Calculus Demystified Part 2
Derivatives and Differentiation
The process of finding a derivative is called differentiation. So far we are clear on that.
What the heck were thinking back then our friends Leibniz and Newton? All we can say, they were chosen to help to change the world for good. A quick study of their minds tells us that they had the drive and the inquisitiveness to go beyond than any man has gone..until then. Let's go back to business:
change of rates and its relationships
An independent X number is put to a test and we noticed a logical output called Y. But we studied limits and knew that for a variation of X, there was an expected 'move' on Y.
Newton preferred the use of this notation for differentiation f '(x)
And der Herr Leibniz used the more accepted notation dy/dx to let us know that for an infinitesimal change in x...there will be an slight change in y as a function of x.
The graph illustrates clearly the relationship and the consequent slope obtained. From our studies of Trigonometry, we know that this ratio is a tangent. Let's be more practical since we might loose a critical understanding of the basics.
We have a giraffe, a monkey and a Pomeranian looking over a hill. Our three animals will have a different angle of view. Poor Pomeranian will have to look further up compared with the tall giraffe.
Slope 'm' will vary for our three friends as we already know.
For a function..let's say Y=X2
Assuming that x=3 and we apply limits and binomials for h reaching or getting close to zero (h->0)
f '(3)= 6 on the range y=9 and x=3 as domain.
DERIVATIVE OF AN ELEMENTAL FUNCTION
Now we know how we did get 6 as an answer, but in practice we already have tables that give us the result in just seconds... avoiding us the tenuous process using limits.
In our last example y=x2
for x=3 y=9
f '(3)= 2(3)2-1
f ' (3)= 2x3=6
From here we can study and memorized the derivatives of other well known functions, vectors and even trigonometrical formulas. The applications are extended to statistics and even approximations and optimization problems.
suppose you have 250 cars for sale and your profit per month is given by the formula:
P(X)= -8X2 + 3200x - 80,000
First we get the derivative of P(x) and be careful with critical points
P'(X)= -16X+ 3200
Now for a critical point we assume that P'(x)=0
then our formula will get reduced to:
solving this linear equation will give us X=200
To satisfy the best results we will have to limit our sale to 200 cars.
For a point in a curve where the tangent equals zero, we know that is the point where a curve is changing. Let's go to a practical example:
You are roller skating on a sunny day and you are going up a steeply road...in time you will reach the top and eventually you will descend. When you reached the top, you were looking straight to the horizon, after that split of a second, you were heading down...easier that this?
Determine all the critical points for the function.
Damn! What a big daddy we have in here, but let's apply the derivative and head into the solution.
We know already that in order to get a critical point we should equal F'(X)=0
Luckily this derivative is a polynomial, and the only critical points will be those values of X which make the derivative zero.
From there is pretty easy to identify the three critical points...which are
X=0 X=-5 X=3/5
Another example of derivatives is in the foreign exchange of our currency. If Warren buffets knows it then it's a good tip to know. Everyday the rate of the dollar depreciation captures the interest of investors...like we have done it with our functions, there will be peaks and lows and studying those fluctuations can save investors lots of green cash. The formulas used by the top guys from Wall St. will be part of another hub.
Part 3--calculus demystified!
- Calculus demystified Part 3
A third part..that mostly gives a nice introduction to Integrals and antiderivatives.