Calculus I - Limits and Continuity

I've decided to venture out and take an online course via SAIL. You guessed it, Calculus! I'm excited about this one, so I hope you are too. 

 Limits and Continuity 

Limits:

Do not care about what happens at a point, simply care about what happens as you get infinitely closer to the point.

Methods to find the limit:

1.) Plug in the value the limit approaches. 

2.) If an error occurs, try:

- Factoring

- Mulitiplying by the conjugate

- Foiling or Expanding

- Creating common denominators

- For indeterminant forms (0/0 or inf/inf) use L'Hopital's Rule [below]

L'Hopitals Rule: derive the numerator and denominator (keep it in fractional form), then plug in the value the limit approaches. If after plugging in the value, you come back to the original limit, that means that there may be no proper conclusion. 

Properties of Limits:

Rules for limit as x approaches 0 for sin(x) / x:

Limit as x approaches 0 for (sin(x)/ x) is always 1.

Limit as x approaches 0 for (sin(ax)/ sin(bx)) is always a/b. 

Limit as x approaches 0 for (sin(ax)/bx) is always a/b. 

Limit as x approaches 0 for sin(1/x) is always non-existent. We don't really know what happens at x = 0.

Limits that never exist:

Limit as x approaches 0 (1/x^2). However, some may argue this is infinity. 

Limit as x approaches 0 for sin(1/x)

Limit as x approaches 0 for x/|x| (left and right limits will never 

Continuity

Types:

Removable Discontinuity:

If you have get a zero in the denominator (undefined) and you can use a method to get rid of the "undefined" , then you have a removable discontinuity. This would appear as an open circle on the graph:

Jump Discontinuity:

The left and right side limits exist, and are finite, but aren't equal. You will find the graph approaches the same x-value at two different y-values:

Infinite Discontinuity:

Either one or both the left and right side limits do no exist or are infinite. You will find that one or both sides will go to plus or minus infinity, or it will be terribly hard to figure out what is happening at a point:

Rules for Continuity:

For a function to be continuous at "c":

- f(c) should be defined at c. 

- Limit as x approaches c of f(x) should exist.

- f(c) = limit as x approaches 0 of f(x). 

Intermediate Value Theorem:

At an interval [x1, x2], if the function is continous, the function should be defined (have y-values for all x-values) within that interval. 

Factorials - Permutations and Combinations

Combination: probabilities selections where the order DOES NOT MATTER.

Permutation: probabilities where the order MATTERS.

Formulas:

Why is 0! = 1?

Say we take permutation formula from above, and set n = 3 and r = 3. The permutation formula with these number inputs would be:

P(3,3) = (3!)/(3-3)!

This equals 3! which means the denominator must've been 1. 

Another way to approach it:

_ _ _

We have above 3 possible slots, and we only have 3 options (A, B, C). So, our possible outcomes (when order matters) would be as follows:

A B C

B C A

C A B

B A C

C B A

A C B

*if we put the first option at the back and scoot everything over by 1 again we'll just have made a repetitive set*

B A C

Anyways, we now see we have 6 options which is indeed 3! = 1 * 2 * 3

Therefore, 0! = 1.

 :)

Math/CS blog made from an intrigued 11th grader perspective.

(3-5 minute read)

Hello world! Glad you found my blog. Not quite sure about the journey you took to get here (as there's only a 1 in about 2 billion probability of landing on this page) unless you intended to search it up because I told you!! Nevertheless, hope you choose to stay. I'm here to try to make my love for math (even on the worst days) contagious. If you find yourself crying over a problem, just remember it means you actually care and that's true interest [dedication]. So, I completely get it if math isn't your favored or strongest subject; I designed this website to provide several ways of memorizing/solving math problems, with some examples from real text-books/ websites. If anything here helps you out, or you would like to ask a question I am more than happy to respond back to your comments. 

___________________________________________________________________________________

Always remember that it's better to be surrounded by smarter people that help you grow.  

Please don't be afraid to reach me, I come from a student perspective too! :)

I don't bite, if you prefer to email private questions or interesting websites for me to check out relating to math/CS, you are welcome send them my way! 

It's indianpenny28@gmail.com, please be aware that I may not respond back ASAP. 

