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Concepts of Calculus

Calculus was invented independently by two mathematicians, Issac Newton of England and Gottfried Wilhelm Leibniz of Germany in the 1680s. Leibniz published his research in the journal ¡¥Acta Eruditorum¡¦ in 1684 and Newton ¡¥s treatise was published in ¡¥Principia Mathematica¡¦ in 1687.
Calculus was mainly developed to solve two types of problems ¡V the determination of tangents to curves, fig.1, and the calculation of area, fig. 2, especially for irregular shapes. Both of these problems are closely related to the rate of change of continuous function and the fundamental concept of calculus (also known as the ¡¥limit¡¦). The following shows what limit is and what are its applications.

Given a function f(x) and a point P(x,y) on its graph, find an equation of the line tangent to the graph at P.

Given a function f(x), find the area between the graph of f(x) and an interval [a,b] on the axis.

In plane geometry, a line is called a ¡§tangent¡¨ to the circle if it meets the circle at precisely one point, fig.3a. However, this definition is not correct for all kinds of curves and irregular shapes. In fig.3b, the line meets the curve exactly once, yet it is not a tangent. Meanwhile in fig.3c, the line is a tangent, yet it meets the curve more than once.

For tangent that applies to curves other than circles as in fig.4a, point P on curve in the xy-plane and Q is any point on the curve different from P, the line through P and Q is the ¡¥secant¡¦ line (the secant line is a line that intercepts a curve at at least two points). If Q moves along the curve toward P, fig.4b., the secant line will rotate toward a ¡¥limiting¡¦ position. The line occupying this limiting position is considered to be the tangent line at P.


The area of the plane regions also leads to the concept of a ¡§limit¡¨ (the approximated area of a curve). Area of some plane regions can be calculated by subdividing them into finite number of rectangles or triangles, then adding the areas of its parts, fig.5. In calculus the area of an irregular shape can be approximated by inscribing rectangles of equal width under the curve and adding the areas of these rectangles, fig.6. If the process was to repeat using more and more rectangles, then the rectangles will tend to fill in the gaps under the curve and the exact area under the curve as a ¡§limiting value¡¨.

Fig.5 Fig.6
In general limit can be written as:

lim f(x) = L which means that the value of a continuous function f(x)
x "žÂ³ a
will approach f(a) or L and x is approached ¡§a¡¨.

Calculus is divided into two main categories. The portion of calculus arising from the tangent problem is called differential calculus. From fig.7, if P(xo ,yo ) and Q(x1 ,y1 ) are distinct points on the curve y = f(x) then the secant line connecting P and Q has a slope of:


If x1 approaches xo then Q will approach P along the graph of f(x) and the secant line through P and Q will approach the tangent line at P which is the same as the slope M secant with limit of x1 approach xo .

If this formula is rewritten in term of a variable h where

Paper and textbook define M tangent as the derivative f `(x) with respect to x of the function f(x).

Note: The notation f(x), f `(x), d f(x)/dx, means the same thing, derivative of
. .
f(x) with respect to x and f(x), f ``(x), d2f(x)/dx2 means the same thing as the second derivative of f(x) with respect to x.

From this definition more techniques of differentiation has been developed which used to solve related rate problems, relative extreme such as first and second derivative for maximum and minimum problem, motion along a line (rectilinear motion by Newton).

The other portion of calculus is called integral calculus which is to find the area under a curve or continuous function f(x) between interval [a,b], fig.8.
Area = total sum of all small area of rectangles

ƒ´ x k is the width and f(xk) is the height
the symbol ƒÃ means the total sum of all rectangles started from 1 to n.
The area of any shape, f(x), is the sum of all small areas of the rectangles
(the width times the height of the rectangle with the width being very, very, small which approaching 0 but not equal to zero).

A special notation has been used to represent the limit of the area above is

The expression on the right side of this equation is called the definite integral of f(x) from interval a to b; where b is the upper and a is the lower limits of integration.


Calculus Made Easy Third Edition
Author: Silvanus P. Thompson

Calculus Fifth Edition
Author: Howard Anton

Encyclopedia Britannica
The history of mathematics

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