Test: The AP Calculus Essay

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Matthew Mannetta
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Differentials: Differentials can be defined as a subdivision of calculus dealing with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus. Differential calculus consists of the application of a functions derivative at a chosen input value which describes the rate of change of the function near the input value. The process of finding the functions derivative is called differentiation. Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration. Given a function y=f (x) we call dy and dx differentials and the relationship between them is given by, dy = f ’(x) dx.
Note that if we are just given f (x) then the differentials are df and dx and we compute them in the same manner. df = f ‘(x) dx Differentiation has applications to nearly all quantitative disciplines. For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. Newton’s second law of motion states that the derivative of the momentum of a body equals the force applied to the body. The reaction rate of a chemical reaction is a derivative. Derivatives are also used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. The concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometers such as Euclid, Archimedes, and Apollonius of Perga. However, the modern development of calculus is usually credited to Isaac Newton and Gottfried Leibniz. A differential equation is a relation between a collection of functions and their derivatives. Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example, Newton’s second law, which describes the relationship between acceleration and force, can be stated as the ordinary differential equation. The mean value theorem gives a relationship between values of the derivative and values of the original function. If f(x) is a real-valued function and a and b are numbers with a < b, then the mean value theorem says that under mild hypotheses, the slope between the two points (a, f(a)) and (b, f(b)) is equal to the slope of the tangent line to f at some point c between a and b. In other words .
Accountant:
The daily work of accountants revolves around the use of math. Accountants prepare and examine financial records, ensuring they are accurate and in compliance with applicable laws. They also prepare tax returns, calculate the amount of taxes owed, analyze and maintain financial statements, assess company financial operations and recommend efficiencies. Becoming an accountant requires at least a bachelor’s degree in accounting or a related field, such as business administration or finance. Accounting majors can expect their field’s coursework to include a lot of math such as algebra and statistics.
Derivatives and Antiderivatives: The derivative of a function measures the sensitivity to change of a quantity which is determined by another quantity, while the antiderivative is the primitive, or indefinite, integral of a function f whose function F whose derivative is equal to the F ’= f.
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