Historical, Visual and Theoretical Calculus

I like this definition.

Quote:

A branch of mathematics divided into two general fields: differential calculus and integral calculus. Differential calculus can be used to find rates of change, like orbits of planets, satellites, and spacecraft. Integral calculus is a method of calculating quantities by splitting them up into a large number of small parts. It can be used to find the surface area of irregular objects. You can find out the total surface area of your car (even the round parts) by using integral calculus. Source: Children's Encyclopedia Britannica vol. 3, p. 308-309, 1989.

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Calculus is basically a means of plotting a curve, or a slope (which is the same thing). Initially it was used to plot the orbits of planets but many phonomania or objects can be represented as curves, or collections of curves so it has wide application in science, economics, and design.

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Caprice,
Hm. Thinking. Lots of thinking...

A branch of mathematics divided into two general fields: differential calculus and integral calculus.
How does this work? Looking at calculus at a whole, how can it cover two entire different fields? Do these fields intertwine with eachother - or in other words, what is the relationship between these two fields?

Differential calculus can be used to find rates of change, like orbits of planets, satellites, and spacecraft.
What is a rate of change? How do you distinguish a rate of change? Can you physically see a rate of change? How?

Integral calculus is a method of calculating quantities by splitting them up into a large number of small parts. It can be used to find the surface area of irregular objects. You can find out the total surface area of your car (even the round parts) by using integral calculus.
How does this work? What measurements need to be known?

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Acquiunk,
Why do you need calculus to plot a curve...?

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It's algebra in motion.

If you consider algebra as a relation between two (or more) variables, calculus provides a relationship between the change of one variable with respect to the other.

In differential calculus you look at the rate of change (1st derivative) and in integral calculus you look at the cumulative effect of change (1st integral).

If you have any knowledge of analytic geometry and the concept of limits I would recommend looking at the fundamental theorems of the calculus.

Rap c∫ Confused

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Rap,

Lets say we have an equation. The equation simply represents the relationship between the variables.

I take it's derivative, and then I take its integral. Do these numbers work together? How are these numbers represented visually?

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Can we work backwards? Lets say we are given an object, and we are told to find the equation to represent it. How would we do that? Would calculus be involved? How can a slope be measured if its not consistant?

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Linear approximation..
.. planes are made from surfaces, and lines from curves..

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Sure, if you have some reference point for integration.

When you integrate an equation you get what is known as a constant of integration. That's a constant factor in the integrated equation that pops out of the integration process (e.g. if K is a constant its derivative is zero, consequently integral of zero is some constant K).

Consequently, some point has to be available to specify which equation out of the family of equations is the integral.

Look at it this way, if I have a line (y=mx+b) the differential of that line (dy/dx) is m (the slope?-lines have constant tangents). Now if I only have the slope (tangent) and integrate to get the line (e.g. ∫dy/dx dx =mx+K where K is come constant) I don't get the line I started with unless I have some reference for K. IOW I need to know some value of x at a particular y to determine that value of K that specifies the line in question (y=mx+b) where K=b.

The same thing exists for curves with changing tangents (slopes). Say I have a continuous Integra table* expression (df/dx) and some reference point (@ x=c, F(c )=M) then ∫df/dx dx =F(x)+K, the K=M-F(c )
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As for the second question, that is effectively what Newton did with that falling apple. He observed that the velocity increased with its position over time and that velocity was the rate of change (derivative) of position. Through application of the calculus he was able to determine equations that would specify the position and the velocity at any point after the initial condition (@t=0 x=0).

Rap c∫ Confused

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raprap wrote:

Sure, if you have some reference point for integration.

When you integrate an equation you get what is known as a constant of integration. That's a constant factor in the integrated equation that pops out of the integration process (e.g. if K is a constant its derivative is zero, consequently integral of zero is some constant K).

Consequently, some point has to be available to specify which equation out of the family of equations is the integral.

Look at it this way, if I have a line (y=mx+b) the differential of that line (dy/dx) is m (the slope?-lines have constant tangents). Now if I only have the slope (tangent) and integrate to get the line (e.g. ∫dy/dx dx =mx+K where K is come constant) I don't get the line I started with unless I have some reference for K. IOW I need to know some value of x at a particular y to determine that value of K that specifies the line in question (y=mx+b) where K=b.

The same thing exists for curves with changing tangents (slopes). Say I have a continuous Integra table* expression (df/dx) and some reference point (@ x=c, F(c )=M) then ∫df/dx dx =F(x)+K, the K=M-F(c )
____________________________________________________________________
As for the second question, that is effectively what Newton did with that falling apple. He observed that the velocity increased with its position over time and that velocity was the rate of change (derivative) of position. Through application of the calculus he was able to determine equations that would specify the position and the velocity at any point after the initial condition (@t=0 x=0).

Rap c∫/

Rap,

What is a tangent line??
What does it look like visually?
Why can't we see them in real life?

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How do you measure speed and position when the object is moving too fast to be measured without technology?

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Willy Mays could. He was the greatest practical mathematician that ever lived. He could solve that differential equation describing the trajectory of that baseball better than any living human. A mere glance and he saw the tangent, the changing velocity and solved for the endpoint, then he did the differential for the best path for putting his glove there.

If I wanted to see calculus in practice I'd go to a football game, or better yet practice and watch some practice drills.

Rap c∫ Confused

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A visual and simple way to think of it is to imagine the graph of an equation which gives one variable as a function of another. There is a calculus of many variables, but I will consider this simple example of one independent and one dependent variable. Now, what makes this really useful is that you can use this for any relationship between physical quantities, but in this example, let's just consider generic variables - y as a function of x. If you take the derivative (differential calculus) of the function, you get another function which is equal to the rate of change of y with respect to x. If you integrate the function, you get another function which gives you the area under the original curve up to any value of x. This turns out to have countless applications in the sciences.

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Good thread as I'm thinkin of taking Calc next year and I'm not quite sure what it is.

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Acquiunk wrote:

Calculus is basically a means of plotting a curve, or a slope (which is the same thing). Initially it was used to plot the orbits of planets but many phonomania or objects can be represented as curves, or collections of curves so it has wide application in science, economics, and design.

The impression I've always had was that the original major use of calculus and early approaches to something like integral calculus was for shipbuilding in the first couple of centuries of the so-called great age of European seafaring.

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Historical, Visual and Theoretical Calculus

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