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Pauls NotesClose menu Pauls NotesPauls Notes * Home * Classes * Open submenu (Algebra)Algebra * Open submenu (Calculus I)Calculus I * Open submenu (Calculus II)Calculus II * Open submenu (Calculus III)Calculus III * Open submenu (Differential Equations)Differential Equations * Extras * Open submenu (Algebra & Trig Review)Algebra & Trig Review * Open submenu (Common Math Errors)Common Math Errors * Open submenu (Complex Number Primer)Complex Number Primer * Open submenu (How To Study Math)How To Study Math * Misc Links * Contact Me * Links * MathJax Help and Configuration * Privacy Statement * Site Help & FAQ * Terms of Use No results found. Close submenu (Algebra)AlgebraPauls Notes/Algebra * Open submenu (1. Preliminaries)1. Preliminaries * Open submenu (2. Solving Equations and Inequalities)2. Solving Equations and Inequalities * Open submenu (3. Graphing and Functions)3. Graphing and Functions * Open submenu (4. Common Graphs)4. Common Graphs * Open submenu (5. Polynomial Functions)5. Polynomial Functions * Open submenu (6. Exponential and Logarithm Functions)6. Exponential and Logarithm Functions * Open submenu (7. Systems of Equations)7. Systems of Equations Close submenu (1. Preliminaries)1. PreliminariesPauls Notes/Algebra/1. Preliminaries * 1.1 Integer Exponents * 1.2 Rational Exponents * 1.3 Radicals * 1.4 Polynomials * 1.5 Factoring Polynomials * 1.6 Rational Expressions * 1.7 Complex Numbers Close submenu (2. Solving Equations and Inequalities)2. Solving Equations and InequalitiesPauls Notes/Algebra/2. Solving Equations and Inequalities * 2.1 Solutions and Solution Sets * 2.2 Linear Equations * 2.3 Applications of Linear Equations * 2.4 Equations With More Than One Variable * 2.5 Quadratic Equations - Part I * 2.6 Quadratic Equations - Part II * 2.7 Quadratic Equations : A Summary * 2.8 Applications of Quadratic Equations * 2.9 Equations Reducible to Quadratic in Form * 2.10 Equations with Radicals * 2.11 Linear Inequalities * 2.12 Polynomial Inequalities * 2.13 Rational Inequalities * 2.14 Absolute Value Equations * 2.15 Absolute Value Inequalities Close submenu (3. Graphing and Functions)3. Graphing and FunctionsPauls Notes/Algebra/3. Graphing and Functions * 3.1 Graphing * 3.2 Lines * 3.3 Circles * 3.4 The Definition of a Function * 3.5 Graphing Functions * 3.6 Combining Functions * 3.7 Inverse Functions Close submenu (4. Common Graphs)4. Common GraphsPauls Notes/Algebra/4. Common Graphs * 4.1 Lines, Circles and Piecewise Functions * 4.2 Parabolas * 4.3 Ellipses * 4.4 Hyperbolas * 4.5 Miscellaneous Functions * 4.6 Transformations * 4.7 Symmetry * 4.8 Rational Functions Close submenu (5. Polynomial Functions)5. Polynomial FunctionsPauls Notes/Algebra/5. Polynomial Functions * 5.1 Dividing Polynomials * 5.2 Zeroes/Roots of Polynomials * 5.3 Graphing Polynomials * 5.4 Finding Zeroes of Polynomials * 5.5 Partial Fractions Close submenu (6. Exponential and Logarithm Functions)6. Exponential and Logarithm FunctionsPauls Notes/Algebra/6. Exponential and Logarithm Functions * 6.1 Exponential Functions * 6.2 Logarithm Functions * 6.3 Solving Exponential Equations * 6.4 Solving Logarithm Equations * 6.5 Applications Close submenu (7. Systems of Equations)7. Systems of EquationsPauls Notes/Algebra/7. Systems of Equations * 7.1 Linear Systems with Two Variables * 7.2 Linear Systems with Three Variables * 7.3 Augmented Matrices * 7.4 More on the Augmented Matrix * 7.5 Nonlinear Systems Close submenu (Calculus I)Calculus IPauls Notes/Calculus I * Open submenu (1. Review)1. Review * Open submenu (2. Limits)2. Limits * Open submenu (3. Derivatives)3. Derivatives * Open submenu (4. Applications of Derivatives)4. Applications of Derivatives * Open submenu (5. Integrals)5. Integrals * Open submenu (6. Applications of Integrals)6. Applications of Integrals * Open submenu (Appendix A. Extras)Appendix A. Extras Close submenu (1. Review)1. ReviewPauls Notes/Calculus I/1. Review * 1.1 Functions * 1.2 Inverse Functions * 1.3 Trig Functions * 1.4 Solving Trig Equations * 1.5 Trig Equations with Calculators, Part I * 1.6 Trig Equations with Calculators, Part II * 1.7 Exponential Functions * 1.8 Logarithm Functions * 1.9 Exponential and Logarithm Equations * 1.10 Common Graphs Close submenu (2. Limits)2. LimitsPauls Notes/Calculus I/2. Limits * 2.1 Tangent Lines and Rates of Change * 2.2 The Limit * 2.3 One-Sided Limits * 2.4 Limit Properties * 2.5 Computing Limits * 2.6 Infinite Limits * 2.7 Limits At Infinity, Part I * 2.8 Limits At Infinity, Part II * 2.9 Continuity * 2.10 The Definition of the Limit Close submenu (3. Derivatives)3. DerivativesPauls Notes/Calculus I/3. Derivatives * 3.1 The Definition of the Derivative * 3.2 Interpretation of the Derivative * 3.3 Differentiation Formulas * 3.4 Product and Quotient Rule * 3.5 Derivatives of Trig Functions * 3.6 Derivatives of Exponential and Logarithm Functions * 3.7 Derivatives of Inverse Trig Functions * 3.8 Derivatives of Hyperbolic Functions * 3.9 Chain Rule * 3.10 Implicit Differentiation * 3.11 Related Rates * 3.12 Higher Order Derivatives * 3.13 Logarithmic Differentiation Close submenu (4. Applications of Derivatives)4. Applications of DerivativesPauls Notes/Calculus I/4. Applications of Derivatives * 4.1 Rates of Change * 4.2 Critical Points * 4.3 Minimum and Maximum Values * 4.4 Finding Absolute Extrema * 4.5 The Shape of a Graph, Part I * 4.6 The Shape of a Graph, Part II * 4.7 The Mean Value Theorem * 4.8 Optimization * 4.9 More Optimization Problems * 4.10 L'Hospital's Rule and Indeterminate Forms * 4.11 Linear Approximations * 4.12 Differentials * 4.13 Newton's Method * 4.14 Business Applications Close submenu (5. Integrals)5. IntegralsPauls Notes/Calculus I/5. Integrals * 5.1 Indefinite Integrals * 5.2 Computing Indefinite Integrals * 5.3 Substitution Rule for Indefinite Integrals * 5.4 More Substitution Rule * 5.5 Area Problem * 5.6 Definition of the Definite Integral * 5.7 Computing Definite Integrals * 5.8 Substitution Rule for Definite Integrals Close submenu (6. Applications of Integrals)6. Applications of IntegralsPauls Notes/Calculus I/6. Applications of Integrals * 6.1 Average Function Value * 6.2 Area Between Curves * 6.3 Volumes of Solids of Revolution / Method of Rings * 6.4 Volumes of Solids of Revolution/Method of Cylinders * 6.5 More Volume Problems * 6.6 Work Close submenu (Appendix A. Extras)Appendix A. ExtrasPauls Notes/Calculus I/Appendix A. Extras * A.1 Proof of Various Limit Properties * A.2 Proof of Various Derivative Properties * A.3 Proof of Trig Limits * A.4 Proofs of Derivative Applications Facts * A.5 Proof of Various Integral Properties * A.6 Area and Volume Formulas * A.7 Types of Infinity * A.8 Summation Notation * A.9 Constant of Integration Close submenu (Calculus II)Calculus IIPauls Notes/Calculus II * Open submenu (7. Integration Techniques)7. Integration Techniques * Open submenu (8. Applications of Integrals)8. Applications of Integrals * Open submenu (9. Parametric Equations and Polar Coordinates)9. Parametric Equations and Polar Coordinates * Open submenu (10. Series & Sequences)10. Series & Sequences * Open submenu (11. Vectors)11. Vectors * Open submenu (12. 3-Dimensional Space)12. 3-Dimensional Space Close submenu (7. Integration Techniques)7. Integration TechniquesPauls Notes/Calculus II/7. Integration Techniques * 7.1 Integration by Parts * 7.2 Integrals Involving Trig Functions * 7.3 Trig Substitutions * 7.4 Partial Fractions * 7.5 Integrals Involving Roots * 7.6 Integrals Involving Quadratics * 7.7 Integration Strategy * 7.8 Improper Integrals * 7.9 Comparison Test for Improper Integrals * 7.10 Approximating Definite Integrals Close submenu (8. Applications of Integrals)8. Applications of IntegralsPauls Notes/Calculus II/8. Applications of Integrals * 8.1 Arc Length * 8.2 Surface Area * 8.3 Center of Mass * 8.4 Hydrostatic Pressure * 8.5 Probability Close submenu (9. Parametric Equations and Polar Coordinates)9. Parametric Equations and Polar CoordinatesPauls