Monthly Archives: February 2012

Studying.. Story of College Kid’s Lives!

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In high school, I never had to study much. Teachers baby fed us and gave us all the information on the test. I was only forced to memorize about ten definitions and didn’t have to memorize the true concepts behind things such as definitions. I could memorize one word in the definition that would match with the actual word and be good to go. Thirty minutes of studying for a test would allow me to easily pass the test and make an easy A!

In college, studying is a whole different story. To pass a test you have to study for at least 2 hours. To make a B on the test you better study at least 4 and to make an A you have to study at least 5+ hours. This is a huge shock when you come to college. This wasnt as big of a surprise for me because I had older sisters who told me I was going to have to put in the hours, but I still hate having to spend so much time on one subject.

Here are some, I think are helpful when studying for an exam!

First off, you need to have time management. You are going to have to say no to stuff you  really want to go to. If you are debating on whether or not to go somewhere, think that school is your future and that however you do in school determines what type of job you have. School is more important than partying for one night! Party once you have finished the test and know you excelled on it! DO NOT START STUDYING THE NIGHT BEFORE…BEGIN EARLY AND BREAK UP YOUR STUDYING!!!

Secondly, you need to get enough sleep. This ties in with number one, but still needs to be its own tip. Some people think it  is cool to pull an “all nighter” of studying before a big exam. This is one of the worst things you can do. Not getting enough sleep makes your mind not function properly. Just think when you are hanging out on the weekend with your friends and y’all are staying up late talking and y’all start laughing for no reason! This is because your brain is not functioning properly and is tired. This is the exact same thing that is happening when you take an exam totally blank out.

Thirdly, it is always good to study with people. When you study with people you will learn things that you would never learn by yourself. When you are studying by yourself you only get one perspective on how to approach problems. When you study with someone or with a group you get many perspectives, which can help you have a broader mindset when approaching problems. It also helps to study with people because you realize how much you actually know. If you are unable to explain a problem to someone, then in reality you do not know the material well enough. Quizzing and helping each other can make you be very successful on test.

These are some tips I use! Hope they help! We have a test next Wednesday! I VOTE YOU START STUDYING NOW!

 

Derivatives of Exponential Functions

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HERE IS SOME BRIEF HELP FROM THE BOOK YOU CAN USE TO STUDY FOR NEXT CLASS! THIS IS REVIEW FROM PAGES 234-237. I HOPE THIS HELPS SOME!

First off, what is an exponential function?

A exponential  function is a function whose value is a constant raised to the power of the argument.

For example, f(x)=a^{x} , where a > 0 and a \neq 1

FINDING THE DERIVATIVE:

We can find the derivative of exponential functions by using the definition of the derivative.

\frac {d(a^x)}{dx}\overset {lim}{ h \rightarrow 0} \frac {a^{x+h}-a^x}{h} (assuming a > 0)

=\overset {lim}{ h \rightarrow 0} \frac {a^{x}a^{h}-a^x}{h}

=\overset {lim}{ h \rightarrow 0} \frac {a^{h}-1}{h}

In this last step, since a^x does not involve h, we were able to bring a^x in front of the limit. The result says that the derivative of a^x  is a^x times a constant that depends on a, namely \overset {lim}{ h \rightarrow 0} \frac {a^{h}-1}{h} .

We could also find the derivative with the following short cuts:

derivative of a^x

D_x [a^x]= (ln a) a^x

This formula becomes particularly simple when we let a = e, because of the face that ln e = 1.

Derivative of e^x

D_x[e^x] = e^x

We now see why e is the best base to work with: it has the simplest derivative of all exponential functions. Even if we choose a different base, e appears in the derivative anyway through the ln a term. (Recall that ln a is tha logarithm of a to the base e.) In fact, of all the functions we have studied, e^x is the simplest to differentiate, because its derivative is just itself.

Derivative of a^{g(x)}  and e^{g(x)}

D_x[a^{g(x)}] = (ln a)a^{g(x)} g'(x)

and

D_x[e^{g(x)}] = e^{g(x)}g'(x)

EXAMPLES:

(A) y= e^{5x}

Solution:  let g(x) = 5x, with g'(x) =5 then

\frac {dy}{dx} = 5e^{5x}

(B) y = e^ {x} \sqrt {x}

Solution: use the product rule

\frac {dy}{dx} = e^x \cdot \frac {1}{2 \sqrt x}  + \sqrt {xe^x}

= \frac {e^x}{2 \sqrt x} + e^x \sqrt {x} \cdot \frac {2 \sqrt x}{2 \sqrt x}

= \frac{e^x}{2 \sqrt x} (1 + 2x)

(C) f(x)= \frac {100,000}{1+ 100e^{-0.3x}}

Solution: use the quotient rule.

f(x) = \frac {(1 + 100 e^{-0.3x})(0) - 100,000 (-30e^{-0.3x})}{(1 + 100 e^{-0.3x})}

= \frac {3,000,000e^{-0.3x}}{(1 + 100 e^{-0.3x})^ {2}}

 

******Here is a joke that is actually in our textbook:***** 

 A deranged mathematician who frightened other inmates at an insane asylum by screaming at them, “I am going to differentiate you!” But one inmate remained calm and simply responded, ” I don’t care; I’m e^x.”

