Investigating the Ratios of Areas and Volumes around a Curve
Investigating the Ratios of Areas and Volumes around a Curve Introduction: In this investigation I will examine how changing the power of a function changes the ratio of the areas between it and the two axes. I will begin by making assumptions to simplify the mathematics, and then move on to more general cases. The curve used throughout will be, and I will aim for an expression of an area or volume ratio in terms of n. Notes: As the end goal in my investigation is to find a (unitless) ratio, I have left units out of my calculations. For all areas these can be given as 'units2', in whatever unit the particular graph is drawn. Volumes would be measured in 'units3'. All diagrams have been produced on Autograph V3.20, and modified slightly to add labels/shading in Microsoft Paint. A simple case: The curve is drawn between x=0 and x=1. The region between y=0, y=1, the curve and the y-axis will be known as A, and the region between x=0, x=1, the curve and the x-axis will be known as B. The ratio A:B can be calculated using the power rule for integration; to calculate A the equation is rearranged into the form ; the limits remain the same when raised to this power: Thus for n=2, A:B=2. 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
PROTECTION DE L'ORGANISME
PROTECTION DE L'ORGANISME Microbes : - Les Bactéries : o Appelées coque ou bacille o Se multiplient dans les liquides intérieurs riches en nutrients (ex : glucose) et en dioxygene - Les Virus : o 10 a 1000 fois plus petits que les bactéries o Appelé parasites o Parasitent nos cellules qui vont alors les multiplier ; les cellules meurent à la naissance des nouveaux virus. o Se transmettent par : * l'air * les aliments et la boisson * les liquides de l'organisme (sang, sperme, salive) * le toucher Protections : - Peau : o Constitue une barrière physique aux agents pathogènes o Accrue d'une protection chimique grâce a la sueur et d'une protection bactérienne - Muqueuses : o Respiratoire et digestives o Protégées par des secrétions de type : * Mucus * Acide o Empêchent les bactéries de se multiplier - Bonne hygiène corporelle, - la désinfection des places avec des antiseptiques (ex : alcool) limitent l'entrée des microbes dans nos corps - la stérilisation - Si attaqué par les bactéries, on peut utiliser des antibiotiques pour les tuer. Mes ceux ci sont sans action sur les virus. - SIDA : présent dans tout le monde et s'agit d'une pandémie comme MST (Maladies, Sexuellement Transmissibles). Seule l'utilisation d'un préservatif (condom) empêche sa propagation Réactions : - Après une blessure, apparaît
Type 1 Portfolio: Matrix Binomials
Type 1 Portfolio: Matrix Binomials Hudson Liao 2/7/2008 I was given the expression X = and Y = , where I calculate X2,X3,X4;Y2,Y3,Y4. Below I calculated X2 and made my way up to X4, where I also did the same with Y2 to Y4. X2= X3 or X2 * X1 = X4 or X3 * X1 = Y2 = Y3 or Y2*Y1 = Y4 or Y3+ * Y1 = Xn X1 = 1X = 20X X2 = 2X = 21X X3 = 4X = 22X X4 = 8X = 23X Now I am going to find and expression for: [Xn, Yn, (X+Y)n], by inputting different 'n' values. By doing this I can find a correlation between each variable. Expression: Xn = 2(n-1) X This general statement was found by finding a relationship through values from X1 to X4. In the Xn table, a pattern begins to form from 1X, 2X, 4X and 8X. If we simplify these numbers by using a constant value such as 1X = 20X we can find a general statement for this expression. Yn Y1= 1Y = 20Y Y2= 2Y = 21Y Y3= 4Y = 22Y Y4= 8Y = 23Y Expression: Yn = 2(n-1) Y The same method to determine the general statement for the expression Xn = 2(n-1) X was also used for Yn = 2(n-1) Y. (X+Y)n 2I 4I 8I 6I I am going to determine the expression for (X+Y)n by letting X = and Y = . Therefore the expression would look like: The resultant matrix is I am going to prove that this expression works with (X+Y)n : Calculation: (X+Y)n n=2 My expression: (X+Y)2 = 22I = With (X+Y)n: As well, since 'X' and 'Y'
We see and understand things not as they are but as we are. Discuss this claim in relation to at least two ways of knowing.
