Free Calculus Questions and Problems with Solutions
Free calculus tutorials are presented. The analytical tutorials may be used to further develop your skills in solving problems in calculus. Also topics in calculus are explored interactively, using apps, and analytically with examples and detailed solutions. Calculus problems and questions are also included in this website. Mutlivariable Functions and partial derivatives are included.
Calculus Problems and Questions
 Calculus 1 Practice Question with detailed solutions.
 Optimization Problems for Calculus 1 with detailed solutions.
 Linear Least Squares Fitting. Use partial derivatives to find a linear fit for a given experimental data.
 Minimum Distance Problem. The first derivative is used to minimize distance traveled.
 Maximum Area of Rectangle  Problem with Solution. Maximize the area of a rectangle inscribed in a triangle using the first derivative. The problem and its solution are presented.
 Maximum Radius of Circle  Problem with Solution. Find the size of an angle of a right triangle so that the radius of the circle inscribed is maximum; for a constant hypotenuse.
 Find The Area of a Circle Using Integrals in Calculus.
 Find The Area of an Ellipse Using Calculus.
 Volume of a Spherical Cap using integrals.
 Find The Volume of a Sphere Using Calculus.
 Find The Volume of a Frustum Using Calculus.
 Find The Volume of a Square Pyramid Using Integrals.
 Maximum Area of Triangle  Problem with Solution. The first derivative is used to maximize the area of a triangle inscribed inside a circle.
 Maximum Area of Rectangle in a Right Triangle  Problem with Solution. Maximize the area of a rectangle inscribed in right triangle using the first derivative
 Maximize Volume of a Box. How to maximize the volume of a box using the first derivative of the volume.
 Maximize Power Delivered to Circuits. The first derivative is used to maximize the power delivered to a load in electronic circuits.
 Use Derivative to Find Quadratic Function. Use the first derivative to find the equation of a quadratic function given tangent lines to the graph of this function.
 Mean Value Theorem Problems. Problems, with detailed solutions, where the mean value theorem is used are presented.
 Rolle's Theorem Questions and Examples
 Use First Derivative to Minimize Area of Pyramid. The first derivative is used to minimize the surface area of a pyramid with a square base. A detailed solution to the problem is presented.
 Solve Tangent Lines Problems in Calculus. Tangent lines problems and their solutions are presented.
 Solve Rate of Change Problems in Calculus. Calculus Rate of change problems and their solutions are presented.
 Use Derivatives to solve problems: Distancetime Optimization. A problem to minimize (optimization) the time taken to walk from one point to another is presented.
 Use Derivatives to solve problems: Area Optimization. A problem to maximize (optimization) the area of a rectangle with a constant perimeter is presented.
 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.
 First, Second Derivatives and Graphs Of Functions. A tutorial on how to use the first and second derivatives, in calculus, to graph functions.
Analytical Tutorials
Limits and Continuity
 Introduction to Limits in Calculus. Numerical and graphical examples are used to explain the concept of limits.
 Limits of Absolute Value Functions Questions.
 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.
 Properties of Limits in Calculus. Main theorem in limits and its uses in calculating limits of functions.
 Continuous Functions in Calculus. Introduction definition of the concept of continuous functions in calculus with examples.
 Continuity Theorems and Their use in Calculus. Theorems, related to the continuity of functions and their uses in calculus, are presented and discussed with examples.
 Non Differentiable Functions. Graphical and analytical explanations.
 Questions on Continuity with Solutions.
 Use of Squeezing Theorem to Find Limits. The squeezing theorem is used to find limits of functions such as sin x/x a x approaches 0.
 Calculate Limits of Trigonometric Functions. Many examples with detailed solutions and exercises with answers on calculating limits of trigonometric functions or functions involving trigonometric 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 ∞  ∞.
Differentiation and Derivatives
 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 Definition to Find Derivative. The derivative is found using its definition. The difference quotient is first calculated then its limit computed as h > 0.
 Proof of Derivative of e^x. The definition of the derivative is used to calculate The derivative of e^x.
 Proof of Derivative of ln(x). The derivative of ln(x) is calculated using the definition.
 Proof of Derivative of sin x. The derivative of sin (x) is calculated using the definition of the derivative as a limit.
 Proof of Derivative of cos x. The derivative of cos (x) is calculated using the definition of the derivative as a limit.
 Derivative of tan(x). The derivative of tan (x) is computed using the quotient rule and the derivatives of sin(x) and cos(x).
 Proof of Derivative of cot(x). The proof of the derivative of cot (x) is presented using the quotient rule and the derivatives of sin(x) and cos(x).
 Proof of Derivative of sec(x). The proof of the derivative of sec (x) is presented.
 Proof of Derivative of csc(x). The proof of the derivative of csc (x) is 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.
 Rules of Differentiation of Functions in Calculus. The basic rules of differentiation of functions in calculus are presented along with several examples.
 Use the Chain Rule of Differentiation in Calculus. The chain rule of differentiation of functions in calculus is presented along with several examples.
 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.
 Find Derivative of f(x) = arccos(cos(x)) and graph it.
 Find Derivative of f(x) = arcsin(sin(x)) and graph it.
 Find Derivative of f(x) = arctan(tan(x)) and graph it.