Pascal’s Triangle

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Pascals Triangle is one of the most incredible cheat sheets, in my opinion. The brilliance behind this work is magnificent! The triangle was named after Blaise Pascal, but it was first used and studied by the Persians and Chinese long before Pascal was born. Somehow, Pascal gained the credit for the triangle. I decided to explain some of its interesting patterns that occur in the triangle. Once you learn the rules, you find yourself trying to find an error in the triangle, but the triangle will prove its efficiency every time. It’s almost too good to be true.  

The Basics:

The very  top point of the triangle is known as the zeroth row. This row starts with the number 1. As we move onto row two, the numbers are 1 and 1. The second row is made by adding the two numbers to the left above the number and to the right above the number together. Anything outside the triangle is a zero. A good easy example of this pattern in pascals triangle is if you look at the number two. The top left number above 2 is 1 and the top right number above 2 is 1. If you do the math, 1 + 1 = 2. If you look at any number in the triangle, this rule stands true for all of them. I found this cool animation to help the confusion!

The Triangle is Symmetrical:

The graph is symmetrical. The numbers on the right side correspond to the numbers on the left side.

Pascal's Triangle Symmetry

Sum of The Rows:

If you add the numbers in each row together, each row will equal to 2^{n} . Where n is equal to the row number. You could also look at this rule as simply just the next row in the sequence is a double of the row before. Example: row 4 is double the sum of row 3. You should try it yourself! Here is some proof:

20 = 1
21 = 1+1 = 2
22 = 1+2+1 = 4
23 = 1+3+3+1 = 8
24 = 1+4+6+4+1 = 16

 Awesome Elevens:

11 is a unique number in the triangle, because similar to the number 2, each row answers what 11^ {n} equals. N is equal to the row.  The trick is you have to ignore the 1’s along the outside of the triangle and focus on the number between the 1s. Here is proof of the pattern:

Row # Formula = Multi-Digit number Actual Row
Row 0 110 = 1 1
Row 1 111 = 11 1 1
Row 2 112 = 121 1 2 1
Row 3 113 = 1331 1 3 3 1
Row 4 114 = 14641 1 4 6 4 1
Row 5 115 = 161051 1 5 10 10 5 1
Row 6 116 = 1771561 1 6 15 20 15 6 1
Row 7 117 = 19487171 1 7 21 35 35 21 7 1
Row 8 118 = 214358881 1 8 28 56 70 56 28 8 1

Hockey Stick Sequence:

If you start at a one of the number ones on the side of the triangle and follow a diagonal line of numbers. You can find the sum of the certain group of numbers you want by looking at the number below the diagonal, that is in the opposite direction of the diagonal you made. The picture below shows three good examples to make this pattern easier to understand.

1+6+21+56 = 84
1+7+28+84+210+462+924 = 1716
1+12 = 13

I hope you enjoyed this blog about Pascals Triangle! Once I was introduced to this triangle, I was very fascinated and I hope you are too now. There is many more patterns to the triangle, but some are complicated to explain. Some of them I am trying to figure out on my own.

10 Easy Things You Can Do To Improve Your Life Today

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I was reading some blogs and stumbled upon this one! All of us are overwhelmed and busy with school, so here are some ways to help your day run smoother! 

THIS POST IS A BLOG FROM ANOTHER USER ON WORDPRESS (NOT MY OWN WORK):

“10 Easy Things You Can Do To Improve Your Life Today

Changing or improving your life doesn’t have to be a long, frustrating, drawn-out process. I’m always in search of the smallest changes that yield the biggest returns, and below is my collection of quick tips that can make your life easier, more fulfilling, or just less of a pain in the ass.

Try one or two of these and tell me how great they worked.

