Fundamental Functions





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02) 2D Vector: Ratios

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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...
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04) Differentiation: Basic Rules Theory

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 The first rule, the Power Rule, is the rule used upon functions of x that are bases for non-variable values. The second rule, the Reciprocal Rule, is used on the reciprocal of certain functions or identities, however this rule can be derived (aka found from) the Power Rule. A reciprocal is merely 1 divided by a function, this can be rewritten as f(x)^(-1), thus the function becomes the base of the power of "-1". The third rule, the product rule, is the rule used on the products of multiple functions and identities. This rule is based off the initial theory used to formulate differentiation, the use of the difference of hypothetical infinitesimally small differences. As the derivative is the change, we only consider the two blue boxes of uv' and u'v as a result of the change, we can ignore the u'v' since both values are small, we can consider it negligible. The fourth rule, the Quotient Rule, derivable from the Product Rule The fifth rule, the Chain rule, a si...
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