Pengertian Kalkulus

BAB I

INTRODUCTION

1.1.  What Is Calculus?

To a Roman in the days of the empire a “calculus” was a pebble used in counting and in gambling. Centuries later “calculare” came to mean “to compute”, “to reckon”, “to figure out”. To the engineer and mathematician of today calculus is elementary mathematics (algebra, geometry, trigonometry) enhanced by the by limit process. 

Calculus takes ideas from elementary mathematics and extends them to a more general situation. Here are some example. On the left-hand side you will find an idea from elementary mathematics; on the right, this same idea as extended by calculus.

ELEMENTARY MATHEMATICS	CALCULUS
Slope of a line Y=mx+b	Slope of a curve Y=f(x)
Tange  nt line to a circle	Tangent line to a more general curve
Average velocity, Average acceleration.	Instantaneous velocity, Instantaneous acceleration.
Distance moved under A constant velocity.	Distance moved under Varying velocity.
Area of a region bounded by line segments	Area of a region bounded by curves.
Sum of a finite collection of numbers,	Sum of an infinite series.
Average of a finite collection of numbers.	Average value of a function on an interval.
Length of a line segment	Length of a curve
Center of a circle	Centroid of a region
volume of a rectangular solid	Volume of a solid with a curve boundary
Surface area of a cylinder	Surface area of a more general solid
Tangent plane to a sphere	Tangent plane to a more general surface
work done by a constant force	Work done by a varying force
mass of an object of constant density	Mass of an object of varying density
Center of a sphere	Center of gravity of a more general solid

It is fitting to say something about of discovery of calculus and the people who made it. A certain amount of  preliminary work was done during the first half of the seventeenth century, but the actual discovery can be credited to Sir Isaac Newton (1642-1727) and to Gottfried Wilhelm Leibniz (1646-1716), an Englishman and a German. Newton’s discovery is one of the few good turns that the Great Plague did mankind. The plagued forced the closing of Cambridge University in 1665 and young Isaac Newton  of Trinity College returned to this home in Lincolnshire for eighteen months of meditation, out of which grew his method of fluxions, his theory of gravitation ( Philosophiae Naturalis Principia Mathematica, 1687) and his theory of light (Opticks, 1704). The method of fluxions is what concerns us here. A treatise with this title was written by Newton in 1672.

Leibniz started his work in 1673, eight years after Newton. In 1675 he initiated the basic notation: dx and . His first publications appeared in 1684 and 1686. These made little stir in Germany, but the two brothers Bernoulii of Basel (Switzerland) took up the ideas and added profusely to them. From 1690 onward calculus grew rapidly and reached roughly its present state in about a hundred years. Certain theoretical subtleties were not fully resolved until the twentieth century.

1.2.Nations and formulas Elementary Mathematics

The following outline is presented for reviewnand reference.

1.      Sets

The object x is in the set A (x  is an element of A) The object x is not in the set A Subset, containtment Uniion Intersection Empty set Cartsian products

(These are the only nation form set theory that you will need for this book. If you are not familiar whit them, see Appendix A. I at the back of the book.)

2.      Real Number