AN INTRODUCTION TO CALCULUS THE TANGENT Calculus is divided into two main parts – differential calculus and integral calculus. These two branches of calculus were founded on two very familiar concepts respectively – tangents and areas, and although these two concepts seem completely unrelated, calculus has revealed that there is in fact, a close correlation between the two; a relationship that is defined by the Fundamental Theorem of Calculus. We'll explore this relationship when we start integral calculus. We begin this tutorial by exploring the 'founder' of differential calculus – the tangent. So, how do we define the term 'tangent'? Generally, a tangent could be said to be a line that touches a curve. But there is a serious flaw in that definition. If a tangent is indeed a line that merely touches a curve, then it means that a tangent could potentially touch the curve in any number of ways. It could pass through a curve, or intersect a curve twice, or possibly three times. The point is, this definition of a tangent is simply inaccurate. Euclid gives an acceptable definition of a tangent: a line that touches a curve once and only once . We will expand on this definition. Let's assume you have a point P(0.5, 2) lying on the curve y = 1/x, and we want to find the equation of the tangent line at point P. To solve this problem, we apply some basic coordinate geometry. To find the equation of a tangent, we use the slope-point form equation:
y – y1 = m(x – x1) Before we can move any further, we need to know what m (i.e. the slope) is. That's the first problem. This has led to another problem: we need two points to compute a slope, but we have only one point at the moment. The solution is to pick another point Q(x, 1/x) which has to be considerably close to P, and then compute the slope of the secant line PQ.