Calculus and Philosophy

I keep hearing people say “I’m never going to use this in real life.” I hear it in all of my classes. I heard it today in Calculus. I heard it the other day in Philosophy, and I often hear it in Chemistry (and mildly agree with them if they aren’t planning on going to med school like I am). 

Here’s the thing. You actually are going to use it, and I’m going to break it down for you.

At the beginning of this semester, I had no clue that Calculus and Philosophy could have anything to do with each other or have anything in common, but they do.

In order to persuade others to see your point of view in philosophy, you need to have a logical, clear explanation of why you see things the way you do. For example, if you believe the death penalty is wrong, you need to be able to explain why. If you think the death penalty is right, you need to explain why. How do you improve your ability to logic and reason? One word: MATH. The higher your ability to think in a mathematically advanced way, the better you’re brain will be able to logic. You don’t have to believe me on this, I know it to be true (unless you have some sort of social phobias or severe social handicaps). I know it to be true because in 1 semester of calculus, I have improved my ability to express my ideas in philosophy despite the fact that I struggle with most of the reading and find the teacher to be less than fantastic at teaching us debate skills (that’s not her job, her job is to teach us ethics). I have shut down more people in my class arguing against my own personal beliefs to know that it’s no coincidence. I wasn’t this good before. I even find myself winning arguments with my husband, the most logical man I have ever met. 

Back to the topic at hand. You will use philosophy in life. I promise you. When you see a $20 bill drop out of a stranger’s pocket, you’ll either choose to ignore what you learned in your ethics class, or you’ll choose to use it and decide what your next move will be. Will you pocket the money, or give it back to it’s rightful owner. Even bigger, you need a job. You want to work at this really great place, but don’t have much in the way of experience. Do you lie on your resume, or do you tell the truth and hope you’re charisma will get you in the door? Philosophy helps us determine what our morals are, it helps us figure out what kind of people we want to be. If you don’t think you use ethics, look back on all the decisions you have to make. I guarantee you will use it your entire life.

Calculus is a bit different. It is possible you will never need to find the derivative of a function ever again once you leave this class. It is possible you won’t need to know what the greatest rate of change of something is if you become an English teacher. However, you might use the math knowledge elsewhere. It’s possible that learning calculus, and becoming better and better with numbers will help me as a doctor. Simple mathematical mistakes can turn a lifesaving drug into a death certificate. Knowing how to use a derivative to quickly find the greatest profit margin for a company, or the quickest way to a destination can be useful in life. Learning math will only improve your mental prowess and aid you in other aspects of your life. I would rather know how to use calculus than end up helpless, feeling like an idiot. There’s nothing wrong with exercising your brain, trust me, I spent too many years letting mine sit dusty on the shelf. 

 

Calculus with Trig Functions

We got our tests back today. Whew! I got a 92. It’s my first A without test corrections.. but you know what, I might just be a dill hole and do test corrections anyway, because you know what sounds better than a 92? A 97.

I feel that I have been slacking on my blog. I have found myself lost for what to post, but I know our portfolio is due this Friday, so I am going to do my best not to drop the ball as we near the finish line.

So on to the subject at hand:

Calculus with Trig Functions:

I felt like today we had a lot of confused people in class. Perhaps it’s just the fact that we haven’t worked with “h” in a while, but I was shocked at the bombardment of questions at our Professor today. She was showing us exactly how she got the derivative of the sin, tan, cos, cot, sec and csc. It made sense to me. The reason we don’t need to derive it on the test is because we can now memorize the derivative of each trig function and use it (I imagine) with more difficult forms of each function.

I’m going to walk through how we get the derivative of one or two of these functions, and then I’m going to make a pretty little chart of memorization for myself to refer to for the FINAL.

Find the Derivative:

One of the rules our professor provided is really key in this next step:

sin(a+b) = sinacosb + sinbcosa (this will be provided for the exam so we don’t have to memorize this particular rule)

– here we simply separated the equation out so that we could factor out the sinx.

We were able to factor out the sinx and cos x here because they did not affect h, which is what we are trying to find the limit of.

Earlier we had already solved for the following equations: and

we can now plug those into the equation we have, and get the following:

See? It’s just using the earliest method we learned to find the derivative of these trig functions. If you can’t remember that method, please read this blog post of mine.

So here’s a list of the need to memorize derivatives:

Here’s a list of useful trig identities:

is shorthand for

sin (a+b) = sin(a)cos(b) + sin(b)cos(a)

cos (a+b) = cos(a)cos(b) + sin(a)sin(b)

*also remember to keep your calculator in radians for trig functions
That’s all I’ve got for you folks today. I am (confusingly) excited to see what our last Calculus class entails.  Then on to Calc 2 in a month!

This is hilarious

Test 3 & Nearing the Final

Today we got our tests back. Still don’t know the grade, but I got all of the free answer questions correct, so I feel really good about it.

I’m finding it harder to post right now because most of what we’re learning is pretty self explanatory. I hope that Calc 2 is this awesome. I have been afraid of this class forever, and I kept waiting to get to that part everyone complains about, you know, when you get completely lost and don’t know where to turn? I never got there.