"We see and understand things not as they are but as we are." Discuss this claim in relation to at least two ways of knowing. Because of our different upbringings, prejudices, and viewpoints, the way we perceive or understand certain things may be different from how they are in reality. The prescribed title implies that instead of perceiving or understanding things for the way they truly are in reality, there are things that get in the way of our doing so. This essay will examine three such things that get in the way: our sense perception, emotions and language as knowers. In order to examine how these three ways of knowing affect reality or truth, the following knowledge issue will be used: To what extent does a knower's sense perception, language, emotion, or reason shape reality? Firstly, Sense Perception, in this case, is simply what the eyes see, ears hear, nose smells, or body feels. It is what the knower is experiencing through his or her sense perceptions. Secondly, Emotion refers to the emotions that a knower may feel or experience at any given point. In other words, it is the emotional state of the knower. Lastly, what reality refers to is the world of objective knowledge or truth. Furthermore, the reality of something is not subjective, but rather objective. What this essay examines is how sense perception, language, and emotion interact with reality and how, in
determining the empirical of magnesium oxide
MAGNESIUM OXIDE-EMPIRICAL FORMULA CHEMISTRY HL JAIME CASTRO A. 0-2 PRESENTED TO: KEITH RIGBY ANGLO COLOMBIAN SCHOOL EXPERIMENTAL SCIENCES DEPARTMENT RESULTS: The following table shows the mass recorded for the crucible, the magnesium and the magnesium oxide. Table1. Showing the masses recorded during the practical Crucible /g ±0.01g Crucible and magnesium /g ±0.01 g Crucible and magnesium oxide /g ±0.01g 25.50 25.67 25.80 If we subtract the mass of the crucible from the masses recorded in table 1 we can get the values of the masses of the magnesium and the magnesium oxide, the next table shows the exact masses of both magnesium and magnesium oxide. Table2. Showing the mass of Magnesium and the mass of magnesium oxide Magnesium /g ±0.01 g Magnesium oxide /g ±0.01g 0.17 0.30 ANALYSIS OF RESULTS: Taking in count the data recorded in table 2 we can calculate the mass of the oxygen from the air that reacted with the magnesium in order to form magnesium oxide, this can be done by just finding the difference between the values of the two masses, like shown below. Mass of Magnesium: 0.17g(±0.01 g) Mass of Magnesium oxide: 0.30g(±0.01 g) Mass of oxygen = (Mass of Magnesium oxide - Mass of Magnesium) = (0.30 - 0.17) = 0.13g Mass of oxygen = 0.13g Now that we know that the mass of the oxygen involved in the reaction we
The knowledge that we value the most is the knowledge for which we can provide the strongest justifications. To what extent would you agree with this claim?
Humans possess many kinds of knowledge; knowledge that was arrived at logically, knowledge based on perceptions, and religious knowledge are all some examples. This combination of knowledge gives us an advantage and separates us from the rest of the species on Earth, so it can be said to be of great value to mankind. However of all the knowledge that we have individually, some are more valuable than others. I believe the knowledge that we value the most is the knowledge for which we can provide the strongest justifications. The definition of knowledge itself is justified true beliefs ; therefore the more justifications we have for a true belief, the more proof we have for the knowledge to be valid. With strong justifications, we are also less likely to question the truth behind certain knowledge, and would be more compelled to believe it. Arriving at valid knowledge through logic is an example that offers strong justifications. For example, if all of A is B, and all of B is C, then we can logically deduce that all of A must be C. There would be no way to question this conclusion because its justifications allow no room for doubt, so we can positively accept that all A must be C. This knowledge holds its value in that its justifications are so concrete that it would be impossible for us to question it, and the fact that we can now accept it as knowledge with little doubt
Funeral Blues Analysis
Funeral Blues The poem begins with the line "Stop all the clocks, cut off the telephone." This introduction makes it clear that the narrator is depressed and a tragic event has taken place. The first half of the sentence portrays her wish for time to stand still and her unwillingness to carry on with her life. She feels that her situation will not improve in the future and she has nothing to live for or look forward to. The second half of the sentence: "cut off the telephone" shows that she is unwilling to communicate. She is inconsolable and perhaps there are many people around her trying to comfort her but they are unsuccessful. It could also show that she wants to isolate herself from the outside world and ignore the situation she is in as she cannot deal with it. The first stanza shows a wish for silence, as she wants all the objects around her that make noise to stop. This shows that the noise is stopping her from being able think properly, which is a metaphor for how the events in her life are stopping her from moving on or thinking in a rational way. The way all these noisy objects are mentioned together signifies how too many problems are distressing her at once. The last line of the stanza shows us the reason for her distress; someone close to her has died. She wants everyone to be quiet as a mark of respect for the person she has lost. She also says that there are
