The Calculus
Calculus is a branch of mathematics which uses derivative to analyze the way in which the values of a function vary. Developed on 17th century, Calculus has now applications almost in all areas of human endeavor: engineering, physics, business, economics, astronomy, chemistry, biology, psychology, sociology, etc. Sir Isaac Newton (1642 – 1727) and Gottfried Wilhelm Leibniz (1646 – 1716), working independently from each other, developed the Calculus in connections with their work. Newton used Calculus in finding the velocity of a moving body, the work done by force and the centroid of mass of a body. Leibniz on the other hand focused on geometric calculation like finding the tangent and normal to a curve, area bounded by two or more curves, and volume of a solid. Leibniz is the one who initiated the modern notation of dx and $\displaystyle \int$.
 

Differential Calculus
Calculus is divided into 5 major branches namely: Differential Calculus; Integral Calculus; Differential Equations; Calculus of Variations; and Calculus of Errors. As for this section, we are only concerned with the Differential Calculus. Differential Calculus is a branch of Calculus involving application such as the determination of maximum and minimum points and rate of change.
 

Relation and Function

Not all relations are function but all functions are relation. A good example of a relation that is not a function is a point in the Cartesian coordinate system, say (2, 3). Though 2 and 3 in (2, 3) are related to each other, neither is a function of the other.
 

Function is a relation between two variables that inhibits an apparent connection. If the variables are x and y, then y can be determined for some range of values of x. We call this, y as a function of x denoted by y = f (x). Differential Calculus is limited only to those relations that are functions defined by equations.
 

Limits

We can redefine Calculus as a branch of mathematics that enhances Algebra, Trigonometry, and Geometry through the limit process. Calculus simply will not exist without limits because every aspect of it is in the form of a limit in one sense or another. To illustrate this notion, consider a secant line whose slope is changing until it will become a tangent (or the slope of the curve) at point P (see figure below). Then we can say that the slope of the curve at any point P is the limit of the slope of the secant through P.
 

tangent-limit-of-secant.gif

 

Another is by considering the area of a region bounded by curve shown in figures (a), (b), and (c) below. The area can be approximated by summing up the areas of series of rectangles. As the number of rectangles increases, the sum of their areas will be close enough to the area in (c). We can then say that the area of the region bounded by a curve is the limit of the sum of areas of approximating rectangles.
 

limit-area.gif

 

We can therefore define limit as a number such that the value of a given function remains arbitrarily close to this number when the independent variable is sufficiently close to a specified point.
 

Theorems on Limits

  1. If $f(x) = c$, a constant, then $\displaystyle \lim_{x \to a} f(x) = c$.
     
  2. $\displaystyle \lim_{x \to a} k \, f(x) = k \, \lim_{x \to a} f(x)$, k being constant.
     
  3. $\displaystyle \lim_{x \to a} \left[ f(x) \pm g(x) \right] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$
     
  4. $\displaystyle \lim_{x \to a} \left[ f(x) \times g(x) \right] = \left[ \lim_{x \to a} f(x) \right] \times \left[ \lim_{x \to a} g(x) \right]$
     
  5. $\displaystyle \lim_{x \to a} \left[ \dfrac{f(x)}{g(x)} \right] = \dfrac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, provided $\displaystyle \lim_{x \to a} g(x) \ne 0$
     
  6. $\displaystyle \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$, provided $\sqrt[n]{\lim_{x \to a} f(x)}$ is a real number.
     
  7. $\displaystyle \lim_{x \to 0} \dfrac{\sin x}{x} = 1$, x is in radians.

 

L'Hospital’s Rule for Indeterminate type 0/0

If $a$ is a number, if $f(x)$ and $g(x)$ are differentiable and $g(x) \ne 0$ for all $x$ on some interval $0 \lt \left| x - a \right| \lt \delta$, and if $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, then, when $\displaystyle \lim_{x \to a} \dfrac{f'(x)}{g'(x)}$ exists or is infinite,
 

$\displaystyle \lim_{x \to a}\dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f'(x)}{g'(x)}$

 

where $f'(x)$ and $g'(x)$ are derivatives of $f(x)$ and $g(x)$, respectively.
 

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