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Chaithanya R.
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Pre-Calculus
TutorMe
Question:

F(x)=1/(10^x) +4. Find the value of y when x = 0.

Chaithanya R.
Answer:

y = 1/(10^x) +4 y = 1/(10^0) +4 y= 1/1 +4 y 5

Writing
TutorMe
Question:

Humans are pattern seeking animals and we are adept at finding patterns whether they exist or not. Discuss knowledge questions raised by this idea in two areas of knowledge.

Chaithanya R.
Answer:

Patterns, which are defined by the Cambridge dictionary as “regular and intelligible forms or sequences discernible in the way in which something happens or is done”, may be observed all around us. The validity or existence of these patterns which we supposedly observe is a question which may be widely debated. Finding patterns is of exceptional importance to us as humankind since we can use established relationships to predict phenomenon and build technical tools which may further assist us and our cause. A brilliant quote made by a famous philosopher acts as a wonderful example for this question and it goes “make living creatures out of mud so the mud can see what we have done and god created every living creature that moveth and man alone could speech. What is the purpose of all of this asked he to god and god says “then I leave it to you to find the pattern” This quote made me think about a very important question that is “is it obsession or necessity of humans finding patterns in each and every aspect of their life, no matter how insignificant it might be?” I claim that throughout history humans have made sense of everyday events by making patterns that may or may not have any connection with each other, but these patterns are far away from reason or purpose. At evolutionary standpoint humans had to make ambiguous observations and connect them to each other in order for their survival. During the Stone Age there was no concept of calendars. Due to this sudden climatic changes and farming cycles were a huge concern and this put a risk at the survival of the species. To prevent this, early cave men created shapes in the night sky connecting stars and gave them names. With the help of this they could take a look in the night sky and gain a rough understanding of what point in the modern year they were in. With this they could prepare for the upcoming future which was critical especially in western countries just before winter. Without any preparation the Stone Age humans would not have the slightest chance of survival. We all know that closely grouped stars do not have any meaning by them self but we see that threw these stars humans made patterns from ambiguity for the survival of the species. This clearly shows the human mind set as se see that the human mind is developed in order to find patterns on even create patterns where they may or may not exist in order for their wellbeing. But on the other hand we frequently see that humans create patterns from ambiguity but there must be some or the relation for these patterns to have any meaning in the real world. When it comes to day-to-day life humans are constantly looking for patterns that govern their life. Humans constantly look for behavioural patterns in other humans to determine their actions. This quality of humans has evolved since man’s first steps on the earth. In those days humans lived in small, closed groups. When a new member was to be added to that group, the existing members observed for and behavioural patterns that would have reason for mistrust in him. This unique quality is seen even in today’s world. When it comes to job applications and interviews, the interviewers look at the behavioural of the candidate just as much as his/her knowledge in their field. Subtle movements like the kind of hand shake, biting of nails, fidgeting of fingers or even the tone of voice are enough to form an opinion about the character of the person. In a study it was found the people can look at the behavioural patterns in others and form a correct prediction more than 93% of the times about their next action and often even predictions of their personality as an individual. If someone does the math it is easy to see that it is not possible to be correct 93% of the times about something as complicated as human behaviour. Humans had to have evolved to seeing the subtle but yet prominent patterns. This led me to explore another question that is “is it human nature for all humans to act and behave, following rigid patterns?” Schema Theory states that humans comprehend every aspect of their lives by creating patterns and connecting them to some or previous encounter they had. Because of this nature, humans are hardwired to break down knowledge into smaller units they recognise set knowledge by linking it to similar experiences set in the past. Due to this human categorize all their actions in order to make meaning of what is happening and predict what might happen. This helps humans survive in the vastly competitive world by creating a head start due to previous situations that they might have encountered that had the same encounters leading up to the situation. A counter perspective in given threw a vast topic in Mathematics known as fractals. The topic is about similar shapes but has a chaos or randomness when it is evolved. When seeing these fractals people tend to see patterns and yet chaos at the same time. Due to this humans have a tough time classifying it as a well-defined pattern or not. Through this we can see that humans have a tendency of relating any two figures that look similar and assuming that there is a pattern present in it. This is very impractical at times but we can see how humans tendency of making assumptions from complete ambiguity. This is often interpreted as intuition or the gut feeling that humans base actions and judgements. Quoting Louise Smith “your intuition knows what to do. The trick is getting your head to shut up so that you can hear”. Hence, even in personal life, I have observed that only when I pay attention to my thoughts and think through an unbiased third person perspective, I understand the meaning behind patterns like the pigeonhole principle in mathematics. Another way that humans calculate or determine intelligence is though tests IQ which generally test thought during problem solving and design making. These so called perfect tests for interest that are supposedly based on perfect patterns are often inaccurate which totally goes against what patterns stand for. This can only mean one thing that either humans are forming patterns that do not exist or they consider something a pattern when it should not be considered as one. There is no argument on the fact that humans threw ought the world suffers for the phenomenon of mind priming or priming of the mind. It is the nature of a human mind to develop to soot its immediate surrounding which is completely dependent on situation to situation of ones experiences. This development of their mind is entirely based on their life style or the patterns that they experience day today. But we overwhelmingly see that humans have a tendency to act differently then what one would expect. We see that patterns are seeing in their actions but there are always uncertainties within them and one could even go to the extent of saying that there is chaos even within patterns. If this is the case can one truly says that they have discovered a pattern as that same pattern I going to be obsolete in situation that it was meant to be for. Methods can be employed to find correct patterns and rectify any non-existing patterns though. For example there are several methods employed in the mathematics which allow for the removal of any fallacies that might creep into the work that they have produced. These allow for the verification of the validity of the patterns and relationships these works establish. The most important and indeed useful such technique is peer review. This is particularly important since the patterns we notice differ due to our personal knowledge. Since our personal knowledge is shaped by ways of knowing such as emotion, language employed in society, faith, diverse differential regional rhetoric’s and so on, it can never be similar for two separate individuals. Thus, in the peer review process where the experts from journals where the research is published check the validity of the work and have different backgrounds with different personal knowledge, the patterns can be corrected. Thus, as some patterns formed with personal knowledge transgress into shared knowledge and become established patterns, they can be verified and corrected to a great extent through this process. This accounts for the most significant difference in factual subjects such as science and mathematics opposed to opinion based subjects like religion where an individual’s work does not depend on their position, rather the validity of the arguments made and patters developed. In conclusion, humans are indeed pattern seeking animals. Such behaviour is ingrained into us because of both evolutionary and social causes and makes us find patterns in everything where they sometimes exist and sometimes don’t. The usefulness of a correct pattern far outweighs the harm caused by an incorrect one and hence we must continue to find patterns and relationships but correct them through processes such as peer review. Hence, if one gains is put through rigorous treatment by one, the patterns developed are truer and more applicable to the real world. Not only does a skeptic hence have a more developed outlook in scientific and naturalistic areas, upon much thought, even personal perspectives on topics such as politics are derived.

Calculus
TutorMe
Question:

Find the integral of sin^3(x) * cos(x) + sin(x) cos^3(x) dx.

Chaithanya R.
Answer:

∫sin^3⁡(x) cos⁡(x) +sin⁡(x) cos^3⁡(x) dx ∫ sin⁡(x) cos⁡(x) (sin^2⁡(x)+cos⁡^2(x)) dx =∫sin⁡(x) cos⁡(x)dx Substituting u = sin⁡(x), du = cos(x) dx =∫ u du = ( u^2 /2) + c = 0.5 sin^2⁡(x)+ c

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