Notes/Calculus II/9. Parametric Equations and Polar Coordinates * 9.1 Parametric Equations and Curves * 9.2 Tangents with Parametric Equations * 9.3 Area with Parametric Equations * 9.4 Arc Length with Parametric Equations * 9.5 Surface Area with Parametric Equations * 9.6 Polar Coordinates * 9.7 Tangents with Polar Coordinates * 9.8 Area with Polar Coordinates * 9.9 Arc Length with Polar Coordinates * 9.10 Surface Area with Polar Coordinates * 9.11 Arc Length and Surface Area Revisited Close submenu (10. Series & Sequences)10. Series & SequencesPauls Notes/Calculus II/10. Series & Sequences * 10.1 Sequences * 10.2 More on Sequences * 10.3 Series - The Basics * 10.4 Convergence/Divergence of Series * 10.5 Special Series * 10.6 Integral Test * 10.7 Comparison Test/Limit Comparison Test * 10.8 Alternating Series Test * 10.9 Absolute Convergence * 10.10 Ratio Test * 10.11 Root Test * 10.12 Strategy for Series * 10.13 Estimating the Value of a Series * 10.14 Power Series * 10.15 Power Series and Functions * 10.16 Taylor Series * 10.17 Applications of Series * 10.18 Binomial Series Close submenu (11. Vectors)11. VectorsPauls Notes/Calculus II/11. Vectors * 11.1 Vectors - The Basics * 11.2 Vector Arithmetic * 11.3 Dot Product * 11.4 Cross Product Close submenu (12. 3-Dimensional Space)12. 3-Dimensional SpacePauls Notes/Calculus II/12. 3-Dimensional Space * 12.1 The 3-D Coordinate System * 12.2 Equations of Lines * 12.3 Equations of Planes * 12.4 Quadric Surfaces * 12.5 Functions of Several Variables * 12.6 Vector Functions * 12.7 Calculus with Vector Functions * 12.8 Tangent, Normal and Binormal Vectors * 12.9 Arc Length with Vector Functions * 12.10 Curvature * 12.11 Velocity and Acceleration * 12.12 Cylindrical Coordinates * 12.13 Spherical Coordinates Close submenu (Calculus III)Calculus IIIPauls Notes/Calculus III * Open submenu (12. 3-Dimensional Space)12. 3-Dimensional Space * Open submenu (13. Partial Derivatives)13. Partial Derivatives * Open submenu (14. Applications of Partial Derivatives)14. Applications of Partial Derivatives * Open submenu (15. Multiple Integrals)15. Multiple Integrals * Open submenu (16. Line Integrals)16. Line Integrals * Open submenu (17.Surface Integrals)17.Surface Integrals Close submenu (12. 3-Dimensional Space)12. 3-Dimensional SpacePauls Notes/Calculus III/12. 3-Dimensional Space * 12.1 The 3-D Coordinate System * 12.2 Equations of Lines * 12.3 Equations of Planes * 12.4 Quadric Surfaces * 12.5 Functions of Several Variables * 12.6 Vector Functions * 12.7 Calculus with Vector Functions * 12.8 Tangent, Normal and Binormal Vectors * 12.9 Arc Length with Vector Functions * 12.10 Curvature * 12.11 Velocity and Acceleration * 12.12 Cylindrical Coordinates * 12.13 Spherical Coordinates Close submenu (13. Partial Derivatives)13. Partial DerivativesPauls Notes/Calculus III/13. Partial Derivatives * 13.1 Limits * 13.2 Partial Derivatives * 13.3 Interpretations of Partial Derivatives * 13.4 Higher Order Partial Derivatives * 13.5 Differentials * 13.6 Chain Rule * 13.7 Directional Derivatives Close submenu (14. Applications of Partial Derivatives)14. Applications of Partial DerivativesPauls Notes/Calculus III/14. Applications of Partial Derivatives * 14.1 Tangent Planes and Linear Approximations * 14.2 Gradient Vector, Tangent Planes and Normal Lines * 14.3 Relative Minimums and Maximums * 14.4 Absolute Minimums and Maximums * 14.5 Lagrange Multipliers Close submenu (15. Multiple Integrals)15. Multiple IntegralsPauls Notes/Calculus III/15. Multiple Integrals * 15.1 Double Integrals * 15.2 Iterated Integrals * 15.3 Double Integrals over General Regions * 15.4 Double Integrals in Polar Coordinates * 15.5 Triple Integrals * 15.6 Triple Integrals in Cylindrical Coordinates * 15.7 Triple Integrals in Spherical Coordinates * 15.8 Change of Variables * 15.9 Surface Area * 15.10 Area and Volume Revisited Close submenu (16. Line Integrals)16. Line IntegralsPauls Notes/Calculus III/16. Line Integrals * 16.1 Vector Fields * 16.2 Line Integrals - Part