Free interactive tutorials that may be used to explore a new topic or as a complement to what have been studied already. The analytical tutorials may be used to further develop your skills in solving problems in calculus. Topics in calculus are explored interactively, using large window java applets, and analytically with examples and detailed solutions. Calculus problems are also included in this website. Mutlivariable Functions and partial derivatives are included.

Derivative of tan(x). The derivative of tan (x) is explored interactively to understand the behavior of the tangent line close to a vertical asymptote.

Concavity of Graphs. The definition of the of graphs is introduced along with inflection points.

Concavity of Polynomial Functions. The concavity of the graph of a polynomial function of the form f(x) = x^{ 3} + a x ^{ 2} + bx + c is explored using an applet.

Vertical Tangent. The derivative of f(x) = x ^{ 1 / 3} is explored interactively to understand the concept of vertical tangent.

Fourier Series Of Periodic Functions. A tutorial on how to find the Fourier coefficients of a function and an intercartive tutorial using an applet to explore, graphically, the same function and its Fourier series.

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.

Find Limits of Functions in Calculus. Find the limits of various functions using different methods. Several Examples with detailed solutions are presented. More exercises with answers are at the end of this page.

Limits of Basic Functions. Limits of the basic functions f(x) = constant and f(x) = x. Examples, exercises, detailed solutions and answers.

Calculate Limits of Trigonometric Functions. Many examples with detailed solutions and exercises with answers on calculating limits of trigonometric functions or functions involving trigonomatric functions.

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 ∞ - ∞.

Find Derivatives of Functions in Calculus. Find the derivatives of various functions using different methods and rules. Several Examples with detailed solutions are presented. Also exercises with answers are included at the end of the page.

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.

Use Definiton to Find Derivative. The derivative is found using its definition. The difference quotient is first calculated then its limit computed as h ---> 0.

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.

Derivatives Involving Absolute Value. Examples on how to find the derivative of functions involving absolute value. Exercises with answers are also included.

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, in calculus, are presented along with several examples involving products, sums and quotients of trigonometric functions.

Differentiation of Exponential Functions. Formulas and examples of the derivatives of exponential functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.

Differentiation of Logarithmic Functions. Examples of the derivatives of logarithmic functions, in calculus, are presented. Several examples, with detailed solutions, involving products, sums and quotients of exponential functions are examined.

Differentiation of Hyperbolic Functions. A table of the derivatives of the hyperbolic functions is presented. Examples, with detailed solutions, involving products, sums, power and quotients of hyprbolic functions are examined.

Newton's Method to Find Zeros of a Function. Newton's method is an example of how differentiation 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. Linear approximation is another example of how differentiation is used to approximate functions by linear ones close to a given point. Examples with detailed solutions on linear approximations are presented.

Find Critical Numbers of Functions. Tutorial on how to find the critical numbers of a function. Several examples with detailed solutions and exercises with answers.

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.

Integration by Parts. Tutorials with examples and detailed solutions and exercises with answers on how to use the technique of integration by parts to find integrals.

Integration by Substitution. Tutorials with examples and detailed solutions and exercises with answers on how to use the powerful technique of integration by substitution to find integrals.

Find Area Under Curve. How to find the area under (and between) curves using definite integrals; tutorials, with examples and detailed solutions are presented.

Find The Volume of a Solid of Revolution. How to find the volume of a solid of revolution generated by revolving a region bounded by the graph of a function around one of the axes using definite integrals?

Properties of Integrals - Tutorial. A tutorial, with examples and detailed solutions, in using the properties of indefinite integrals in calculus is presented. A set of exercises with answers is presented after the tutorial.

Evaluate integrals. Evaluate integrals: A tutorial, with examples and detailed solutions. A set of exercises with answers is presented after the tutorial.

Solve First Order Differential Equations. How to solve first order differential equations. The general solution is discussed and examples with detailed solutions are presented.

Solve Second Order Differential Equations - part 2. Tutorials on how to solve differential equations of the second order where the auxiliary equation has two equal real solutions. Detailed examples and exercises with answers included.

Solve Second Order Differential Equations - part 3. Tutorials on how to solve differential equations of the second order where the auxiliary equation has two complex conjugate solutions. Detailed examples and exercises with answers included.

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.