There are rules if you want to succeed in this class. Clarissa Audrey (51844165), Mariana Paspuel (44069169), Tong Cui/Chelsea (31032162), Hongjiang Ye/Violet (52433166) present to you these guidelines, follow them (and some lucks) and you may be the top star!

1. If you’re confused, don’t be embarrassed to ask simple math question, like

Seriously, don’t wait until last minute to ask, you might (and always) forget.

2. If you wanna skip, skip the lecture, not the tutorial

Tips: Work best if you don’t skip at all.

3. Late to class? It’s better late than…

…never

4. Office hour is “ask everything you want” time

5. If you’re looking at your homework like this

or like this

come for help!

6. You will likely fail this class if you cheat, so don’t.

P.S. I never cheat and still fail this class.

7. If you fail on a test, don’t be down! Because 95% chance you’re not alone.

One for all, all for one!!

8. The question on the test is as simple as: If John have 3 apples and Tim give him another 5 apples, then Jess eat 2 of the apples. How many apples does Terry have?

Who the hell is Terry?

Seriously, most of the exam questions don’t make sense at all.

9. If you don’t know what to write, just write anything you know


and may God bless the teacher who grades your test.

10. Last minute studying before exam won’t work

11. Math needs

12. Look who were there for you when you’re at your worst. Your goddamn self.

So, encourage yourself!

14. Tomorrow is a mythical land, do it now.

15. HA! You don’t realize that I skipped 13 right? Come on man, this is math. Gotta be careful with numbers.

13. Math is not always step forward. Sometimes you gotta step back to fully understand. So, here’s your 13.

16. Pray to Baby Jesus.

17. Piazza is there if you’re stuck on maths!

18. If you’re terrified when Professors ask a question, instead of

Stare into their eyes

until they’re uncomfortable.

19. Enough sleep!

20. There always be those students that itch you to the core. In that case,

or in some cases,

21. When the question is too easy, most of the time it’s because you do it wrong

22. You may look at your teachers like this

But remember, they’re there to help you. Always.

23. Friday Quizzes!! L Not sure if the math test was easy or I got everything wrong

Note: Not always Math is that hard as you think. Trust yourself!!

24.First Math class, “Limits” find epsilon and delta. What in the world is that?


Do not scared about this, it is just two first weeks of class :’) :’)

25.Spend like 3 hours typing written assignments is totally fine at the beginning 

Never ever present handwritten homework.

26. Avoiding eye contact with your instructor because you do not know the answer to the question. Usually happens!!

Do not be scared to be wrong, that’s part of learning.

27. Do not be late for lectures, Dr. Leung will realize everything. (e-v-e-r-y-t-h-i-n-g)

Consider Math as the most important course ever 

28. When you finally understand what the class was about. You are like…

Keep trying and always give your best.

29. When you don’t know how to start written homework. Try!! Is not difficult as it looks.

30. When your instructor says that the power series (1/n^2)= (pi^2/6) and you need to used the Taylor series of sin(x). You are like…

Assigment_7

Integration 3 examples

Integration and Antidifferentiation

What is Integration?

The simplest way to answer this is by saying that integration is a way of adding small slices or small parts to find the whole. And it can be used to find areas, volumes, and central points of many things. So, how can we use those slices to find the area or volume? Imagine that we have a glass and tap water. Integration is like filling the glass from a tap. Adding up the little bits of water to the bucket gives us the volume of water in the bucket.

In more mathematical way, if we are being asked to find the area of a triangle or a circle or a square, it’s easy since we already know the formula of those shapes. But, imagine if we are being asked to find area of irregular shapes, e.g an irregular curve.

A good starting point will be to draw rectangles in the area of this shape because as we said before, we already knew how to calculate the area of the basic shapes.
Then, we can calculate the area of both rectangles and add them together. However, to be noted, the problem with this method is that it’s only a rough approximation of the area (not the factual area).

So, the solution is to make the rectangles narrower as you can see in the picture below.