I’ve managed to follow the professor through the entire class, sure I hit a few speed bumps here and there, but I always got it relatively quickly. Either we have an exceptional professor (which I kind of suspect that we do), or I’m a genius. But seriously, I think our professor just has a wonderful gift for explaining math in a way I get.

The scariest thing to me right now is our comprehensive Final. We only have Wednesday and Friday (and possibly monday? I need to check on that) left to learn. Our Final is on May 4th, right after my Philosophy final. I’m a little sad I have to take 2 finals in one day, especially such different topics, but it won’t be anything new. I’ve taken Philosophy and Calculus back to back all semester. It’s cumulative, so I’ve got to review all the old stuff, and get to know the new stuff really well.

Luckily for us, she isn’t making us memorize the unit circle, nor the chart that goes along with it. That’s a huge relief to me. It was a lot to memorize.

My plan of attack for the final is to start reviewing from the very beginning, review all my old tests, and new material and hope I haven’t forgotten anything. I’ll probably post my full study plan before the big day, even after the blog is officially over. I kind of want to keep writing in it for Calc 2… does that make me weird?

Graphing the Sine, Cosine, Tangent, etc

Using the unit circle, we were able to map out different angle degrees and radians and their corresponding points, tangents, cotangents, sines, cosines, secant and cosecants.

With this information, we were able to actually graph the sine, cosine, tangent and cotangent of a unit circle. The graphs ended up looking like this:

y=sin(x)

y=cos(x)

Notice here that the cosine and sine functions are continuous and periodic, with a period of . Notice also that they are almost identical except that the graph of the cosine is shifted to the left by .

y=tan(x)

y=cot(x)

As you can see, both the cotangent and tangent of x are periodic, meaning the continually repeat in the same pattern, with a period (the length of the graph that repeats) of .

I also wanted to see what the secant and cosecant would look like:

y=csc(x)

y=sec(x)

As you can see, these functions are also periodic with a period of .

If you want to learn more on these graphs, I suggest going to this website: http://www.intmath.com/trigonometric-graphs/1-graphs-sine-cosine-amplitude.php. It has some great info.

I’m guessing we will learn more on this topic. It’s nice to get the review on trig, since I completely forgot most of what I learned in high school. It’s relatively straight forward, so I don’t have much left to say on the topic.

The test went well. I enjoyed having 2 days, I felt good when I finished each test half because I was able to review all of my work and take my time. I wasn’t as nervous this time, and I really felt like I nailed it. I guess we’ll see on Monday.

Hope you’re all enjoying your weekend!

Exam 3: Guidelines and Study Tips a’ la Caitlin

As our Professor explained:

The test will take place over two class periods: Monday, April 16 and Wednesday, April 18.
The problems will come from the following sections:

3 Questions on Previously Covered Material (M)
5 Questions on Curve-Sketching (M)
3 Questions on Absolute Extrema (M)
3 Questions on Optimization
2 Questions on Differential Approximation
2 Questions on Angles and Intro to Trig

Personally, I plan on going back through my notes from the beginning of class, forward, because I feel I’ve forgotten a lot from the very beginning of class. Not forgotten how to do it, necessarily, but forgotten specifics that I need to know. Since it’s not an open-note test, I should probably refresh. I also plan on reviewing the last 2 exams and specifically looking at the questions I missed so that I can get those down perfectly.

Here’s a list of things I plan on doing in bullet point, study tips from a girl with (so far) an A in this class:

• Go over notes, rework practice equations
• Go over old tests, rework missed problems
• Memorize quadratic equation
• Go to MathXL.com and rework most of the homework problems covered by this Exam (and possibly older homework if there is time)
• Do some of the Optimization problems in the book (so I can look them up in the answer key to know I’m doing it right).
• Get a lot of sleep both Sunday and Tuesday night
• Eat a balanced breakfast before class, and probably something sugary right before the test (like a banana or candy in a pinch) – it’s proven to boost memory

That’s pretty much it. Testing can be anxiety-inducing if you let it (thus you shouldn’t eat TOO much candy or it will make it worse), but if you are prepared, you have nothing to worry about, and the anxiety factor kind of goes out the window. I’m planning on doing a post (if I have time) on the Trig stuff we just learned. Hope you all are doing well with the pre-exam studying!

Fun Math Memes

Optimization (Time/Distance Equation)

So I finally figured out the last question on the packet we had in class, I figured it out the morning before our Friday class, and when the professor went over it I was happy to see I was right, but I had to go to the book for help, which is the first time ever I have had to use the book (other than homework).

The equation involved a man throwing a stick for his dog into a lake, and then figuring out how long he had to run on land, and then swim in the water to get their the fastest/most efficiently.

Today I am going to show a similar problem, and how it would be solved.

 

Example: A man is taken hostage in a vehicle driving along a desert road. The man knows the desert well, because he (being wealthy and very paranoid) built a hidden apocalyptic bomb shelter/hideout along this very road. The man realizes that his only option for escape is to wait for an opportunity to open the car, roll out and sprint to his shelter. The vehicle is traveling at 60 miles per hour, The man knows he can sprint at exactly 9 miles per hour (he’s been marathon training at this speed, so he can maintain it for well over an hour). How long does the man need to stay in the car before he jumps out in order to make it to his secret shelter in the shortest (thus safest) amount of time? (For our purposes pretend the rolling takes such a minimal amount of time there is not a need to factor in time lost for it).

