Entry tags:

calculus, math, whatido

Calculus Tutorial
In my last post, I talked about how I want to do research on how to build better calculators. However, I was being a bit misleading. Your basic hand-held calculator can solve all the math problems that most people encounter in their daily lives, so that problem has already been well solved. When I say I want to build better calculators, I mean that I want to build computers that solve harder math problems.

Here's where I come to a dilemma. My research usually tries to solve problems written in the language of calculus, and just the word "calculus" scares most people. Calculus seems to have this mystique of difficulty. For 85% of the population, calculus means "hard math that you never had because you chose a non-calc path in high school." For the other 14.9% of the population, calculus means "The highest (and hardest) math I know." My high school physics teacher never took calculus. Whenever he talked about it, his voice was filled with equal parts fear and awe, as if calculus were some black art that could rip the physical planes asunder if used improperly.

I want to dispel this myth. Today, I'll try to explain the notation of calculus and how it helps us. Then, I'll explain half of the results of calculus in 5 minutes. The target for this posts is the absolute lay person-- an amateur. I sometimes expect simple geometry, simple high school algebra, and that's about it. So if you don't understand these posts, please let me know in the comments.




First, the motivation. Around the time of Newton, physicists were making all sorts of graphs to describe the motion of planets and objects. After staring at the laboriously hand-drawn graphs, the physicists noticed strange relationships between the slope of graphs to other phenomina. For example, take a look at these graphs of the total distance traveled by a fictional car. The driver begins by punching the gas, slows down to a crawl, accelerates again, then starts backing up.



The top graph shows the total distance traveled by the car in time, and the bottom graph shows the velocity of the car in time. Also, notice that the slope of the line that touches the distance graph is more vertical when the velocity of car is higher. Physicists noticed this, so they invented a notation to describe it. Let f(t) be an abbreviation for "total distance traveled by the car at time t" and let f'(t) be the "slope of the line that just touches the graph at time t", where the apostrophe means "slope of." Using this notation, physicists could then perfectly describe the relationship between distance and velocity: "velocity at time t" = f'(t). It turns out that most physics phenomina can be quickly described using this apostrophe notation.

However, is this notation really useful? After all, the whole point of physics is to translate something in the real world into a math equation we can solve. How do we find the slope of the line just touching a curve? The answer lies in this fact, the first tenant of calculus: If you "zoom in" close enough to any graph, the graph looks like a straight line. Therefore, once you "zoom in" you can just measure the slope of the line.

Take the function f(t) = t*t, for example. Say we want to find the slope when zoomed into t=1. The trick is to pick points that are really close together near t=1, then pretend the function is a line and calculate the slope. Let "b" be a point near "a". If "b" is close enough, we can write:


so for a=1 and b=1.1, we get:


If we zoom in further with b=1.01 and a=1, we get


See the pattern? For a general point, you can prove that for any time t and small jump in the future h,



Therefore, as h gets closer to 0, you get that the f'(t)=2t.

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Why do we care about this? First of all, the "Slope of" notation allows us to summarize physics laws. Secondly, we can use calculus to solve maximization problems. For example, assume we have an equation for the height of a thrown football: g(t)=t*t-t. The ball is thrown at t=0, and lands at t=1. When is the ball at it's maximum height? Before calculus, you spend all your time trying to solve equations like ax+b=0. No one ever tells you how to solve this type of problem.

The trick here is to notice on the graph above that we hit our maximum when the slope is equal to zero. Therefore, we need to solve g'(t)=0. Using the same "small h" trick, we can find that g'(t)=2t-1. Therefore, the ball reaches it's maximum height at t=0.5, because g'(0.5)=1-1=0.

In other words, calculus allows us to take a new problem and put it in a form that we already know how to solve.

This new ability is insanely practical. Lets say you work in business, and you have an equation for expected profit versus requested price, or profit=f(price). Using calculus, you can solve for the maximization of this function and find the price that will earn the most money possible!




This, in a nutshell, is the first 3 months of calculus. You spend all your time calculating "slopes", or "derivatives" of many different functions. You learn some tricks that allow you to quickly calculate the "slope" of arbitrary functions. Then, you drill solving 100 maximization problems until you can do it in your sleep. The reason that you had to learn all those complicated simplification procedures in Algebra II was because calculating the "slope" can sometimes involve some tricky simplifications. From a conceptual standpoint though, you can sum up the first three months of calculus with the following ideas:

• The "slope of a graph at a point" is called the derivative at that point.
  
• If you zoom into any function, it becomes a straight line.
  
• Once you zoom in, you can just measure slope at that point.
  
• Whenever you are at a maximum or minimum, the slope of the graph is zero.
  
• Calculus is the language of physics. If there is a Heaven where you can learn the mysteries and wonders of the universe, then you can be sure that everyone in Heaven knows calculus.
  

Next post, I'll talk about the other half of calculus: integration. I'll show you why the "slope of" aka derivative operation is magically related to the area under a curve.

(Edit, fixed the statistics as per this this graph from the NSF.)