APPLY IT!!!
Math exists to help solve problems that arise in real-life. and it has proven immensely useful. Applied mathematicians use math to make people's lives easier. They are the ones who use calculus to explain and predict events in real-life. Let's take a peek at the most basic stuff they do.
Using calculus, one can:
- Find the maximum and minimum values of a function.
- Solve equations and approximate functions.
- Draw rudimentary graphs of equations.
- Apply the concept of instantaneous change to real life.
- Find the maximum and minimum values of a function.
- Solve equations and approximate functions.
- Draw rudimentary graphs of equations.
- Apply the concept of instantaneous change to real life.
MATHEMATICAL APPLICATIONS
ExtremE ValuEsProbably the most important application of derivatives is to find the extreme values of a function. Using derivatives, one can readily convert the problem of finding the extreme value into solving an equation. Then other methods, including using calculus again, can be employed.
So. Think about the derivative as the slope of the tangent line at the given point. What if the slope is 0? This could only mean one thing: the tangent line is flat at that point. It must be a peak or a trough. And peaks and troughs are what we are looking for if we want to find extreme values of functions. Is that right? Now there is just two things left to do: examine whether this is a peek or trough, then compare it to the other peaks or troughs found also by solving the equation: derivative=0. However, consider the following scenario: what if we have found a peak, and it is the only peak we found. Is it sure to be the maximum of the function? NO. Consider the following graph: The two black lines at either side means it is cut off from a bigger curve. Now suppose we would like to find the maximum value in this section. Obviously, there is only one peak, and that is in the middle. However, its value is no match compared to the values on either side of the selected curve. why?
This is because on either side, the function could be increasing rapidly, but was cut off. In the middle, the function could be in a transition from a low growth into a low decay. So, this brings something more into our equation: the value of the function on both endpoints of the curve we are studying. Another thing to consider is gaps and isolated points in the curve. suppose we have a parabola: y=x^2. (when x=0, y=-1) Everything is nice. We have the derivative 2x, we solve it and find the minimum to occur at x=0, and y=0. WRONG! Look at the parentheses next to y=x^2. You will find that y=-1 when x=0. since -1<0, we safely conclude out smallest value is -1. However, this is a lucky scenario. What if y=1 at x=0? Then the minimum no longer occurs at x=0. it occurs on either side of it. (it is noncontinuous at x=0.) Below is a step by step approach to attack extreme value problems: 1. Calculate derivative. (usually the easiest in real-life) 2. Solve "derivative=0". (usually the toughest in real-life) 3. Compare the values obtained by evaluating the function at the x's that make the derivative 0 with the values of x's that are on the boundary lines, or make the function noncontinuous. 4. Conclude your results. And now you can brag that you have finished studying the most important application of derivatives! Newton's Way of Solving EquationsNewton's method, undoubtedly names after one of the greatest masters of all times, Issac Newton, is a way to solve equations. Well, you might know about the quadratic formula, the cubic formula, or even the quartic formula. And you say all you have to do to solve equations is to plug numbers in.
WRONG. Try to solve x^5+4x^4+7x^3-10x^2+5=0 for example. There is no "pentagic" formula. No. Nothing to solve the Pentagon. No. Would require something much stronger than that. No. Nothing for the fifth power or above. Then income Mr. Newton and his idea. Let the function we are to solve be f(x). Then we need to first find a value that makes this function quite small. In other words, we have to first find an approximate solution. Don't ask me how. Guesswork's power is unlimited. Then, carry out the following steps repeatedly: This is, until f(x_n)=0.
Now, why does this work? Try to draw a graph, and draw the tangent lines as derivatives when you carry this out. You will find that the lines are flatter and flatter, until... |
MonotonicityWe know what happens when the derivative is zero. It's a peak or trough.
(I secretly wish that I could apply this to the stock market!) Now, suppose we say that the derivative is bigger than or smaller than 0. what is going on here? If the derivative is bigger than zero, the tangent line is pointing upwards. (slope>0) Now, using the same logic, we conclude that when it's smaller than 0, the tangent line points down. It takes no Einstein to notice than when the tangent line goes up, so does the function. if it goes down, the function follows. (sort of like the economy and the stock market. the market is a good representation of the trend of an economy.) To summarize: 1. When derivative>0, we have an increasing function. 2. When derivative<0, we have a decreasing function. 3. At derivative=0, we have an extreme value. Remember, the monotonicity is a powerful weapon in analyzing functions. ConcavityUp there, we learned about monotonicity. We still have one big question to ask ourselves-- is the monotonicity increasing or decreasing? I know this is technically incorrect but think of it as this-- we only know from monotonicity whether a function is increasing or decreasing. How do we know if it's increasing at an increasing pace, or increasing at a decreasing pace?