I * 16.3 Line Integrals - Part II * 16.4 Line Integrals of Vector Fields * 16.5 Fundamental Theorem for Line Integrals * 16.6 Conservative Vector Fields * 16.7 Green's Theorem Close submenu (17.Surface Integrals)17.Surface IntegralsPauls Notes/Calculus III/17.Surface Integrals * 17.1 Curl and Divergence * 17.2 Parametric Surfaces * 17.3 Surface Integrals * 17.4 Surface Integrals of Vector Fields * 17.5 Stokes' Theorem * 17.6 Divergence Theorem Close submenu (Differential Equations)Differential EquationsPauls Notes/Differential Equations * Open submenu (1. Basic Concepts)1. Basic Concepts * Open submenu (2. First Order DE's)2. First Order DE's * Open submenu (3. Second Order DE's)3. Second Order DE's * Open submenu (4. Laplace Transforms)4. Laplace Transforms * Open submenu (5. Systems of DE's)5. Systems of DE's * Open submenu (6. Series Solutions to DE's)6. Series Solutions to DE's * Open submenu (7. Higher Order Differential Equations)7. Higher Order Differential Equations * Open submenu (8. Boundary Value Problems & Fourier Series)8. Boundary Value Problems & Fourier Series * Open submenu (9. Partial Differential Equations )9. Partial Differential Equations Close submenu (1. Basic Concepts)1. Basic ConceptsPauls Notes/Differential Equations/1. Basic Concepts * 1.1 Definitions * 1.2 Direction Fields * 1.3 Final Thoughts Close submenu (2. First Order DE's)2. First Order DE'sPauls Notes/Differential Equations/2. First Order DE's * 2.1 Linear Equations * 2.2 Separable Equations * 2.3 Exact Equations * 2.4 Bernoulli Differential Equations * 2.5 Substitutions * 2.6 Intervals of Validity * 2.7 Modeling with First Order DE's * 2.8 Equilibrium Solutions * 2.9 Euler's Method Close submenu (3. Second Order DE's)3. Second Order DE'sPauls Notes/Differential Equations/3. Second Order DE's * 3.1 Basic Concepts * 3.2 Real & Distinct Roots * 3.3 Complex Roots * 3.4 Repeated Roots * 3.5 Reduction of Order * 3.6 Fundamental Sets of Solutions * 3.7 More on the Wronskian * 3.8 Nonhomogeneous Differential Equations * 3.9 Undetermined Coefficients * 3.10 Variation of Parameters * 3.11 Mechanical Vibrations Close submenu (4. Laplace Transforms)4. Laplace TransformsPauls Notes/Differential Equations/4. Laplace Transforms * 4.1 The Definition * 4.2 Laplace Transforms * 4.3 Inverse Laplace Transforms * 4.4 Step Functions * 4.5 Solving IVP's with Laplace Transforms * 4.6 Nonconstant Coefficient IVP's * 4.7 IVP's With Step Functions * 4.8 Dirac Delta Function * 4.9 Convolution Integrals * 4.10 Table Of Laplace Transforms Close submenu (5. Systems of DE's)5. Systems of DE'sPauls Notes/Differential Equations/5. Systems of DE's * 5.1 Review : Systems of Equations * 5.2 Review : Matrices & Vectors * 5.3 Review : Eigenvalues & Eigenvectors * 5.4 Systems of Differential Equations * 5.5 Solutions to Systems * 5.6 Phase Plane * 5.7 Real Eigenvalues * 5.8 Complex Eigenvalues * 5.9 Repeated Eigenvalues * 5.10 Nonhomogeneous Systems * 5.11 Laplace Transforms * 5.12 Modeling Close submenu (6. Series Solutions to DE's)6. Series Solutions to DE'sPauls Notes/Differential Equations/6. Series Solutions to DE's * 6.1 Review : Power Series * 6.2 Review : Taylor Series * 6.3 Series Solutions * 6.4 Euler Equations Close submenu (7. Higher Order Differential Equations)7. Higher Order Differential EquationsPauls Notes/Differential Equations/7. Higher Order Differential Equations * 7.1 Basic Concepts for nth Order Linear Equations * 7.2 Linear Homogeneous Differential Equations * 7.3 Undetermined Coefficients * 7.4 Variation of Parameters * 7.5 Laplace Transforms * 7.6 Systems of Differential Equations * 7.7 Series Solutions Close submenu (8. Boundary Value Problems & Fourier Series)8. Boundary Value Problems & Fourier SeriesPauls Notes/Differential Equations/8. Boundary Value Problems & Fourier Series * 8.1 Boundary Value Problems * 8.2 Eigenvalues and Eigenfunctions * 8.3 Periodic Functions & Orthogonal Functions * 8.4 Fourier Sine Series * 8.5 Fourier Cosine Series * 8.6 Fourier Series * 8.7 Convergence