Comparative commentary
Comparative Commentary For most of the people, bears are cute and look somehow friendly. Even though we know, that those animals can be very dangerous, the bears are often underestimated. Not only because we cannot find them in our near environment, and therefore don't know which impact they have, but also because we grow up watching series like Winnie the Poo (1977) or Brother Bear (2003), where bears have the hero and victim roles. Both these texts are based on the relation between the bear and man. The first is a newspaper article, concerned with the bear's extinction. The second text is an extract from a nineteenth-century novel, elevating the status of the bear and forgiving its misdemeanors. The theme of the first text is the survival of the bear. It explains how urgent it is to do something against its possible extinction. The article states the causes of depletion and includes various statistics supporting their arguments. Besides the text is concerned about the bears all over the world and mentions a range of species, including the brown bear and the panda. The second text is more about the invincibility of the bear and its mythic dimensions. It has to do with a conflict which arises when trying to analyze the bear and putting it into the "right" category. The text notes that there can be a mutual respect between a man and a bear but moreover it notes a
This extract is from Chapter 7 from the book Heidi by Johanna Spyri. The extract talks about the first morning in Frankfurt for Heidi and her actions and feelings about her new surroundings.
This extract is from Chapter 7 from the book "Heidi" by Johanna Spyri. The extract talks about the first morning in Frankfurt for Heidi and her actions and feelings about her new surroundings. Heidi is a German name and Frankfurt is also in Germany. This extract seems realistic as it talks about the normal actions that a person would do if he/she finds himself/herself in a new, unknown place. The title of the book "Heidi" tells us that the book talks about the life or autobiography of Heidi. Normally when a book is named after one of its characters, that character is the true protagonist of the book and the story is about him/her. The extract is written in 3rd person and uses a bit of direct speech also. "She could not think where she was" This phrase sounds like a sarcastic phrase and actually means that she didn't know where she was. It was as if she had been unconscious when she arrived at this place and didn't know how she reached here or what place this was. "She was sitting up in a high white bed, on one side of a large, wide room, into which the light was falling through very, very long white curtains; near the window stood two chairs covered with large flowers, and then came a sofa with the same flowers, in front of which was a round table; in the corner was a washstand, with things upon it that Heidi had never seen in her life before." This is a very long sentence
"What I tell you three times is true." (Lewis Carroll) Might this formula - Or a more sophisticates version of it actually determine what we believe to be true?
"What I tell you three times is true." (Lewis Carroll) Might this formula - Or a more sophisticates version of it actually determine what we believe to be true? "What is Truth? Said jesting Pilate, and would not stay for an answer." 1 Had he stayed for an answer he would still be waiting. The term 'truth' has no entire satisfactory definition; however truth is generally accepted as having three key characteristics. These being; that one, truth is something that is public and true for everybody. Two that truth is independent of anyone's beliefs, meaning that a statement can be completely and utterly false though; everyone still believes it to be true, and finally that truth is eternal, meaning that if something is true, it will always be true. Although 'truth' is so uncertain, it is still said to be the basis for all of our knowledge. Looking closely at the truth, the factors affecting the truth and the reliability of the above formula, I will show in this essay that there is an underlying uncertainty in the word 'truth.' Based on a series of examples, I will show how truth can and cannot gain validity through the formula; "What I tell you three times is true." As shown in the aforementioned characteristics of truth, truth is independent of anyone's beliefs. For example, the truth of my statement 'my car is fast' is independent of whether one thinks it to true or not. I
