The concept of the limit of any function is the basis of all calculus. It is used to define derivation and integration, which form the basis of calculus. Consider a line on a graph where y= f(x). Here, when f(x) goes closer to a particular number L, x gets closer and closer to c from … Read more

Derivative introduced as rate of change both as that of distance function and geometrically A rate can be explained as the comparison between two quantities of different kinds. A common rate that we use as a physical quantity is a speed. Speed is defined as the rate of distance traveled per unit of time. Also, … Read more

Introduction to Three–dimensional Geometry: the distance between two points and the section formula The familiar formula for the distance between two points in a plane is easily extended from 2D geometry to the following 3D formula. To prove this equation, construct a rectangular box as shown, where: P1 and P2 are opposite vertices.The faces of the … Read more

Introduction to Three–dimensional Geometry: Coordinates of a point In a 3D plane, assume that P is any point in space. In this place, let a be the (directed) distance from the yz-plane to P, let b be the distance from the xz-plane to P, and let c be the distance from the xy-plane to P. … Read more

Introduction to Three–dimensional Geometry: Coordinate axes and coordinate planes in three dimensions To locate a point in a plane, two numbers are needed. Recall that any point in the plane can be represented as an ordered pair (a, b) of real numbers, where a is the x-coordinate and b is the y-coordinate. For this reason, … Read more

Conic sections: Standard equation and simple properties of Hyperbola The standard form of the equation of a hyperbola is developed in a similar methodology to an ellipse. Note, however, that a, b and c are related differently for hyperbolas than for ellipses.For a hyperbola, the distance between the foci and the centre is greater than … Read more

Conic sections: Hyperbola − a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. When we pull the two foci of an ellipse so far apart that they moved outside the ellipse, the hyperbola, another conic section is formed. Although ellipses and hyperbolas have completely different … Read more

For a parabola with the axis of symmetry parallel to the y-axis and vertex at (h, k), the standard form is (x – h)2 = 4p(y – k) Example 1: Find the equation of the parabola that has a minimum at(-2, 6) and passes through the point (2, 8). The axis of symmetry is parallel … Read more

Geometric Definition of a Parabola: The collection of all the points P(x,y) in a plane at the same distance from a fixed point, the focus, as they are from a fixed line called the directrix. Note that “p” represents the distance from the focus to the vertex or the distance from the vertex to the … Read more

The equation of an ellipse with its centre at the origin has one of two forms: Horizontal Ellipse Foci:                   Vertical Ellipse:        Foci: For a horizontal Major Axis and C(0,0): major axis = 2a and minor axis = 2b For a Horizontal Major Axis … Read more