of Fourier Series Close submenu (9. Partial Differential Equations )9. Partial Differential Equations Pauls Notes/Differential Equations/9. Partial Differential Equations * 9.1 The Heat Equation * 9.2 The Wave Equation * 9.3 Terminology * 9.4 Separation of Variables * 9.5 Solving the Heat Equation * 9.6 Heat Equation with Non-Zero Temperature Boundaries * 9.7 Laplace's Equation * 9.8 Vibrating String * 9.9 Summary of Separation of Variables Close submenu (Algebra & Trig Review)Algebra & Trig ReviewPauls Notes/Algebra & Trig Review * Open submenu (1. Algebra)1. Algebra * Open submenu (2. Trigonometry)2. Trigonometry * Open submenu (3. Exponentials & Logarithms)3. Exponentials & Logarithms Close submenu (1. Algebra)1. AlgebraPauls Notes/Algebra & Trig Review/1. Algebra * 1.1 Exponents * 1.2 Absolute Value * 1.3 Radicals * 1.4 Rationalizing * 1.5 Functions * 1.6 Multiplying Polynomials * 1.7 Factoring * 1.8 Simplifying Rational Expressions * 1.9 Graphing and Common Graphs * 1.10 Solving Equations, Part I * 1.11 Solving Equations, Part II * 1.12 Solving Systems of Equations * 1.13 Solving Inequalities * 1.14 Absolute Value Equations and Inequalities Close submenu (2. Trigonometry)2. TrigonometryPauls Notes/Algebra & Trig Review/2. Trigonometry * 2.1 Trig Function Evaluation * 2.2 Graphs of Trig Functions * 2.3 Trig Formulas * 2.4 Solving Trig Equations * 2.5 Inverse Trig Functions Close submenu (3. Exponentials & Logarithms)3. Exponentials & LogarithmsPauls Notes/Algebra & Trig Review/3. Exponentials & Logarithms * 3.1 Basic Exponential Functions * 3.2 Basic Logarithm Functions * 3.3 Logarithm Properties * 3.4 Simplifying Logarithms * 3.5 Solving Exponential Equations * 3.6 Solving Logarithm Equations Close submenu (Common Math Errors)Common Math ErrorsPauls Notes/Common Math Errors * 1. General Errors * 2. Algebra Errors * 3. Trig Errors * 4. Common Errors * 5. Calculus Errors Close submenu (Complex Number Primer)Complex Number PrimerPauls Notes/Complex Number Primer * 1. The Definition * 2. Arithmetic * 3. Conjugate and Modulus * 4. Polar and Exponential Forms * 5. Powers and Roots Close submenu (How To Study Math)How To Study MathPauls Notes/How To Study Math * 1. General Tips * 2. Taking Notes * 3. Getting Help * 4. Doing Homework * 5. Problem Solving * 6. Studying For an Exam * 7. Taking an Exam * 8. Learn From Your Errors Paul's Online Notes View Quick Nav Download × search Custom Search * This menu is only active after you have chosen one of the main topics (Algebra, Calculus or Differential Equations) from the Quick Nav menu to the right or Main Menu in the upper left corner. * Classes * Algebra * Calculus I * Calculus II * Calculus III * Differential Equations * Extras * Algebra & Trig Review * Common Math Errors * Complex Number Primer * How To Study Math * Cheat Sheets & Tables * Misc * Contact Me * MathJax Help and Configuration * This menu is only active after you have chosen a topic from the Quick Nav menu to the left or Main Menu in the upper left corner. * Print Page in Current Form (Default) * Show all Solutions/Steps and Print Page * Hide all Solutions/Steps and Print Page Paul's Online Notes Home Show Mobile Notice Show All Notes Hide All Notes Mobile Notice You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width. Welcome to my online math tutorials and notes. The intent of this site is to provide a complete set of free online (and downloadable) notes and/or tutorials for classes that I teach at Lamar University. I've tried to write the notes/tutorials in such a way that they should be accessible to anyone wanting to learn the subject regardless of whether you are in my classes or not. In other words, they do not assume you've got any prior knowledge other than the standard set of prerequisite material needed for that class. In other words, it is assumed that you know Algebra and Trig prior to reading the Calculus I notes, know Calculus I prior to reading the Calculus II notes, etc. The assumptions about your background that I've made