FIRST ENCOUNTER WITH ARITHMETIC SEQUENCES
Consider the following sequences of real numbers. Sequence 1: 2, 5, 8, 11, 14, … Sequence 2: 11, 16, 21, 26, 31, … Sequence 3: 15, 13, 11, 9, 7, … They are examples of what weRead More
APPLICATIONS OF THE MEAN-VALUE THEOREM IN ESTIMATION
The mean-value theorem is one of the most important theorems in calculus. However, despite its roles in the development of calculus itself, the mean-value theorem can be used in estimation-related problems. This article provides some examples of how it isRead More
HOW TO DETERMINE QUADRATIC FUNCTIONS
In the article Quadratic Functions, we infer the properties of quadratic functions based on given equations of the form f(x) = ax2 + bx + c where a ≠ 0. In the opposite direction, this article describes how to findRead More
ONE-SIDED DERIVATIVES
Just as the concept of one-sided limits in the study of limits of functions, we may find the concept of one-sided derivatives when we learn calculus. In the article The Limit of A Function at A Point, it is claimedRead More
THE LENGTHS OF THE MEDIANS OF A TRIANGLE
Given the lengths of the sides of a triangle, how to find the lengths of the medians? There are several formulae for determining the lengths of the medians. They are easily derived by applying Stewart’s Theorem. Stewart’s Theorem Let ABCRead More
THE CIRCUMRADIUS OF A CONCYCLIC QUADRILATERAL
Suppose that a quadrilateral ABCD be given with the following lengths of sides: AB = 3 cm, BC = 8 cm, CD = 6 cm and AD = 5 cm. (See Figure 1.) What is the radius of the circumcircleRead More
FINDING THE EQUATION OF A TANGENT LINE
Let f be a function defined on to and C be the curve whose equation is y = f (x). This article shows us how to determine the equation of the tangent line to C at the point K(a,b) whereRead More
BIJECTIVE FUNCTIONS
In the article Injective and Surjective Functions, the concepts of injective functions and surjective functions are discussed separately. Some functions are injective but not surjective while others are surjective but not injective. In fact, It is possible for a functionRead More
CRAMER’S RULE
This article elaborates on how to solve systems of linear equations by applying a theorem called the Cramer’s rule. This rule is an alternative to the elimination and substitution techniques. At the first stage of applying the rule, we haveRead More
TECHNIQUES FOR SOLVING SYSTEMS OF LINEAR EQUATIONS IN TWO UNKNOWNS
An arbitrary system of m linear equations in n unknowns has the following general form: where x1, x2, …, xn are the unknowns and the subscripted a’s and b’s denote constants. As with the systems of linear equations inRead More