What is the purpose of calculus and how can it be applied?

[mitch bass]

Did Newton invent calculus to help him work out his theories on gravity? What is calculus used for? What does it do? Excuse my utter ignorance on the subject please, but I would very much like to know the purpose of calculus and what it has the potential to do.

 

[HallsofIvy]

It's much more general than that but, in part, Newton developed the calculus to deal with gravity. Newton, along with many other people, recognized that, if one assumed that the force of gravity depended on the distance, then the motion of the planets could be explained by the same laws as objects falling on earth. However, there was a "conceptual" problem with that.

Do this "thought" experiment: Imagine yourself in a rocket ship high above the ecliptic. Take a photograph that includes both the Earth and the moon at a specific instant. You could use that photograph to measure the distance from the moon to the earth. You could NOT, of course, determine the speed or acceleration of the moon relative to the Earth from the photograh: the photograph is at a specific instant. You can't measure motion at a specific instant.

(A common high school physics experiment: use strobe lights to get a picture of a ball at several points in its trajectory- use that to measure the speed and acceleration of the ball. Of course, that won't work if you only have ONE picture of the ball.)

That's the conceptual problem: If the gravitational force depends on distance, then it would be theoretically possible to measure the distance between two bodies AT A GIVEN INSTANT, calculate the gravitational force and then use "F= ma" to calculate the acceleration AT THAT INSTANT. Yet the very concept of acceleration (or speed) AT A SINGLE INSTANT doesn't make sense. That was part of "Zeno's paradox".

Calculus was developed to make sense of the concept of "speed" and "acceleration" at a given instant.

 

[FZ+]

Originally posted by mitch bass
Did Newton invent calculus to help him work out his theories on gravity? What is calculus used for? What does it do? Excuse my utter ignorance on the subject please, but I would very much like to know the purpose of calculus and what it has the potential to do.

Have you ever drawn a curved graph? Suppose you pick two points on the graph curve, P1 and P2, and draw a line between them. Now, you move P2 along the curved steadily towards P1, until you get to the point where P1 is infinitely close to P2. Of cause, you never get there but by mathematical methods you can work out what this value tends to, and this gives you the instantaneous derivative, or rate of change at that point. It gives you the gradient of the tangent, so to speak. Suppose you have an equation for the speed of an object using time. Differentiation can tell you the rate of change of speed, it's acceleration at any given moment.

Integration does the opposite. By taking the integral of a curve, you effectively split the graph's are into an infinite number of trapeziums, each with 0 width. By mathematical method, you take the sum of them, and hence get a perfect value for the area under the curve. With the previous equation, you can work out the total distance traveled at each moment.

 

[Sting]

Newton applied calculus for his work in physics but Liebniz's work was more theoretical.

One example of integration is work. In a non-Calculus based Physics course, work is defined as force times displacement (W = fs)

That's great for the ideal world but in the real world the force is not going to be constant, so what do we do? Here's where Calculus comes into place. First we take a very small distance and find the work performed there. We do this over the interval and then add up each individual work. This should give us the work performed on the interval where a variable force is involved.

 

[STAii]

Mainly, calculus was found to solve three problems :
1-The instaneous speed problem.
That is the problem that HallsofIvy exaplained, in fewer words, it is the problem of finding the velocity at a certain instant knowning the graph of displacemet versus time.
2-The tangent line problem.
This problem is very well connected to problem1, it is problem of finding a tangent line of a graph on a certain point.
3-The area under graph problem.
The part of calculus that solves this problem is called integration. The idea of integration (mainly) is to find the area between a graph and the X-axis (normally), this can turn very usefull in problems like the one described by Sting (also -for example- in finding equations of gravitational potential energy between objects in space)