Free Calculus Questions and Problems with Solutions

Free calculus tutorials including problems and questions with solutions are presented.

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 the distance traveled.
• Maximum Area of 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.
• Maximize the Area of Rectangle in a Right Triangle inscribed in a right triangle using the first derivative. Problem with Solution presented.
• Maximize Volume of a Box using the first derivative of the volume.
• Maximize Power Delivered to Electronic Circuits.
• Use Derivative to Find Quadratic Function given tangent lines to the graph of this function.
• Mean Value Theorem Problems are presented where the mean value theorem is used.
• Rolle's Theorem Questions and Examples
• Use First Derivative to Minimize Area of Pyramid with a square base. A detailed solution to the problem is presented.
• Solve Tangent Lines Problems in Calculus. Problems and their solutions are presented.
• Solve Rate of Change Problems in Calculus. Problems and their solutions are presented.
• Use Derivatives to solve problems: Distance-time 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 (optimize) 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.
• Use of First and Second Derivatives to Graphs Functions.

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.
• Limit of Arctan(x) as x Approaches Infinity .
• Find Limits of Functions in Calculus of various functions using different methods. Several Examples with detailed solutions are presented.
• Limits of Constant and Linear Functions.
• 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 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 of functions such as sin x/x as x approaches 0.
• Calculate Limits of 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 ∞ - ∞.
• Convergent and Divergent Series.

Differentiation and Derivatives

• Product Rule of Differentiation with Examples.
• Quotient Rule of Differentiation with Examples.
• Taylor and Maclaurin Series with Examples.
• Find Derivatives of Functions in Calculus using different methods and rules. Several Examples with detailed solutions are presented.
• 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 difference quotient is first calculated then its limitas h approaches zero is computed .
• Proof of Derivative of a^x.
• Derivative of Logarithm function to Any Base : Log_a (x) .
• 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 are presented along with several examples.
• Use of the Chain Rule of Differentiation in Calculus is presented along with several examples.
• Derivatives Involving Absolute Value. Examples on how to find the derivative of functions involving absolute value.
• 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 are presented along with several examples involving products, sums and quotients of trigonometric functions.
• Find Derivative of y = x^x for x > 0.
• Derivative of a Function Raised to the Power of Another Function.
• 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.

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 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.
• Find Critical Numbers of Functions. 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

• Evaluate Integrals of Functions. Examples with detailed solutions.
• Integration by Parts to find integrals. Examples with solutions.
• Integration by Substitution is used to find integrals. Examples with solutions.
• Integral of ln x .
• Integral of Logarithmic Function to Any Base : Log_a (x) .
• Improper integrals with Infinite Intervals . Examples with detailed solutions.
• Trigonometric Substitution in Integrals . Examples with detailed solutions.
• Integral of Absolute Value of x .
• Integral of a^x .
• Evaluate Integrals Involving Quadratics Using Completing Square. Examples with solutions.
• 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.
• 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. Find the area under (and between) curves using definite integrals. Examples and detailed solutions are presented.
• Find Area Between Curves. How to find the area between curves using definite integrals.
• Length of a Curve. Examples with detailed solutions.
• Calculate The Area in Polar Coordinates Curve. Examples with detailed solutions.
• Area of an Ellipse in Polar Coordinates .
• 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?
• 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.
• Partial Fractions Decompositions into simpler ones for integration.
• Integrals of Rational Functions using decomposition of Fractions.
• Online Partial Fractions Decomposition Calculator.
• Table of Integrals. A table of indefinite integrals of functions is presented below.
• Evaluate Integrals Involving Logarithmic functions.
• Rules of Integrals with Examples including solutions and detailed explanations and exercises.
• Table of Laplace Transforms. A comprehensive table of Laplace transforms.
• Table of Fourier Transforms. A table of Fourier transforms.
• Multiple Integrals Calculations and Applications.

Integrals of Power of Trigonometric Functions

Detailed solutions and explanations of
• Integral of sec(x)
• Integral of csc(x)
• Integral of sin^2(x)
• Integral of cos^2(x)
• Integral of sec^3(x)
• Integral of cos^3(x)
• Integral of sin^3(x)
• Integral of sec^4(x)
• Integral of tan^3(x)
• Integral of csc^3(x)
• Integral of sin^2(x)cos^2(x)
• Integral of sin^2(x)cos^3(x)
• Integral of sin^3(x)cos^2(x)
• Integral of sin^4(x)

Differential Equations

• Introduction to Differential Equations. What are differential equations?
• Applications of Differential Equations in modeling real life situations.
• Order and Linearity of Differential Equations with examples and exercises.
• Solve Simple 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. 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 differential equations of the second order.
• Solve Second Order Differential Equations (part 1) where the auxiliary equation has two distinct real solutions.
• Solve Second Order Differential Equations ( part 2) where the auxiliary equation has two equal real solutions.
• Solve Second Order Differential Equations ( part 3) where the auxiliary equation has two complex conjugate solutions.

Parametric Equations and their Applications

• Parametric Equations. Examples and questions with solutions.
• Derivative of Parametric Equations and Applications. Examples and questions with solutions.

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.
• Calculate Partial Derivatives; examples with detailed solutions and exercises with answers are presented.
• Determine the Critical Points of Functions of Two Variables. Examples 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. 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..
• Calculate Second Order Partial Derivatives in Calculus; examples and detailed solutions are presented.

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 app.
• Proof of Derivatives of Quadratic Functions. The proof of the derivative of quadratic functions, using the limits, is presented with explanations.
• 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 of the form f(x) = a x ² + bx + c is explored interactively.
• Concavity of Polynomial Functions of the form f(x) = x ³ + a x ² + bx + c is explored using an app.
• 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.
• 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 app.
• 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 app 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 interactive tutorial using an applet to explore, graphically, the same function and its Fourier series.

More Links and References

• Elementary Statistics and Probability Tutorials and Problems
• Mathematical Formulas and Identities
• Engineering Mathematics
• Home Page