Thanks again and take care!!

Financial Exponential Functions - Compound Interest

Compound Interest and Several Examples of Exponential Functions

Here, we will go over some examples of Compound Interest. It is important to know that Compound Interest is just a exponential function that can be solved using simple algebra. 

Labeled Formula of Compound Interest

P = the principal amount, otherwise known as the original/ initial amount.

r = rate at which the money grows per year; interest rate (written in decimals)

m = number of times compounded in a year (ex. daily, biweekly, weekly, monthly, semimonthly...)

t = after how many years (amount of years)

A = the total amount after 't' years

If 'm' equals the following numbers, that is the way we are compounding in a year.

m = 1 only counted once per year, every year

m = 2 semiannually, because it is only counted 2 times a year; meaning after every 6 months (12/2=6)

m = 4 quarterly, because it is only counted every 4 times a year; meaning after every 3 months (12/4=3)

m = 12 monthly, because there are 12 months in a year, so after every month for 12 total counts in a year.

m = 52 weekly, because there are 52 weeks in a year, so after every week for 52 total counts in a year.

m = 365 daily, because there are 365 days in a year, so after every day for 365 total counts in a year.

Great, let's try an example.

So, we have to solve for x in terms of 'm'.

Let's rephrase the question into a formula.

Original formula: A = P(1 + r/m)^mt

Our equation: A = 200(x)^mt 

where x = (1 + r/m)

We know that 'r' or rate has to be written as a decimal. So the rate HAS to be a decimal.

The answer is A.)

See? No math required on that one!

--

Here's another example:

Peter starts with $5,000. That is his principal value. His rate is written as a percentage; 9.2%. Since we are given percentage, we would need to divide by 100 = (9.2/100). Rate is 0.092. Now we just plug the numbers we have to find total amount (A)

Original formula: A = P(1+r/m)^mt

A = 5000(1+0.092/1)^1*2

P= 5000

r= 0.092

m= 1, it's calculated per year

t= 2, total amount after 2 years

A= 5000(1.092)^2

A = 5000(1.192464)

A = $5962.32

Sine, Cosine, Tangent and their Reciprocals formulas and Trigonometry examples

 

Sine (Sin), Cosine (Cos), Tangent (Tan)



How to remember the sides of a triangle:

This is called "beta"

Simple, the angle with the beta symbol, represents the viewpoint.

The opposite side will always face right across from the angle "beta".

The hypotenuse will always be across from the 90°. 

The adjacent side will always be right below the hypotenuse; the line between the angle "beta" and 90°. 

Diagram for Reference

Awesome! Now we can learn the Sin, Cos, and Tan formulas! They make life a bit more easier.

 

Naming each makes it easier to remember:

Soh for Sin because Sin= Opposite/Hypotenuse

Cah for Cos because Cos=  Adjacent/Hypotenuse 

and Toa for Tan because Tan= Opposite/Adjacent

 

Sin is used for right triangles that relate to the values of opposite side and hypotenuse to find the value of the triangle's "beta" angle.

For reference purposes

Ex.1

o = 5, h = 10 find beta.

Press "2nd" button on calculator.

Following, press the "SIN" button

Next, enter the value of your opposite side. In this case, 5.

Then press the division sign. This makes a ratio 

Lastly, enter the value of your hypotenuse side. In this case, 10.

Your result should be "Sin of 'beta' = 30°."

Ex.2 Only beta and one side are given

  o = 1, "beta"= 90° find hypotenuse.

This is better written and understood as Sin 90° = 1/h, otherwise "1/Sin 90° = h/1"

Cross multiplication.

Start by entering your opposite value. In this case 1.

Then press the division sign

Press the "SIN" button.

Enter the value of "beta". In this case 90.

Your result should be h = 1.

Cos is used for right triangles that relate to the values of adjacent side and hypotenuse to find the value of the triangle's "beta" angle.

For reference purposes

 Ex.1 

a = 5, h = 10 find beta.

Press "2nd" button on calculator.

Following, press the "COS" button

Next, enter the value of your adjacent side. In this case, 5.

Then press the division sign. This makes a ratio 

Lastly, enter the value of your hypotenuse side. In this case, 10.