Zack’s Not-So-Secret List of Simple Ways to Increase Happiness, Productivity, or General Awesomeness

1. Get rid of anything you haven’t worn in the past 2 months.

We both know that you should really donate everything you haven’t worn in the past two weeks, but I’ll cut you some slack. This isn’t about throwing away clothes you hate – it’s about getting rid of things that you don’t love. Having excess clothes that you’re never going to wear clogs up your closet, makes decision making more difficult, and serves as a constant reminder of unfinished business. Take two hours out of your weekend, tear everything out of your closet, put it in a big pile and start sorting. I’ll give you an allowance for three nostalgic t-shirts that you’ll never wear again (fraternity Beer Pong Championship shirt, etc).

When you’re done with that, think about the deep metaphor that you’ve just learned and apply it to other parts of your life. Shed yourself of activities, TV shows, people, work, and other obligations that you don’t love. You’ll feel lighter without a life that’s cluttered with fluff, and you’ll quickly fill those voids with meaningful things (you can think of me as a modern-day Mr. Miyagi, Daniel-san).

2. If something takes less than 2 minutes to do, do it immediately.

This is my favorite way to improve personal productivity. Whenever you encounter a task/obligation that requires less than 2 minutes of your time, just do it now. You’ll be amazed as how much this will reduce stress – both because you have less to remember and because you have a higher personal output.

3. Schedule a dinner with a group of 5 or more friends.

It’s actually scientifically proven that people who have dinner in groups of 5 or more at least once per week are happier than people who don’t. Planning an event yields more return than just the event itself – the anticipation of the event gives you something to look forward to. It takes less than 2 minutes to send an email, so you have to do this one right away.

4. Make a donation.

$5, $10, $50, $100. Whatever you can afford. If you don’t do this often, you’ll be surprised at how great it feels. I recommend checking out the charity first to see what percentage of your donation actually makes it to the end of the line – vs. being eaten up by big salaries, expensive fundraisers, and general wastefulness.

If you want some instant impact, give $20 to the next homeless person you see.

5. Practice saying “no.”

I had a friend in high school, Taylor, who had perfected the art of the “no.” When he didn’t want to do something, he’d look you in the face and respond, “I’d ’bout rather shoot myself in the face than do that.” I don’t recommend trying approach without a thick Southern accent, which has the magical power of making rude comments sound hilarious and acceptable.

Most people I know – myself included – get roped into doing things that they really don’t want to do. Saying “no” is liberating – stop giving automatic “yes” responses or, even worse, delaying the inevitable “no” by telling someone you’ll get back to them. If you aren’t instantly drawn to something, try a response like this: “I have plans that day, so I won’t be able to make it. If something changes, I’ll be sure to let you know.”

6. Stop sending open ended emails.

I used to send emails that said things like ”is that shirt available in black, red, or blue?” The other person responds with the answer and thus starts the email chain. Eliminate all this superfluous nonsense with a more comprehensive email that gives instructions for the reasonable contingencies: “I’m looking for a shirt in size medium. My favorite colors are black, red, and blue – in that order. If one of those is in-stock, please create an order, respond with the order number and your phone extension, and I will call you this evening with my credit card number and shipping information. If none are available, please let me know the estimated lead time and email me when it becomes available.” BAM. One and done.

Methods like this allow me to run two automated companies and only work a few hours per week.

7. Use self-scheduling software.

Most people waste an inordinate amount of time scheduling meetings, dinners, family get-togethers, dates, conference calls, and other events. Self-scheduling software, like TimeTrade, is all web-based and most offer free 30-day trials – they allow you to block off parts of your schedule for personal time, work, etc, and your friends/family/blind dates can choose available time slots. This eliminates all the back-and-forth of trying to find a time that works for both of you, and has the added bonus of making you look really important (you may want to consider exaggerating other exciting aspects of your life if a self-scheduling calendar is completely incongruent with your general image). In case you lead an underwhelming social life, be sure to block off a bunch of nights before sending a completely empty calendar to a potential date.

When you’re trying to get a group together, like for Tip #3, try a poll-based scheduling software like Doodle. This allows you to pick a few dates and ask a group of people which date works best.

8. Start one day per week off right.

Remember how much energy you had in high school? Yeah, well that’s because your parents cooked you decent food, you slept normal hours, and you didn’t drink your face off during the week (well, not as often as you do now, at least). Food and sleep are fuel for your body – put crude oil in your car and see how it runs.

Try doing this one day per week and you’ll be blown away at how great you feel – you’ll get more done on that day than the other days combined:

Get in bed at 9:30 and set your alarm for 6:30 – then leave your alarm clock or phone across the room so you actually wake up. In the morning, throw on some sweats and take a brisk 15 minute walk (if it’s cold out, watch Rocky IV the night before to prevent whining). Eat three eggs and have coffee or tea as normal – but leave out the toast, cereal, sugar, juice, and other garbage you usually consume in the morning.

