Minimum, Maximum, First and Second Derivatives. A tutorial on how to use calculus theorems using first and second derivatives to determine whether a function has a relative maximum or minimum or neither at a given point.
L'hopital's Rule And The Indeterminate forms 0 / 0 . Several examples and their detailed solutions and exercises with answers on how to use the l'Hopital's theorem to calculate limits of the indeterminate forms 0/0.
Indeterminate forms of Limits. Several examples and their detailed solutions and exercises with answers, on how to calculate limits of indeterminate forms such as
∞ / ∞ 0^{ 0}, ∞^{ 0}, 1^{ ∞}, ∞^{ o} and ∞ - ∞.
Difference Quotient. We start with the definition of the difference quotient and then use several examples to calculate it. Detailed solutions to questions are presented.
Logarithmic Differentiation. A powerful method to find the derivative of complicated functions. The method uses the chain rule and the properties of logarithms.
Table of Derivatives. A table of derivatives of exponential and logarithmic functions, trigonometric functions and their inverses, hyperbolic functions and their inverses.
Implicit Differentiation. Implicit differentiation examples, with detailed solutions, are presented.
Derivative of Inverse Function. Examples with detailed solutions on how to find the derivative of an inverse function are presented.
Derivative of Inverse Trigonometric Functions. Formulas of the derivatives of inverse trigonometric functions are presented along with several other examples involving sums, products and quotients of functions.
Differentiation of Trigonometric Functions. Formulas of the derivatives of trigonometric functions are presented along with several examples involving products, sums and quotients of trigonometric functions.
Differentiation of Exponential Functions is presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.
Differentiation of Logarithmic Functions is presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.
Differentiation of Hyperbolic Functions is presented. Examples, with detailed solutions, involving products, sums, power and quotients of hyperbolic functions are examined.
Newton's Method to Find Zeros of a Function is used to find zeros of functions and solve equations numerically. Examples with detailed solutions on how to use Newton s method are presented.
Linear Approximation of Functions is used to approximate functions by linear ones close to a given point. Examples with detailed solutions on linear approximations are presented.
Derivative, Maximum, Minimum of Quadratic Functions. Differentiation is used to analyze the properties such as intervals of increase, decrease, local maximum, local minimum of quadratic functions. Examples with solutions and exercises with answers.
Find the Volume by Cylindrical Shells Method of a solid of revolution generated by revolving a region bounded by the graph of a function around one of the axes using cylindrical shells.
Maxima and Minima of Functions of Two Variables. Locate relative maxima, minima and saddle points of functions of two variables. Several examples with detailed solutions are presented. 3-Dimensional graphs of functions are shown to confirm the existence of these points.
Optimization Problems with Functions of Two Variables. Several optimization problems are solved and detailed solutions are presented. These problems involve optimizing functions in two variables using first and second order partial derivatives..
Tables of Mathematical Formulas. Several tables of mathematical formulas including decimal multipliers, series, factorial, permutations, combinations, binomial expansion, trigonometric formulas and tables of derivatives, integrals, Laplace and Fourier transforms.
Vertical Tangent. The derivative of f(x) = x ^{ 1 / 3} is explored interactively to understand the concept of vertical tangent to a graph of a function.
Fourier Series Of Periodic Functions. A tutorial on how to find the Fourier coefficients of a function and an interactive tutorial using an applet to explore, graphically, the same function and its Fourier series.