There, we invented something called concavity. Analogical to monotonicity, concavity is based around the second derivative of a function. (Recall higher order derivatives). So, basically, it's "the derivative of the monotonicity". Now you might see some sense in the question I asked up there. It is defined as follows: 1. A function is concave up at x if f''(x)>0. 2. A function is concave down at x if f''(x)<0. 3. x is called an inflection point if f''(x)=0. The concavity measures the speed the function is increasing or decreasing. For example, an exponential function y=2^x would be concave up for x>0, while a logarithmic function y=log(x) would be concave down for x>0. It is really as simple as that. Now that we have extreme values, monotonicity, and concavity, we can draw the graph of a function even without a TI-84! Amazing? Approximating a FunctionOur friend Euler (pronounced Oil-er) is a great mathematician and one of the "founding fathers" of calculus. He proved to the world how powerful math could be. You are about to witness one of his works.
Suppose I ask you to calculate the square of 40.7892. You may think I'm mad and go for your calculator. However, I knock it out of your hands and tell you the catch: your answer must be within 0.01 of the real answer. So, what do you do? Take an educated guess. How would you guess? Euler did the following: He proposed first that f(x+dx)=f(x)+dy, where dx and dy are tiny increments in the x and y direction. Going a step further, he suggested that f(x+dx) is about f(x)+f'(x)dx. This is basically true as f'(x)dx is just the rise of the tangent run on the run of dx. In other words, Euler approximated a little part of the graph with a tangent line. So, the square of 40.7892 is basically 40^2+0.7892*2. The answer is surprisingly close to the one given by your calculator. But then you ask: why is Texas Instruments not out of business if every problem can be countered by this method? (the Euler method) Because sometimes, it gives horrible approximations. Think about the graph of y=x^10. A tiny increment of x=1 occurred. Now let's approximate y(10) by y(9). The answer is off by 1000 at least. Why? A general rule of thumb says that if a function is increasing very quickly (think about what you have just learned above), the Euler approximation would give a horrible approximation. Vice versa. There is one other catch. Euler said f(x+dx) is about f(x) +f'(x)dx when dx is tiny, tiny, tiny. This usually means 0.01 or 0.1. Keep this in mind. |
NON-MATHEMATICAL APPLICATIONS
PhysicsThe derivative plays a gigantic role in physics. The concept of instantaneous activity is of utmost importance-- using algebra, we can calculate the average speed of an object, yet, with the derivative, we can calculate the instantaneous speed (the speed at any given instant) of an object.
It works like this. Think about a car moving along the x axis. If the location function, or the distance function (the car is f(t) away from the origin after time t) of the car is f(t), then its instantaneous velocity at t is just f'(t). Why does it works like this? So v(t)=f'(t). Probably you have seen distance-time graphs. And the slope=velocity. It is the same here. Think of the derivative=slope of tangent line=slope of a point. Another concept that comes in is acceleration. Everybody knows that F=ma. But what is a? Suppose we go back to our definition of the "position function", f(t). Take a guess using the definition of acceleration. a(t)=f''(t). If your guess is not too far off, it should read a=f''(t). That is, the acceleration is the second derivative of the position function, the derivative of the velocity function. |
EconomicsThe derivative also has significant appliances in the field of economics. Setting complicated models aside, we can look at the simple concept of "marginal benefit". Or "marginal utility".
Well, for those of you who are not familiar, marginal benefit is just a fancy way for economists to say the happiness you get from doing *something* a little bit more. So, perhaps it's a car business. Then, the marginal benefit is the profit you get from selling another car. Yeah? So now, we have the profit function f(t). Don't ask me how do we get it. That's usually the difficult part. But what comes up in tests is ONCE WE HAVE f(t), what is the marginal benefit at say t=1. (Only selling one car makes one a terrible salesman.) Now that would just be f'(1). By the way, there are something in economics called the Law of Diminishing Returns, stating that once something has been done too much, the benefits start to decline. Think about eating stuffed chocolate cakes. It only takes so many to make one puke. |
Something Else to Say
I would not bother with the other myriads of applications because there are so many. It depends on the field in which you wish to study. Chemistry also makes extensive use of calculus. Calculus is used to explain reaction rates, ideal gas expansion, among many other things. Perhaps you have heard the logistic curve. Biologists use it to account for population growth in an environment. The curve came out of a differential equation. Calculus is the basic math requirement if you wish to pursue a career in science.
Even though nowadays, there are computer softwares that can help you do all the math calculations, an understanding of the principles and ideas at work behind the calculations is critical if you wish to apply mathematical models to the real world.
Even though nowadays, there are computer softwares that can help you do all the math calculations, an understanding of the principles and ideas at work behind the calculations is critical if you wish to apply mathematical models to the real world.