are given with each description below. I'd like to thank Shane F, Fred J., Mike K. and David A. for all the typos that they've found and sent my way! I've tried to proof read these pages and catch as many typos as I could, however it just isn't possible to catch all of them when you are also the person who wrote the material. Fred, Mike and David have caught quite a few typos that I'd missed and been nice enough to send them my way. Thanks again Fred, Mike and David! If you are one of my current students and are here looking for homework assignments I've got a set of links that will get you to the right pages listed here. At present I've gotten the notes/tutorials for my Algebra (Math 1314), Calculus I (Math 2413), Calculus II (Math 2414), Calculus III (Math 2415) and Differential Equations (Math 3301) class online. I've also got a couple of Review/Extras available as well. Among the reviews/extras that I've got are an Algebra/Trig review for my Calculus Students, a Complex Number primer, a set of Common Math Errors, and some tips on How to Study Math. I've made most of the pages on this site available for download as well. These downloadable versions are in pdf format. Each subject on this site is available as a complete download and in the case of very large documents I've also split them up into smaller portions that mostly correspond to each of the individual topics. To get the downloadable version of any topic navigate to that topic and then under the Download menu you will be presented an option to download the topic. Here is a complete listing of all the subjects that are currently available on this site as well as brief descriptions of each. Cheat Sheets & Tables Algebra Cheat Sheets - This is as many common algebra facts, properties, formulas, and functions that I could think of. There is also a page of common algebra errors included. There are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper. Trig Cheat Sheets - Here is a set of common trig facts, properties and formulas. A unit circle (completely filled out) is also included. There are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper. Calculus Cheat Sheets - These are a series of Calculus Cheat Sheets that covers most of a standard Calculus I course and a few topics from a Calculus II course. There are four different cheat sheets here. One contains all the information, one has just Limits information, one has just Derivatives information and the final one has just Integrals information. Each cheat sheets comes in two versions. One that is full sized and another that has been reduced, with exactly the same information as the full sized version, that prints two pages on the front and/or back of each page of paper. Common Derivatives and Integrals - Here is a set of common derivatives and integrals that are used somewhat regularly in a Calculus I or Calculus II class. Also included are reminders on several integration techniques. here are two versions of the cheat sheet available. One is full sized and is currently four pages. The other version is a reduced version that contains exactly the same information as the full version except it has just been shrunk down so two pages print of the front and two pages print on the back of a single piece of paper. Table of Laplace Transforms - Here is a list of Laplace transforms for a differential equations class. This table gives many of the commonly used Laplace transforms and formulas. It is currently two pages long with the first page being the Laplace transforms and the second being some information/facts about some of the entries. Class Notes All of the classes, with the exception of Differential Equations, have practice problems (with solutions) you can use for practice as well as a set of assignment problems (without solutions/answers) for instructors to use if they wish. Algebra (Math 1314) [Notes] [Practice Problems] [Assignment Problems] - Topics included in this set of notes/tutorial are : * Preliminaries - Exponent Properties, Rational