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 HISTORY OF CALCULUS

History of Calculus is part of the history of mathematics focused on limits, functions, derivatives, integrals, and infinite series. The subject, known historically as infinitesimalcalculus, constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by thefundamental theorem of calculus. Calculus is the study of change, in the same way thatgeometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, moreadvanced courses in mathematics devoted to the study of functions and limits, broadlycalled mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone isinsufficient.

  HISTORY OF INTEGRAL CALCULUS 

Integral calculus is the study of the definitions, properties, and applications of tworelated concepts, the indefinite integral and the definite integral. The process of finding

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the value of an integral is called integration. In technical language, integral calculusstudies two related linear operators. 

The indefinite integral is the antiderivative, the inverse operation to the derivative. F isan indefinite integral of f when f is a derivative of F. (This use of upper- and lower-case

letters for a function and its indefinite integral is common in calculus.)

The definite integral inputs a function and outputs a number, which gives the areabetween the graph of the input and the x-axis. The technical definition of the definiteintegral is the limit of a sum of areas of rectangles, called a Riemann sum. 

 A motivating example is the distances traveled in a given time.

If the speed is constant, only multiplication is needed, but if the speed changes, then we

need a more powerful method of finding the distance. One such method is toapproximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in thatinterval, and then taking the sum (a Riemann sum) of the approximate distance traveledin each interval. The basic idea is that if only a short time elapses, then the speed willstay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exactdistance traveled.

Integration can be thought of as measuring the area under a curve, defined by f(x),between two points (here a and b).

If f(x) in the diagram on the left represents speed as it varies over time, the distancetraveled (between the times represented by a and b) is the area of the shaded region s.

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To approximate that area, an intuitive method would be to divide up the distancebetween a and b into a number of equal segments, the length of each segmentrepresented by the symbol  Δx. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base  Δx andheight h gives the distance (time  Δx multiplied by speed h) traveled in that segment.

 Associated with each segment is the average value of the function above it, f(x)=h. Thesum of all such rectangles gives an approximation of the area between the axis and thecurve, which is an approximation of the total distance traveled. A smaller value for  Δx will give more rectangles and in most cases a better approximation, but for an exactanswer we need to take a limit as  Δx approaches zero.

The symbol of integration is , an elongated S (the S stands for "sum"). The definiteintegral is written as:

and is read "the integral from a to b of f-of-x with respect to x." The Leibniz notation dxis intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width  Δx becomes the infinitesimally small dx. In a formulationof the calculus based on limits, the notation

is to be understood as an operator that takes a function as an input and gives a number,the area, as an output; dx is not a number, and is not being multiplied by f(x).

The indefinite integral, or antiderivative, is written:

Functions differing by only a constant have the same derivative, and therefore theantiderivative of a given function is actually a family of functions differing only by aconstant. Since the derivative of the function y = x² + C, where C is any constant, is y′ =2x, the antiderivative of the latter is given by:

 An undetermined constant like C in the antiderivative is known as a constant of integration. 

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 HISTORY OF QUADRATIC FUNCTIONS

 A quadratic function, in mathematics, is a polynomial function of the form

The graph of a quadratic function is a parabola whose major axis is parallel to the y -

axis.

The expression ax 2 + bx + c  in the definition of a quadratic function is a polynomial of degree 2 or second order, or a 2nd degree polynomial, because the highest exponent of  x  is 2.

If the quadratic function is set equal to zero, then the result is a quadratic equation. Thesolutions to the equation are called the roots of the equation.

Origin of word

The adjective quadratic comes from the Latin word quadratum for square. A term like  x 2 

is called a square in algebra because it is the area of a square with side  x .

In general, a prefix quadr(i)- indicates the number 4. Examples are quadrilateral andquadrant. Quadratum is the Latin word for square because a square has four sides.

Roots

Further information: Quadratic equation

The roots (zeros) of the quadratic function

are the values of  x  for which f ( x ) = 0.

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When the coefficients a, b, and c , are real or complex, the roots are

where the discriminant is defined as

Forms of a quadratic function

 A quadratic function can be expressed in three formats

  is called the general form,

  is called the factored form, where  x 1 and  x 2 are the

roots of the quadratic equation, it is used in logistic map  is called the vertex form and (also the standard form)

where h and k are the x and y coordinates of the vertex, respectively.

To convert the general form to factored form, one needs only the quadratic formula todetermine the two roots r 1 and r 2. To convert the general form to standard form, oneneeds a process called completing the square. To convert the factored form (or standard form) to general form, one needs to multiply, expand and/or distribute thefactors.

Regardless of the format, the graph of a quadratic function is a parabola (as shown

above).

  If (or is a positive number), the parabola opens upward.  If (or is a negative number), the parabola opens downward.

The coefficient a controls the speed of increase (or decrease) of the quadratic functionfrom the vertex, bigger positive a makes the function increase faster and the graphappear more closed.

The coefficients b and a together control the axis of symmetry of the parabola (also the x -coordinate of the vertex) which is at x = -b/2a.

The coefficient b alone is the declivity of the parabola as it crosses the y-axis.

The coefficient c controls the height of the parabola, more specifically, it is the pointwhere the parabola crosses the y -axis.

 x  –intercepts

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Inspection of the factored form shows that the  x -intercepts of the graph are given by theroots of the quadratic function. These are simply the x-coordinates for which thefunction equals zero.

Vertex

The vertex of a parabola is the place where it turns, hence, it's also called the turning

point. If the quadratic function is in vertex form, the vertex is . By the method of completing the square, one can turn the general form

into

so the vertex of the parabola in the general form is

If the quadratic function is in factored form

the average of the two roots, i.e.,

is the  x -coordinate of the vertex, and hence the vertex is

The vertex is also the maximum point if or the minimum point if .

The vertical line

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that passes through the vertex is also the axis of symmetry of the parabola.

  Maximum and minimum points

Using calculus, the vertex point, being a maximum or minimum of the function, can be

obtained by finding the roots of the derivative: 

giving

with the corresponding function value

so again the vertex point coordinates can be expressed as