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Differential calculus (commonly calculus 1) is used in optimization -- basically finding the best or most likely choice for something. For example, the best movie based on past recommendations, or the best price to sell something at), or the most likely meaning for a word. Series, especially Taylor Series, are used to approximate functions and make computation easier. For example, one can replace e^x with the approximation 1+x+x^2/2, when x is close to 0.
The first thing that come up into my mind is numbers, calculation, integrals and derivatives
It can be used in function approximation, especially in physics and numerical analysis and system & signals. Actually, the essence is that the basis of series is orthorgonal.
I've never taken the Bar exam but I have taken the Series 7 Exam and passed my first try with a 94. The Series 7 is very comprehensive and contains information you are not likely to run into as a practicing financial advisor. You best prepare for 8 weeks or more if you have any hope to pass.
Calculating trigonometric functions, such as sin, cos, tan, requires some fairly involved calculations. If you don't have a calculator, you best use tables. Such functions are calculated with Taylor series; for example, if you want to calculate the sine of an angle, and the angle is specified in degrees, multiply by (pi/180) to convert to radians. Then, having the angle "x" in radians, you can use the formula: sin x = x - x3/3! + x5/5! - x7/7! ... Similarly: cos x = 1 - x2/2! + x4/4! - x6/6! ... Note that, although these are infinite series, they converge pretty quickly, especially for small angles. That means that the individual terms quickly get smaller and smaller.