Exponents, Negative Exponents, Radicals, Polynomials, Factoring, Rational Expressions, Complex Numbers * Solving Equations and Inequalities - Linear Equations, Quadratic Equations, Completing the Square, Quadratic Formula, Applications of Linear and Quadratic Equations, Reducible to Quadratic Form, Equations with Radicals, Linear Inequalities, Polynomial & Rational Inequalities, Absolute Value Equations & Inequalities. * Graphing and Functions - Graphing Lines, Circles, and Piecewise Functions, Function Definition, Function Notation, Function Composition, Inverse Functions. * Common Graphs - Parabolas, Ellipses, Hyperbolas, Absolute Value, Square Root, Constant Function, Rational Functions, Shifts, Reflections, Symmetry. * Polynomial Functions - Dividing Polynomials, Zeroes/Roots of Polynomials, Finding Zeroes of Polynomials, Graphing Polynomials, Partial Fractions. * Exponential and Logarithm Functions - Exponential Functions, Logarithm Functions, Solving Exponential Functions, Solving Logarithm Functions, Applications. * Systems of Equations - Substitution Method, Elimination Method, Augmented Matrix, Nonlinear Systems. The Algebra notes/tutorial assume that you've had some exposure to the basics of Algebra. In particular it is assumed that the exponents and factoring sections will be more of a review for you. Also, it is assumed that you've seen the basics of graphing equations. Graphing particular types of equations is covered extensively in the notes, however, it is assumed that you understand the basic coordinate system and how to plot points. Calculus I (Math 2413) [Notes] [Practice Problems] [Assignment Problems] - Topics included in this set of notes/tutorial are : * Algebra/Trig Review - Trig Functions and Equations, Exponential Functions and Equations, Logarithm Functions and Equations. * Limits - Concepts, Definition, Computing, One-Sided Limits, Continuity, Limits Involving Infinity, L'Hospitals Rule * Derivatives - Definition, Interpretations, Derivative Formulas, Power Rule, Product Rule, Quotient Rule, Chain Rule, Higher Order Derivatives, Implicit Differentiation, Logarithmic Differentiation, Derivatives of Trig Functions, Exponential Functions, Logarithm Functions, Inverse Trig Functions, and Hyperbolic Trig Functions. * Applications of Derivatives - Related Rates, Critical Points, Minimum and Maximum Values, Increasing/Decreasing Functions, Inflection Points, Concavity, Optimization * Integration - Definition, Indefinite Integrals, Definite Integrals, Substitution Rule, Evaluating Definite Integrals, Fundamental Theorem of Calculus * Applications of Integrals - Average Function Value, Area Between Curves, Solids of Revolution, Work. The Calculus I notes/tutorial assume that you've got a working knowledge of Algebra and Trig. There is some review of a couple of Algebra and Trig topics, but for the most part it is assumed that you do have a decent background in Algebra and Trig. These notes assume no prior knowledge of Calculus. Calculus II (Math 2414) [Notes] [Practice Problems] [Assignment Problems] - Topics included in this set of notes/tutorial are : * Integration Techniques - Integration by Parts, Integrals Involving Trig Functions, Trig Substitutions, Integration using Partial Fractions, Integrals Involving Roots, Integrals Involving Quadratics, Integration Strategy, Improper Integrals, Comparison Test for Improper Integrals, and Approximating Definite Integrals. * Applications of Integrals - Arc Length, Surface Area, Center of Mass/Centroid, Hydrostatic Pressure and Force, Probability. * Parametric Equations and Polar Coordinates - Parametric Equations & Curves, Calculus with Parametric Equations (Tangents, Areas, Arc Length and Surface Area), Polar Coordinates, Calculus with Polar Coordinates (Tangents, Areas, Arc Length and Surface Area). * Sequences and Series - Sequences, Series, Convergence/Divergence of Series, Absolute Series, Integral Test, Comparison Test, Limit Comparison Test, Alternating Series Test, Ratio Test, Root