As the rectangles get narrower, the approximation of the area will be more precise, but keep in mind that it is still not the factual area, because we will always have either underestimate or overestimate value, depending on which point (the left point or the right point of the rectangle width) was chosen to make the calculation. One aspect to consider is that when the curve gets more irregular the approximation of the area will be more difficult to calculate.

Just like the examples that show previously, integration could be applied widely no matter in real life or abstract math problems. In order to find the most efficient solution, based on the concept of simplification, we connect some scattered parts to form a valuable whole through a particular way, which is the main idea of integration.

Group members:
Clarissa Audrey 51844165
Mariana Paspuel 44069169
Violet Ye 52433166
Chelsea (Tong Cui) 31032162

a) Determine how much area is removed in steps P1, P2, P3, Pn

b) By considering a suitable series, show that the area of the Sierpinski carpet is equal to 0.

c) Describe a point on the Sierpinski carpet that will never be removed by the algorithm.

In order to solve this problem, you have to analyse what is happening in each square when goes from step 1 to 2,3,4, etc. Look beyond what your eyes can see, be creative. Make assumptions, e.g. from step 1 to 2 think about in how many parts you can divide this square in order to remove just the center of the circle. From step 2 to 3 the same but now how many squares are surrounded to the big white square; note how the number of squares are increasing while you go from step 1 to 3.

As you can see, this exercise looks like a series… Which one? Consider your first term and the ratio (the number of squares that increase) solve by this method, do not forget that the length of the square is 1. Once that you have the result of the sum of the series, you can subtract from the length and you have proved that the area is equal to 0.

Finally, you can conclude that in the square is a part that will never be remove, think about which part you never take in account for solving this problem.

Let´s define series,

A series has a sequence of partial sums, if the sequence of the partial sums converges that means that,when the limits approaches to any numeror L; in this case the value of series is also L. If the sequence of partial sums diverges, then the infite series also diverges.

Here is an example of Converget series.

Suppose that you take a loan to pay your new Porche,  monthly the interest will be low. If you make payments per month of a good amount in a determined moment your loan balance will be zero. As we can see here we are able to know the quatity of months that the person has to pay for the loan, so will be a specific number and as we know if the sequences of this payments have an orden and a pattern that approaches to any number will converge, as well as the partial sums of the series .

Another example is if we eat a pear, first the half, after the half of the half, then the half of the half of the half, etc. we know that this sequences has a logical order and converge to one fished pear; the partial sum of this series will also converge to one pear.

Here is an example of Divergent series.

An interesting example of divergent series is a sunflower, because we find that has several spirals or patterns that are different so we do not have a defined limit, it means that the sequences diverges, this implies that the partial sum will also be infinite so that the series diverges.

Let`s define a sequence,

A sequence is an ordered list of numbers, where each number in the sequence is called term. When we talk about the limit of a sequence we can say that if the terms of a sequence approaches to a number; in the other hand if the terms of that sequence do not approach to any number, has no limit and in  this case the sequences diverges.

Here is an example of convergent sequence.

Irma is having a several pain her arm due to and accident, in order to decrease that pain she takes an amount of ketorolac, this medicament is washed out her bloodstream every hour the same percentage. Now,we know that every hour the amount of the drug will be reducing, this information give us the clue that the amount of ketorolac will be gradually lowered to near 0, when the drug is out of the body, here we can evidence an example in the real life of a sequence that has a logical order and satisfies the requirements of a  converget sequence.

Another example is the moon phases because we can see that depending on the days the moon will be full, the half or less  this changes have a pattern and go in order and are approaching to something, in this case to have a full moon or a new moon.

http://mrvillascienceclass.blogspot.ca/2016/01/moon-phases.html

Here is an example of divergent sequence.

 Carmen, has a cronical disease called Hyperthyroidism , she has to take a pill all days to control this disease, in a determined moment she is not feeling well with this pills, her doctor said that she has to take a pill with more concentration of the medicament to counter the effects. There will be a time when  Carmen needs a stronger medicament; we can evidente here that if the medicament increase, it will do in a logical way (has a pattern)   as well as quantity.  The amount of the medicament will be increasing gradually and has patron. In this case we show that this is divergent sequence.

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