Category: Algebra

After learning the rules (power rule, product and quotient rules, chain rule, etc.) and doing numerous exercises, a student realizes that the mechanics of Differential Calculus are not that difficult. Actually, they are almost boring, if it wasn’t for that one rule learned after all the other rules: implicit differentiation. ...

I am currently learning about this very powerful calculus tool. This tool is of great value in simplifying some functions prior to differentiation. I will try to explain simply in my own words. What are logarithms? Logarithms were invented by John Napier (1550-1617) for the purpose of simplifying calculation; basically ...

The following Algebra problem is taken from one of the American Mathematics Competitions (AMC 10) Preparation books. It is notable that this is one of many books helping 10th grade students get ready for math olympics competition. The problem is, in my estimation, of medium difficulty. It is really an ...

I have been concentrating on vectors lately because I need to obtain a certain level of proficiency with them in order to study college-level physics. The physics textbook I am using gets heavy on vectors right on chapter one. So, here’s another problem using vectors: a 5,000-pound load is lifted ...

Today I want to show an exercise demonstrating the use of vectors that is so complicated that a graphing utility is required to solve it, and yet, the underlying principle is simple once the basic nature of the problem is understood. Two cranes are lifting an object that weighs 20,240 ...

This week’s post has something to do with Physics (position, velocity and acceleration) but it is mostly about good old plain Algebra and about reading the problem carefully. For starters, since this takes place here on Earth, the acceleration of an object due to gravity is \(\mathbf{-32}\) feet per second ...

Every Algebra student knows the distance formula; that is, how to calculate the distance between two points on the Cartesian coordinate plane: \(\boldsymbol{d = \sqrt{\left(x_{2} – x_{1}\right)^{2} + \left(y_{2} – y_{1}\right)^{2}}}\). This week’s blogpost is about finding the distance between a specific point and the closest point(s) in a function. ...

I have used change of variable while trying to solve quadratic equations. Recently, I came across some very clever uses of this technique. Some of the examples below served me as a reminder that it is OK to arbitrarily change the value of one side of an equation… as long ...

Although there’s an element of Calculus in this week’s post, and even more so, of Trigonometry, I am including it in the Algebra category because that is the more interesting aspect of this problem. The following equation is the path of a projectile propelled at an angle \(\boldsymbol\theta\), where: \[y ...