Integration & Differentiation Practical Example

[AypaPhysics]

hi,
I want to know the practical use of Integration and Differentiation. I know what it is. I just want to know how can i explain it with a practical example.

example : for Differentiation - the dy/dx can be explained that y is dependent on independent variable x. How can d2y/dx2 be explained. why is it a second differentiation required. How can it be explained practically with example.

Similarly can anybody explain me integration.

Thanks in advance

 

[lurflurf]

Differentiation is a process by which it can be determined from a function the ratio between changes in independent and dependent variables in the neighborhood of a point.

Consider the point (x,y) if x is changes we have the new point (x+Δx,y+Δy) and an average rate Δy/Δx this is a global statement that considers the function on an interval and the function Δy=f(x+Δx)-f(x) is not in general linear. We can consider an ideal version with the point (x+dx,y+dy) that considers only the local behavior of the function and dy=f'(x)dx is a linear function of dx.

Often we are interested in calculating quatities that are products
I=f(x)dx
Often we calculate a sum of such products
I=f(x1)dx1+f(x2)dx2+f(x3)dx2
written
I=∫f dx
Such sums are simple is f is locally constant, but often we would like to make such a calculation with a function that changes continuously.
To effect this we write f=lim g where ∫g dx is defined and make a definition
∫f dx=∫ lim g dx=lim ∫g dx

Example in the driving of cars one can look at a speedometer and an odometer.
The readings on each meter are related. By integration the behavior of the odometer can be found from the speedometer. By differentiation the behavior of the speedometer can be found from the odometer.

Since the derivative is so important one often wishes study it on its own. That is a function in which we are interested my itsellf be the derivative of another function. In this case the derivative of this function is said to be the secound derivative of the other function.

 

[HallsofIvy]

Perhaps thinking about the problem that lead Newton to the calculus would help.

Imagine that you are in a spaceship high above the eclipitic of the solar system. You take a snapshot of the planets in their orbits. You could, theoretically, use that snap shot to measure the distance from the sun to each planet at that instant. If you have a formula, say F= GmM/r², giving the gravitational force as a function of distance, you could then calculate the gravitatational force on the planet and so its acceleration at that instant.

But what does acceleration at a given instant mean? In order to have motion, we must have some time change: speed equals change in distance divided by change in time and acceleration is worse: acceleration equals change in speeed divided by change in time. "At an instant" there is no change in time and so cannot be any speed or acceleration!

The calculus uses the limit process to define "rate of change" at a given x value so we can define "rate of change" without having to worry about change in the variables. If x(t) is distance as a function of t, then the derivative, dx/dt, is the speed at each instant and the second derivative, d²/dt², is the acceleration at each instant.

 

[AypaPhysics]

hi,

thanks for the response. I am pleased. I want website links where i could learn more practically. Also can u suggest a book? Can anybody help me out.

thanks in advance