What is Calculus? Answers for Everyone

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Discussion Overview

The discussion revolves around the concept of calculus, exploring its definitions, applications, and foundational principles. Participants delve into both differential and integral calculus, discussing their roles in understanding rates of change and areas under curves, as well as their applications in physics and real-life scenarios.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants define calculus as the study of the rate of change of things, with examples illustrating its application to curves like parabolas.
  • Warren explains that calculus allows for the derivation of equations representing slopes, using the parabola y=x^2 as an example, where the slope is given by m=2x.
  • Participants discuss the two main branches of calculus: differential calculus, which focuses on derivatives and rates of change, and integral calculus, which deals with finding areas under curves and reconstructing functions from their derivatives.
  • One participant mentions that calculus is often first applied to equations of motion in Newtonian physics, linking derivatives of position to velocity and acceleration.
  • Another participant elaborates on integral calculus, describing how it approximates areas under curves by dividing them into rectangles and taking limits to improve accuracy.
  • A later reply questions the adequacy of textbook explanations for the slope of the parabola, suggesting that many do not clarify the underlying concepts effectively.
  • One participant notes the historical context of calculus, referencing its development and the use of infinitesimal limits in dynamics.

Areas of Agreement / Disagreement

Participants express a range of views on the definitions and explanations of calculus, with some agreeing on its fundamental concepts while others challenge the clarity and completeness of textbook descriptions. The discussion remains unresolved regarding the sufficiency of existing explanations.

Contextual Notes

Some participants highlight limitations in textbook explanations, suggesting that they may not adequately cover the foundational aspects of calculus, particularly in relation to its historical development and practical applications.

Jack
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Everyone here loves to go on about it but what is it?
 
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Calculus is the study of rate of change of things.

- Warren
 
Originally posted by chroot
Calculus is the study of rate of change of things.

- Warren

[?] Can you explain please
 
Let's say you had a line plotted on a graph. You could measure the slope of that line, and you'd say it's constant everywhere.

Now plot a parabola, like y=x^2. You could measure the slope of that curve, but the slope would be different at every point along the line. You could make a large table of values, like this one:

at x=1 the slope is 2
at x=2 the slope is 4
at x=3 the slope is 6
...

But what if you didn't want to have to pull out the ruler to measure the slope every time? What if you could represent the slope by an equation? Then if you wanted to know the slope at x=4, you wouldn't have to measure it -- you could just solve the equation.

Calculus allows you to find this equation. For the parabola y=x^2, it happens that the slope is given by the equation m=2x. Thus at the point x=4, the slope is m = 2(4) = 8.

It's beyond the scope of this board to explain exactly why the slope of the parabola y=x^2 is m=2x, so I will refer you to any good calculus textbook.

In general, calculus deals with equations in which variables are changing -- in this case, the slope.

To provide another more concrete example, let's say you wanted to calculate how much food a person eats over his/her lifetime. You can't just say "a person eats 1.2 pounds of food every day for 80 years" because the amount of food the person eats per day changes throughout their lifetime. You could use calculus to find the answer.

- Warren
 
Calculus goes into two categories:
(a)Differential calculus deals with calculating equations giving the rate of change of functions with respect to some variable. An example would be establishing the exact gradient of the equation at a certain point. Expressions hence found are called the derivative of an equation. For example, the derivative of velocity with respect to time is acceleration.
(b)Integral calculus does the reverse, going from the derivative to a full function, or from a full function to find things like the area under the curve. Such products of integration are called integrals. For example, the integral of velocity is displacement.
 
The first useful thing you generally do with calculus is apply it to the equations of motion in Newtonian physics. The derivative (slope equation) of the position equation is the velocity equation and the derivative of the velocity equation is the acceleration equation.
 
Just to expand a little bit about integral calculus:

Suppose we have a curve but because of the irregularity of the shape, it is rather difficult to compute the area underneath the curve. However, if we divide the area into small rectangles of equal width, we could add the areas of the triangles to get an approximation of the area. The more rectangles we have, the more accurate our approximation.

What definite integrals do is they take the number of rectangles to approach infinity, thus having a very accurate approximation for the area.
 
Greetings !
Originally posted by chroot
Calculus allows you to find this equation. For the parabola y=x^2, it happens that the slope is given by the equation m=2x. Thus at the point x=4, the slope is m = 2(4) = 8.

It's beyond the scope of this board to explain exactly why the slope of the parabola y=x^2 is m=2x, so I will refer you to any good calculus textbook.
Why ?
(This is a thread that asks for an explanation
of calculus. "Most textbooks" don't explain this
properly.)

Live long and prosper.
 
Yep, one could say that calculus is the *algebraic* study of the rate of change of things. For instance, most historians of calculus refer to book II in the principia, where the rule of integration of monomials is suggested, and they neglect to refer to book I, where infinitesimal limits are used for dynamics.
 

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