My Math Blog

Differentiation is a method to find the "Rate of Change" or "Change Per Unit Time" of an equation, this can mean the change in distance on a journey per hour, the change in volume of water in a leaking tub per minute, or even the change in distance of an accelerating race car per second. These results can be expanded further: finding the velocity of an accelerating ball at a certain distance, finding the current change in volume of a tub of water at a specific height, the finding the maximum area of an object with a constant perimeter, and more. As long as there is a change in a measurement that correlates to the change of another measurement, differentiation can be of use. First and foremost, the differentiated result of a function is known as a derivative. For example, speed can be considered the "Change of Distance Per Unit Time", so speed is the 'Derivative of Distance". Second of all, the amount of times something can be differentiated is inf...

In this scenario, Point P lies on line AB, and ratio AP : PB is 1 : 3. However what does this mean? First, the ratio implies that PB is 3 times the length of AP,. Second, As Points A, P, and B, are all in a line, the direction of AB, AP and PB are the same. Third, seen in the diagram, Vectors OA and OB are known as Vectors "a" and "b" respectively. Therefore, by knowing the statements above, one can find OP. The first step of finding OP is what are it's closest or most relevant connections. Point P lies on line AB, and both OA and OB are known, thus OP will likely be a combination of OA + AP or OB + BP. Second, after finding a path towards OP, next is to find the Vector notation for AB, AP and/or PB. AB can be found by AO + OB = -a + b = b - a. Therefore AP  =  (b - a)/4, and PB = (3/4)(b - a). Thus OP can be found by OP = OA + AP  or OP = OB + BP = OB - PB. The first method would be, OP = a + (b - a)/4 = b/4 + 3a/4. The second method being, OP = b - (3/4)(b - a...

Where Vector means " determining the position of one point in space relative to another". Things to note: 1) While not always the case, Point O is usually considered the origin point, meaning that it's position in 2 dimensional space is (0,0). 2) Due to the limitations of digital text, any Vector going by the naming convention of XY where X and Y are points, implies a Vector Line from Point X to Point Y. Where it would usually look, for OA, like: 3) Vector Lines can be added together, the rule being that the end point of one must be the beginning point of the next, for example: OA + AB = OB, since OA is the vector from Point O to Point A, then AB is the vector from Point A to Point B, adding them together creates a vector that goes from Point O to Point A to Point B. 4) Vectors can also be denoted in a Variable-like fashion, for example OA can be denoted as "a". Thus these Vectors can be arranged in an equation to form the Vector of one point to another. For exa...

 For |ax+b| + c = d, one can square both sides to use the quadratic equation to get both answers of x. First, separate the absolute value from those that are not. Second, square both sides, since squaring anything ensures that the results are always positive when the values are real. This can be shown if |x| can be considered as √[(x)^2]. Thus we can extrapolate that, Therefore, So, We can simplify the Right Hand Side, From this point forward, the Quadratic Equation can be used to find the values of x.