 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.
 Find Derivative of y = x^{x}. A tutorial on how to find the first derivative of y = x^{x} for x > 0.
 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.
Application of Differentiation
 Absolute Minimum and Maximum of a Function, examples with detailed solutions and graphical interpretations.
 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.
 Determine the Concavity of Quadratic Functions. Examples with solutions and exercises with answers.
 Use Derivative to Show That arcsin(x) + arccos(x) = pi/2.
Integrals
 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.
 Evaluate Integrals Involving Quadratics Using Completing Square. Tutorials with examples and detailed solutions and exercises with answers on how to use the techniques completing square and substitution to evaluate integrals. .
 Integrals Involving sin(x) or cos(x) and Exponential. Tutorial to find integrals involving the product of sin(x) or cos(x) with exponential functions. Exercises with answers are at the bottom of the page.
 Integrals Involving sin(x) and cos(x) with odd power. Tutorial to find integrals involving the product of powers of sin(x) and cos(x) with one of the two having an odd power.
 Integrals Involving sin(x) with odd power.
 Integrals Involving sin(x) with even power.
 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 Area Between Curves. How to find the area between curves using definite integrals.
 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?
 Volume by Cylindrical Shells Method. 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 cylindrical shells.
 Partial Fractions Decompositions. How to decompose complicated algebraic fractions into simpler ones for integration?
 Integration Using Partial Fractions. Use decomposition of Fractions in order to evaluate integral.
 Table of Integrals. A table of indefinite integrals of functions is presented below.
 Evaluate Integrals Involving Logarithms  Tutorial. Integrals with integrand containing logarithmic functions.
 Rules of Integrals with Examples including solutions and detailed explanations and exercises.
 Evaluate integrals. Evaluate integrals: A tutorial, with examples and detailed solutions. A set of exercises with answers is presented after the tutorial.
 Evaluate Integrals Involving Logarithms  tutorial. Tutorials, with examples and detailed solutions, exercises with answers, on integrals including logarithmic functions.
 Table of Laplace Transforms. A comprehensive table of Laplace transforms.
 Table of Fourier Transforms. A table of Fourier transforms.
 Multiple Integrals Calculations and Applications.
Differential Equations
 Introduction to Differential Equations. What are differential equations?
 Applications of Differential Equations. Several applications of modeling real life situations applying differential equations.
 Order and Linearity of Differential Equations. Tutorial on the order and linearity of differential equations with examples and exercises.
 Simple Differential Equations. This is a tutorial on solving simple first order differential equations of the form dy / dx = f(x).
 Separable Differential Equations. What are separable differential equations and how to solve them?
 Solve First Order Differential Equations. How to solve first order differential equations. The general solution is discussed and examples with detailed solutions are presented.
 Second Order Differential Equations  Generalities. Review the main definitions and basic ideas behind solving solving differential equations of the second order.
 Solve Second Order Differential Equations  part 1. Tutorials on how to solve differential equations of the second order where the auxiliary equation has two distinct real solutions.
 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.
Multivariable Functions (Functions with several variables)
 Linear Least Squares Fitting. Use partial derivatives to find a linear fit for a given experimental data.
 Introduction to Multivariable Functions. Examples of functions with several variables.
 Partial Derivatives. Examples with detailed solutions and exercises with answers on how to calculate partial derivatives of functions.
 Critical Points of Functions of Two Variables. Many examples to determine the critical points of functions of two variables are presented along with their detailed solution.
 Multivariable Chain Rule.
 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. 3Dimensional 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..
 Second Order Partial Derivatives in Calculus. Tutorials with examples and detailed solutions on how to calculate second order partial derivatives of functions.
Tables of Mathematical Formulas
 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.
Interactive Tutorials
 Online Step by Step Calculus Calculators and Solvers
 The first derivative of a function. Graphical interpretation of the derivative of a function is explored interactively using an applet.
 Derivatives of Quadratic Functions. The derivative of quadratic functions are explored graphically and interactively.
 Derivatives of Polynomial Functions. The derivative of third order polynomial functions are explored interactively and graphically.
 Derivatives of Sine (sin x) Functions. The derivative of sine functions are explored interactively.
 Concavity of Graphs. The definition of the of graphs is introduced along with inflection points.
 Concavity of Graphs of Quadratic Functions. The concavity of the graph of a quadratic function of the form f(x) = a x ^{ 2} + bx + c is explored interactively.
 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.
 Mean Value Theorem. Explore the mean value theorem using an applet.
 Differential Equations  Runge Kutta Method. Explore the Runge Kutta method, a powerful numerical method to approximate solutions to differential equations.
 Definition of the Derivative of a Function. The definition of the derivative of a function in calculus is explored interactively using an applet.
 Definition of Definite Integrals  Riemann Sums. An applet to explore the definition of the definite integral.
 Integral Form of the Definition of Natural Logarithm ln(x). An applet to explore the definition of the natural logarithm ln(x).
 Fourier Series Of Periodic Functions. A tutorial on how to find the Fourier coefficients of a function and an interartive tutorial using an applet to explore, graphically, the same function and its Fourier series.
Mathematical Formulas and Identities
Engineering Mathematics
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