Then I will show what happens when the immune response begins and how to eliminate the viral particles.
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and and be sure that it is true, we are also going to calculate the resulting matrices of using 5,6 and 7 as powers of X and Y. The,;, shown below were calculated by using a GDC. By analyzing the resulting matrices of different powers of X and Y we can develop an expression for and which are shown below. Now to generate an expression for we must first develop the matrix resulting from X+Y so we can then use different powers for (X+Y)
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Knowing the values of and, I can then go on to calculate and. In order to facilitate all my workings out, I will do this on my graphical display calculator. From these calculations we can easily state that there is a pattern within the results of the variations of and when there is an increase in the power of the matrix. I can clearly state that there is a relationship between the power of the matrix and the end product of the matrices. For both the variables we get the same results except for the negative signs that represent inverse matrices depending on the initial matrix.
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Por otra parte, cada l�nea en las gr�ficas esta distinguida con un color diferente. * Las par�bolas, en rojo. * Y = X, en magenta. * Y = 2X, en negro. El software utilizado en todo el trabajo es Graphmatica y el editor de ecuaciones de Microsoft Word. Desarrollo del trabajo. Apartado 1. Considere la par�bola y = (x - 3)2 + 2 = x2 - 6x + 11 y las rectas y = x e y =2x. * Utilizando medios tecnol�gicos, halle las cuatro intersecciones que se muestran en el dibujo. * Rotule los valores correspondientes a la coordenada x de estas intersecciones como x1, x2, x3 y x4 en el orden en el que aparecen, de izquierda a derecha sobre el eje x.
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The CentiKeeper Company has a container that will reduce the temperature of a liquid from 200�F to 100�F in 90 minutes by maintaining a constant temperature of 70�F. (b) The TempControl Company has a container that will reduce the temperature of a liquid from 200�F to 110�F in 60 minutes by maintaining a constant temperature of 60�F. (c) The Hot'n'Cold, Inc., has a container that will reduce the temperature of a liquid from 210�F to 90�F in 30 minutes by maintaining a constant temperature of 50�F.
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