November 2015

The Calculus

The Derivative
Derivative of a function is the limit of the ratio of the incremental change of dependent variable to the incremental change of independent variable as change of independent variable approaches zero. For the function y = f(x), the derivative is symbolized by y’ or dy/dx, where y is the dependent variable and x the independent variable.
 

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.
 

Problem
Evaluate the limit of the following.

1. $\displaystyle \lim_{x \to 4} (x^2 + 3x - 5)$
    
2. $\displaystyle \lim_{x \to -1} (2x^2 + x + 4)$
    
3. $\displaystyle \lim_{y \to 3} (y^3 - 2y + 7)$
    
4. $\displaystyle \lim_{y \to -2} (y^3 + 5y - 1)$
    
5. $\displaystyle \lim_{t \to 0} \dfrac{2t^2 + 1}{t^3 + 3t - 4}$