Here’s a nice diagram for visual aid (and additional info the man already knew.. like distance) :

How to Solve:

1. First we need to figure out the length he’ll be running in the desert. In order to do this, we can use Pythagorean’s theorem, and we get the following equation:

where L stands for the length he’ll be running.

Now that we have that, we can create a time function.

2. If distance = rate x time, Time is equal to the distance over the rate.

If , then

We can use this to make a function for our problem.

3.

4. Now find it’s derivative:

and set it equal to zero.

5. Solve for X (you may need to use the quadratic equation on other problems like this). and you get:

cross multiply

square each side

(approximately)

6. Now that we know what x is, we can solve for distance driving and distance running.

Distance Driving = 15-x = 13.5

Distance Running = which is aprx 10.1 miles

7. How long will it take him? Let’s plug it in:

hours. Let’s hope he can evade them for that long. What if he waited till the last minute and jumped out at 15 miles?

That’s worse… lets say he jumped out at our starting point of 0 miles? Which would be apx 18 miles away (the long side of the triangle) . That would take him 2 hours!

Obviously we can see these in a similar way we see end points and critical points. We had to solve for the derivative to find the critical point, which in this case, was our poor character’s best chance at survival, and the first and last points on the map (aka the triangle) were the end points.

 

Optimization

So we finally get to the practical application of calculus. I find that the optimization process involving fencing around a rectangular perimeter is relatively simple, however I’m currently having a more difficult time with a box equation and, as the professor predicted, the last equation on our homework, where we have to solve an even more complicated equation involving a triangular diagram.

I find myself stumped on the equations we got for homework, yet I sailed through the ones we did as a group in class.

Luckily, I have some useful links that should help me figure it out, so please check out these links if you, too, are struggling. Once we turn in the homework, I will explain how I solved these equations *hopefully* in a video of some sort.

This helped me solve the Box equation: http://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx

And when I couldn’t figure out why my next equation wouldn’t factor right, I freshened up on the quadratic equation…

http://www.purplemath.com/modules/quadform.htm

And finally for the dog problem:

http://www.maa.org/features/elvisdog.pdf

hope these help!

Finding Absolute Extrema without using a Graph

I had to start by saying… did you know that Tom’s makes a CALCULUS SHOE???? Check it out here http://www.toms.com/calculus-101-men-s-classics. They make it in men’s, women’s, and children’s sizes! I would so totally get these and be a nerd if they weren’t so expensive. 

 

So most of this week we learned about how to find absolute extrema using a graph. Friday we learned how to find Absolute Extrema using an equation. It all seems a bit TOO easy compared to what we’ve already learned, but I think this is because our foundation has been so good, the new stuff makes perfect sense using our firmly planted knowledge leading up to it.  Oh, and here’s how our current area of study is relevant to real life:

So, I’ve decided to explain in my own words how you can find the absolute extrema for a function without having to graph it.

First you’re going to need a function. Let’s use, for example’s sake, the function . Typically you are also given end points such as [-5,5] or (0,32) or . All of these have different meanings. [brackets] like these mean that you can use those two numbers as endpoints. (parenthesis) means you can use all the points leading UP to the numbers in parenthesis but not the actual number. When you get infinity to infinity you have to judge by the problem whether or not you’ve got an absolute max or min by looking at its characteristics, which could mean doing a sign chart.

Moving on.. for our problem, we’re going to solve it with [-5,5] as our end points for the ease of discussion.

Now that we’ve got our endpoints and equation we can follow these steps to get our absolute extrema:

1. Find the derivative of the function. For the derivative is

2. Set the derivative to Zero.  in order to find critical points.

3. Find Critical Points.  can be factored to so our critical points are x=0, x=-2, x=2.

4. Solve for f(x) using critical points and endpoints. It’s easier to use a table so that you can immediately see the comparative numbers for each critical and end point, therefore you can instantly identify the absolute max and min. Your table may look something like this:

X	F(x)
-5	1712
-2	-52
0	12
2	-52
5	1712

An easy way to get these numbers without having to put them all in for the same equation over and over, is to press the y= button on your graphing calculator, plug in the original function, and then push the 2nd button and the graph button (which is a table button when you push 2nd first). This will show you a nice table with your x-coordinates and their corresponding y coordinates. Saves a lot of time (especially for testing purposes). This was an idea presented by our professor.

5. Write out your absolute extrema. So for our function and the given end points, we know that our Absolute Maximum is 1712 at x=-5 and x=5, and our Absolute Minimum is  -52 at x=-2 and x=2.

It is important to remember that there isn’t always an absolute max or absolute min. Sometimes you may have one or the other or neither.

Hope you are all having a nice weekend. I’m off to enjoy the sunshine.

 

What We are Currently Doing

 

I felt this video was useful in explaining what we’re doing in math right now.