Mathematics Articles

Derivatives of Inverse Functions

Derivatives of Inverse Functions

We discuss the Implicit Function Theorem and demonstrate its importance through examples. We derive differentiation formulas for exponential, logarithmic, and inverse trigonometric functions. Derivatives of Inverse Functions In this section we state the derivative rules for the natural exponential function and the general exponential function. We also go over several examples of the chain rule …

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Implicit Differentiation

Implicit Differentiation (and Logarithmic Differentiation)

The procedure of implicit differentiation is outlined and many examples are given. Proofs of the derivative formulas for the inverse trigonometric functions are provided and several examples of using them are given. Also detailed is the logarithmic differentiation procedure which can simplify the process of taking derivatives of equations involving products and quotients. Implicit Differentiation …

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Derivatives of Trigonometric Functions

Derivatives of Trigonometric Functions

Formulas for finding the derivative of the six trigonometric functions are given. We assume that the trigonometric functions are functions of real numbers (angles measured in radians) because the trigonometric differentiation formulas rely on limit formulas that become more complicated if the degree measurement is used instead of radian measure. Derivative Formulas Theorem. The trigonometric …

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Writing Mathematical Proofs

Mathematical Proofs (Using Various Methods)

In this section we discuss valid arguments, inference rules, and various methods of proof including direct proofs, indirect proofs, proof by contrapositive, and proof by cases.  Valid Arguments An argument is defined as a  statement $q$ being asserted as a consequence of some list of statements $p_1, p_2, \ldots,p_k.$ The statements $p_1, p_2, \ldots,p_k$ are …

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Logical Discourse Using Rules of Inference

Logical Discourse Using Rules of Inference

Logical Discourse In this section we discuss axiomatic systems and inference rules for quantified statements. To give example, we carry out a simple logical discourse for incidence geometry involving points, lines, and incidence.    Axiomatic Systems An axiomatic (or formal) system must contain a set of technical terms that are deliberately chosen as undefined, called undefined …

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Continuous

Continuous (Its Meaning and Applications)

A function is called continuous whenever sufficiently small changes in the input results in arbitrarily small changes in the output. We discuss continuous functions, one-sided and two-sided continuity, and removable continuity. The infamous Intermediate Value Theorem is considered at the end. Definition. A function is continuous at a point $c$ means $f(c)$ is defined, $\lim_{x\to …

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Find the Limit

Find the Limit (Techniques for Finding Limits)

We demonstrate calculating limits using Limit Theorems. We place importance on examples, especially examples with trigonometric and rational functions. We also discuss rationalization, limits of piecewise defined functions, and the Squeeze Theorem. Using Limit Theorems In this topic we concentrate not on the formal definition of a limit of a function of one variable but …

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Limit Definition

Limit Definition (Precise Definition of Limit)

The history of calculus is interconnected with the history of philosophy and scientific discover. In the 16th Century, French philosopher Rene Descartes combined principles of algebra and geometry by mapping lines and objects using horizontal and vertical planes. By applying numbers to different points on geometric shapes, Descartes created what we now refer to as …

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