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Tutor profile: Berke F.

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Berke F.
Honours Student at Amstedam University College
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Questions

Subject: SAT II Mathematics Level 2

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Question:

Let $$ f(x) = \frac{6x + 10}{2x - 30}$$. As $$x$$ approaches infinity, what will be the value of $$f$$?

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Berke F.
Answer:

One way to visualize this question is to imagine $$x$$ as a very large number. So, let's say $$x$$ is equal to $$10^{8}$$. $$ f(10^{8}) = \frac{6(10^{8}) + 10}{2(10^{8}) - 30} = \frac{600000010}{199999970}$$.Already, we can see the effect of +10 and -30 is not very large in the equation. Even though 10 and 30 are greater than 6 and 2, $$ f(10^{8}) $$ is equal to about 3.0000005. That is more or less equal to $$ \frac{6}{2} $$, where 6 and 2 are the coefficients of $$x$$. As you can imagine, as $$x$$ gets greater and greater, this value gets closer to 3. At some point, the difference becomes so small that we can neglect it. So in questions like this, we can simply ignore the elements other than the one with the highest power of $$x$$. For this case, $$6x$$ and $$2x$$ are the terms with the highest power of x; namely, 1. (Think of $$6x = 6(x^{1})$$ and $$10 = 10(x^{0})$$). When other terms are removed: $$ f(x) = \frac{6x}{2x}$$ Because $$x$$ is not equal to zero (it is approaching infinity), we can simplify the equation further: $$ f(x) = \frac{6}{2} = 3 $$

Subject: SAT II Mathematics Level 1

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Question:

I do not understand why negative powers do not give a negative result. As in, why is $$ 3^{-2} $$ not equal to -9?

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Berke F.
Answer:

I understand the confusion; let's step back for a second and remember how powers work. If I want to multiply the number 3 by itself 5 times, I can denote is as either $$ 3 x 3 x 3 x 3 x 3 $$ or $$ 3^{5} $$. This is equal to 243. If I want to find the value of $$ 3^{4} $$ (or $$ 3^{5-1} $$), all we need to do is to divide it by 3. As in: $$ \frac{3^{5}}{3} = 3^{4} $$ Let's apply the same logic to smaller powers. $$ \frac{3^{2}}{3} = 3^{1} = 3 $$ What if we continued? $$ \frac{3^{1}}{3} = 3^{0} = 1 $$ There you can also see how 3 to the power of 0 is not equal to 0, but rather to 1. $$ \frac{3^{0}}{3} = 3^{-1} = \frac{1}{3} $$ As you can see, no negative numbers are involved in the process. As we cannot really visualize dividing the number 3 by itself a total of -2 times, I can only explain it this way. But, you can think of it this way: $$ 3^{2} = 9 $$ because you multiply 1 with 3, 2 times. (1 x 3 x 3 = 9) $$ 3^{-2} = \frac{1}{9} $$ because you divide 1 with 3, 2 times. ($$ \frac{\frac{1}{3}}{3} = \frac{1}{9}$$)

Subject: Python Programming

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Question:

What is the difference between a list and a tuple in Python?

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Berke F.
Answer:

In Python, both lists and tuples are ordered sequences of elements. For example, if I want to keep a grocery list, I can list the items between square brackets and separated by commas. Assuming my elements are strings, the following is a valid list: ["bread", "cheese", "tomatoes"]. I can also use indexing to get a certain element, or multiple elements at once groceries = ["bread", "cheese", "tomatoes"] groceries[0] should give us "bread" because index numbers start at zero. Similarly, groceries[2] should give us tomatoes. So far, it is the same for tuples, except they are denoted with parentheses. For instance, (3, 4, 5) is a valid tuple. I can still use indexing to get an element or multiple elements. The main difference is that I can "edit" a list, in the sense of adding or removing elements. With the groceries list we have, groceries.append("pepper") would update the list in such a way that "pepper" is added to the end of the list: ["bread", "cheese", "tomatoes", "pepper"]. Also, groceries[0] = "onion" should update the very first element of the list from "bread" to "onion". Because editing is possible and the values are not static in a list, we call this data structure "mutable". In that sense, tuples are "immutable" as we cannot edit the values inside a tuple after it is created. Then another question may be "Why do we need tuples if lists can do anything a tuple can do, and has some properties that tuples don't?". The main answer is, again, tuples are immutable. This may seem as a disadvantage at first, but it may be preferable in certain situations. To put it simply, the "constantness" of tuples make them use less computational power compared to lists. For a one-time use sequence with 10 elements would not make a huge difference. However, for a very long sequence which will not need to be edited anyways, a tuple works faster and more efficiently compared to a list. Imagine I have a basic dataset of people, such that I keep their name, surname and age. If I am using this data for statistics purposes and all I need is to just see the data, I would keep people information as a tuple: ("Name", "Surname", 21). If this is a part of a dataset that I constantly update, e.g. people aging, I would keep them as a list: ["Name", "Surname", 21]. The latter allows me to change their age when needed. Again, there are more efficient data structures to store such data (e.g. Numpy dataframes) but this example should clarify the difference between the two structures.

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