Have trouble sleeping? Try 3mg of melatonin an hour before bed. If you drink coffee out of a fire hose like I do, PharmaGABA can help you from grinding your teeth out of your skull.

9. Know your instrument.

A mentor said this to me and it stuck. The most incredible tool you’ll ever own is your body and, if you want to maximize your effectiveness, you should get to know it well. Most people have a “productivity zone” – a period of a few hours each day when your energy level is at its peak (mine is from 8:30-noon, give or take). I forget the exact statistics, so I’ll make something up: 9 of out 10 people in a Danish study showed a 200% increase in productivity when they worked on their most difficult tasks during their peak-energy hours.

But seriously, give it a try. It works. Spend your low-energy zones doing things you enjoy doing, like watching cat videos on YouTube or thinking about ways to kill your boss.

10. Do your work in order of descending difficulty.

If you start your day off with harvesting FarmVille crops or cruising Facebook to compare your social life to people you barely know, this one goes out to you. Start off with the most difficult task first. Most people start off by opening their email and responding to new messages. Responding to emails is easy, and it’s something that you have to do, so it should be left for low/no-energy periods. Tackle your most difficult item first and you’ll always be rewarded with an easier task to do next.

There’s an added bonus – if you only get one thing done per day, like I usually do, you actually have something to show for it.

I wish I could remember where I picked up each of these tips – I’d say most were somehow born from reading one of my favorite books, which I highly recommend checking out (disclosure: affiliate links):

Wellbeing: The Five Essential Elements

The 4-Hour Workweek: Escape 9-5, Live Anywhere, and Join the New Rich (Expanded and Updated)

Personal Development for Smart People

Getting Things Done: The Art of Stress-Free Productivity

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Latex Help

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I heard a lot of you talking in the hall about how you were having trouble with writing a limit on your blog. I decided to give y’all a tutorial!

Enter all of these codes  in order on one line. starting with the number 1.

Here is how you do it. There are 5 codes you will need. 

  1. $latex
  2.  \overset
  3.  {lim}
  4. {x \rightarrowk}
  5. $
  6. f(x)
  7. =
  8. Then write whatever the limit is. For example “= DNE”
  9. It should look this without numbers 1: \overset{lim}{x \rightarrow k}$ f(x)= DNE
  10. If  you entered it right, it should look like this:  \overset{lim}{x \rightarrow k} f(x)= DNE

Here is help with writing the lim of the difference quotient. There are 8 codes you will need:

  1. $latex
  2. \overset
  3. {lim}
  4. {h \rightarrow 0}
  5. $
  6. $latex
  7. \frac {f(x+h)-f(x)}{h}
  8. $
  9. Then write whatever the limit is equal to. For example “= DNE”
  10. It should look like this without number 1: \overset {lim}{ h \rightarrow 0}$ \frac {f(x+h)-f(x)}{h} 
  11. If you entered it right, it should look like this: \overset {lim}{ h \rightarrow 0} \frac {f(x+h)-f(x)}{h}

Here are some FUN latex codes you can do:

Smiley Face:

  1. $latex
  2. \overset
  3. {\bullet \bullet}
  4. {\smile}
  5. $
  6. It should look like this without number 1: \overset {\bullet \bullet}{\smile} $
  7. \overset {\bullet \bullet}{\smile}

Bow Tie Smiley Face

  1. $latex
  2. \overset
  3. {\overset {\bullet \bullet}
  4. {\smile}}
  5. {\bowtie}
  6. $
  7. It should look like this without number 1: \overset {\overset {\bullet \bullet}{\smile}}{\bowtie} $
  8. \overset {\overset {\bullet \bullet}{\smile}}{\bowtie}

Scared Face

  1. $latex
  2. \overset
  3. {\odot \odot}
  4. {\circ}
  5. $
  6. It should look like this without number 1: \overset {\odot \odot}{\circ} $
  7. \overset {\odot \odot}{\circ}

Mustache Face

  1. $latex
  2. \overset
  3. {\overset {\bullet \bullet}
  4. {\overbrace{}}}
  5. {\smile}
  6. $
  7. It should look like this without number one:  \overset {\overset {\bullet \bullet}{\overbrace{}}}{\smile} $
  8. \overset {\overset {\bullet \bullet}{\overbrace{}}}{\smile}

Heart

  1. $latex
  2. \heartsuit
  3. $
  4. It should look like this without number 1: \heartsuit $
  5. \heartsuit

HOPE THIS TUTORIAL WAS HELPFUL!