Your result should be "Cos of  'beta' = 60°."

Ex.2 Only beta and one side are given

  a = 4, "beta"= 60° find hypotenuse.

This is better written and understood as Cos 60° = 4/h, otherwise "4/Cos 60° = h/1"

Cross multiplication.

Start by entering your adjacent value. In this case 4.

Then press the division sign

Press the "COS" button.

Enter the value of "beta". In this case 60.

Your result should be h = 8.

Tan is used for right triangles that relate to the values of opposite side and adjacent side to find the value of the triangle's "beta" angle.

For reference purposes

Ex. 1

o = 10, a = 10 find beta.

Press "2nd" button on calculator.

Following, press the "TAN" button

Next, enter the value of your opposite side. In this case, 10.

Then press the division sign. This makes a ratio 

Lastly, enter the value of your adjacent side. In this case, 10.

Your result should be "Tan of 'beta' = 45

Ex.2 Only beta and one side are given

  o = 10, "beta"= 45° find adjacent side.

This is better written and understood as Tan 45° = 10/a, otherwise "10/Tan 45° = h/1"

Cross multiplication.

Start by entering your adjacent value. In this case 10.

Then press the division sign

Press the "TAN" button.

Enter the value of "beta". In this case 45.

Your result should be a = 10.

Important Symbols to note!

The Various Symbols and Formulas in Statistics:

Prime and Composite Number Theorem (Divisibility Rules)

How can we check if a number is prime or composite?

Let 'x' be the number you check:

x² = x * x

Say √(x) = y 

√(x) = y

y = a * b 

Now if we were to assign a value to x, like x = 123, then: 

√(123) ~ between √(121) and √(144) closer to √(121) because

123 - 121 = 2

144 - 123 = 21

2 < 21

midpoint

144-121= 23

23/2 = 11.5

and so we know that √(123) ~ is close to 11, a little higher somewhere between 11.1 or 11.09.

y = a * b 

Let's say approximate for y is 11.1

So either a or b have to be less than 11.1

1 * 11.1

2 * 5.55

...

We just have to check the numbers less than the whole number 11 to see if 123 is divisible. 

123 / 2 = WE KNOW THIS WON'T WORK BECAUSE LAST DIGIT ISN'T EVEN.

123 / 3 = WE KNOW THIS WILL WORK BECAUSE 1+2+3= 6, AND 6 IS A MULTIPLE OF 3.

WE CAN STOP HERE.

123 / 4 = WE KNOW THIS WON'T WORK BECAUSE DIDN'T SATISFY "MULTIPLE OF 2"

this is literally just 123 / (2*2) if it helps.

123 / 5 = WE KNOW THIS WON'T WORK BECAUSE LAST DIGIT ISN'T 0 NOR 5.  

123 / 6 = SINCE 123 WAS DIVISIBLE BY 3 BUT NOT 2, IT WILL NOT BE DIVISIBLE BY 6.

this is literally just 123 / (2*3) if it helps.

123 / 7 = WILL NOT WORK; CHECK DIVISIBILITY RULE BY 7 BELOW.

123 / 8 = WILL NOT WORK, NOT DIVISIBILE BY BOTH FACTORS OF 8: 2 NOR 4.

this is literally just 123 / (2*4) if it helps.

123 / 9 = WE KNOW THIS WON'T WORK BECAUSE 1+2+3 = 6, AND 6 IS NOT A FACTOR OF 9.

123 / 10 = WE KNOW THIS WON'T WORK BECAUSE LAST DIGIT ISN'T 0.

123 / 11 = WE KNOW THIS WOULDN'T WORK BECAUSE VALUE IS ONLY 2 AWAY FROM NEAREST MULTIPLE OF 11 WHICH IS 121.

Now we just have to check the first one that works. In this case, check 3.

123/3 = 41

Therefore, 123 is NOT a prime, but a composite number. 

This trick will help you find out till what number you should bother checking (dividing it by) to see if x is prime or composite.. :)

Extras:

EXPLORING THE MULTIPLE OF SEVEN RULE (really cool watch!):

https://youtu.be/UDQjn_-pDSs