What is calculus and how does it work?

Answer 1

Calculus is a central branch of mathematics, developed from algebra and geometry. It is built on two major complementary ideas, both of which rely critically on the concept of limits. The first is differential calculus, which is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behavior of functions. This can be illustrated by the slope of a function's graph. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. These two processes act inversely to each other, a fact delivered conclusively by the fundamental theorem of calculus.

Examples of typical differential calculus problems include:

finding the acceleration and velocity of a free-falling body at a particular moment.
finding the optimal number of units a company should produce to maximize their profit.
Examples of integral calculus problems include:

finding areas and volumes
finding the amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure
finding the amount of parking lot plowed by a snowplow of given power with varying rates of snowfall.
Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimal solution to a problem that can be given in mathematical form.

Fundamental theorem of calculus
Main article: Fundamental theorem of calculus
The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another, continuous, function, then differentiating the newly defined function returns the function you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative.

Here is the mathematical formulation of the Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval [a, b], then


Also, for every x in the interval [a, b],

This realization, made by both Newton and Leibniz, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.

Answer 2

Calculus is a central branch of mathematics, developed from algebra and geometry. It is built on two major complementary ideas, both of which rely critically on the concept of limits. The first is differential calculus, which is concerned with the instantaneous rate of change of quantities with respect to other quantities, or more precisely, the local behavior of functions. This can be illustrated by the slope of a function's graph. The second is integral calculus, which studies the accumulation of quantities, such as areas under a curve, linear distance traveled, or volume displaced. These two processes act inversely to each other, a fact delivered conclusively by the fundamental theorem of calculus.

Examples of typical differential calculus problems include:

finding the acceleration and velocity of a free-falling body at a particular moment.
finding the optimal number of units a company should produce to maximize their profit.
Examples of integral calculus problems include:

finding areas and volumes
finding the amount of water pumped by a pump with a set power input but varying conditions of pumping losses and pressure
finding the amount of parking lot plowed by a snowplow of given power with varying rates of snowfall.
Today, calculus is used in every branch of the physical sciences, in computer science, in statistics, and in engineering; in economics, business, and medicine; and as a general method whenever the goal is an optimal solution to a problem that can be given in mathematical form.

Fundamental theorem of calculus
Main article: Fundamental theorem of calculus
The fundamental theorem of calculus states that differentiation and integration are, in a certain sense, inverse operations. More precisely, if one defines one function as the integral of another, continuous, function, then differentiating the newly defined function returns the function you started with. Furthermore, if you want to find the value of a definite integral, you usually do so by evaluating an antiderivative.

Here is the mathematical formulation of the Fundamental Theorem of Calculus: If a function f is continuous on the interval [a, b] and if F is a function whose derivative is f on the interval [a, b], then


Also, for every x in the interval [a, b],

This realization, made by both Newton and Leibniz, was key to the massive proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives. It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences.

Answer 3

The branch of mathematics called calculus originates from describing the basic physical properties of our universe, such as the motion of cars, planets, and molecules. Calculus approaches the paths of objects in motion as curves, or functions, and then determines the value of these functions to calculate their rate of change, area, or volume. In the 18th century, Sir Isaac Newton and Gottfried Leibniz simultaneously, yet separately, described calculus to help solve problems in physics. The two divisions of calculus, differential and integral, can solve problems like the velocity of a car at a certain moment in time, or the surface area of a complex object like a lampshade.

All of calculus relies on the fundamental principle that you can always use approximations of increasing accuracy to find the exact answer. For instance, you can approximate a curve by a series of straight lines: the shorter the lines, the closer they are to resembling a curve. You can also approximate a spherical solid by a series of cubes, that get smaller and smaller with each iteration, that fits inside the sphere. Using calculus, you can determine that the approximations tend toward the precise end result, called the limit, until you have accurately described and reproduced the curve, surface, or solid.

Differential calculus describes the methods by which, given a function, you can find its associated rate of change function, called the "derivative." The function must describe a constantly changing system, such as the temperature variation over the course of the day or the velocity of a planet around a star over the course of one rotation. The derivative of those functions would give you the rate that the temperature changed and the acceleration of the planet, respectively.

Integral calculus is like the opposite of differential calculus. Given the rate of change in a system, you can find the given values that describe the system's input. In other words, given the derivative, like acceleration, you can use integration to find the original function, like velocity. Also, you use integration to calculate values such as the area under a curve, the surface area, or the volume of a solid. Again, this is possible since you begin by approximating an area with a series of rectangles, and make your guess more and more accurate by studying the limit. The limit, or the number toward which the approximations tend, will give you the precise surface area.

Answer 4

Calculus is the way we tackle situations where one needs to imagine things divided up into zillions of tiny bits in order to deal with them. Newton invented it in order to add up all the tiny gravitational forces of every little part of the earth on the moon in order to prove that it comes out the same as if all the mass of the earth was concentrated all at its center instead of being spread out.

I find integral calculus more intuitive than differential calculus. Integrals are just ways of adding stuff up. Derivatives turn out to be just integrals "in reverse." They represent how one quantity changes with respect to another.

The neat thing about calculus is that you can construct EXACT answers to integrals and derivatives just by learning a few tricks and memorizing patterns. But that isn't the important part of calculus. Unfortunately I see too many courses where all the students do is memorize those things instead of learning the deeper issues.

Answer 5

Calculus is the study of change.
You take an equantion and study the way in which it changes, called taking the derivative. You find the way in which the equation changes at smaller and smaller intervals, until you find the instantaneous rate of change. That's why you use dx/dy, the d stands for change in, so you're studying change in x/change in y. Alot of the time in science the y axis is time, so you'll often see dx/dt. And there are all sorts of tricks to find out how your function is changing. And some shortcuts.
It's really fun.
Then, you can take the rate it's changing and see how much it has changed by finding the integral.
That's not as much fun at first, but it grows on you.
It was invented concurrently by Newton and Liebniz. Newton got the credit for it, but we use Liebniz's notation, and they referred to it as "the calculus". So now I do, too.

Answer 6

CALCULUS IS A TOOL OF THE DEVIL THAT EATS AWAY AT THE CORE OF OUR SOUL AND OUR BRAIN. it slowly devours our very humanity and causes us to become mindless high school teachers and Jr college professors. If you have been exposed to calculus you must go home and watch TV listen to rock, rap, country or jazz for a minimum of 3 hours a day for at least two weeks. You can also play video games as a substitute. I suggest playing for for short periods of time 3times a day and for 1 hour before bedtime. do not read any intellectual material for 2 months after exposure to any for a calculus, college algebra or geometry. also go to the movies and call me in one week.

I'm glad you asked. There are hundreds of thousands of people who are exposed every year and do not report it. You are very brave. So many of my friend became exposed that I have formed a support group. FOFWADTCH (Friends of Friends Who Always Do Their Calculus Homework.)

Answer 7

i'd say structure is component to engineering which signifies that a good draw close on math is major, pre-cal is is like algebra to the subsequent factor with the Pi chart and understanding about radians and perspective measures (which seems major) yet Calculus is the learn of derivatives, (the slope of a line at a particular factor) type of unnecessary whatever field of workd your going to, in case you inquire from me.

Answer 8

it is method of differentiating or integrating with a limit it works with formulas