Test, Estimating the Value of a Series, Power Series, Taylor Series, Binomial Series * Vectors - Basics, Magnitude, Unit Vector, Arithmetic, Dot Product, Cross Product, Projection * Three Dimensional Coordinate System - Equations of Lines, Equations of Planes, Quadratic Surfaces, Functions of Multiple Variables, Vector Functions, Limits, Derivatives, and Integrals of Vector Functions, Tangent Vectors, Normal Vectors, Binormal Vectors, Curvature, Cylindrical Coordinates, Spherical Coordinates The Calculus II notes/tutorial assume that you've got a working knowledge Calculus I, including Limits, Derivatives, and Integration (up to basic substitution). It is also assumed that you have a fairly good knowledge of Trig. Several topics rely heavily on trig and knowledge of trig functions. Calculus III (Math 2415) [Notes] [Practice Problems] [Assignment Problems] - Topics included in this set of notes/tutorial are : * Three Dimensional Coordinate System - Equations of Lines, Equations of Planes, Quadratic Surfaces, Functions of Multiple Variables, Vector Functions, Limits, Derivatives, and Integrals of Vector Functions, Tangent Vectors, Normal Vectors, Binormal Vectors, Curvature, Cylindrical Coordinates, Spherical Coordinates * Partial Derivatives - Limits, Partial Derivatives, Higher Order Partial Derivatives, Differentials, Chain Rule, Directional Derivatives, Gradient. * Applications of Partial Derivatives - Tangent Plane, Normal Line, Relative Extrema, Absolute Extrema, Optimization, Lagrange Multipliers. * Multiple Integrals - Iterated Integrals, Double Integrals, Double Integrals in Polar Coordinates, Triple Integrals, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Change of Variables, Surface Area. * Line Integrals - Vector Fields, Line Integrals With Respect to Arc Length, Line Integrals With Respect to x and y, Line Integrals of Vector Fields, Fundamental Theorem of Line Integrals, Conservative Vector Fields, Potential Functions, Green's Theorem, Curl, Divergence. * Surface Integrals - Parametric Surfaces, Surface Integrals, Surface Integrals of Vector Fields, Stokes' Theorem, Divergence Theorem. The Calculus III notes/tutorial assume that you've got a working knowledge Calculus I, including limits, derivatives and integration. It also assumes that the reader has a good knowledge of several Calculus II topics including some integration techniques, parametric equations, vectors, and knowledge of three dimensional space. Differential Equations (Math 3301) [Notes] - Topics included in this set of notes/tutorial are : * First Order Differential Equations - Linear Equations, Separable Equations, Exact Equations, Equilibrium Solutions, Modeling Problems. * Second Order Differential Equations - Homogeneous and Nonhomogeneous Second Order Differential Equations, Fundamental Set of Solutions, Undetermined Coefficients, Variation of Parameters, Mechanical Vibrations * Laplace Transforms - Definition, Inverse Transforms, Step Functions, Heaviside Functions, Dirac-Delta Function, Solving IVP's, Nonhomogeneous IVP, Nonconstant Coefficient IVP, Convolution Integral. * Systems of Differential Equations - Matrix Form, Eigenvalues/Eigenvectors, Phase Plane, Nonhomogeneous Systems, Laplace Transforms. * Series Solutions - Series Solutions, Euler Differential Equations. * Higher Order Differential Equations - nth order differential equations, Undetermined Coefficients, Variation of Parameters, 3 x 3 Systems of Differential Equations. * Boundary Value Problems & Fourier Series - Boundary Value Problems, Eigenvalues and Eigenfunctions, Orthogonal Functions, Fourier Sine Series, Fourier Cosine Series, Fourier Series. * Partial Differential Equations - Heat Equation, Wave Equation, Laplace's Equation, Separation of Variables. These notes assume no prior knowledge of differential equations. A good grasp of Calculus is required however. This includes a working knowledge of differentiation and integration. Reviews & Extras Algebra/Trig Review - This is an Algebra Review and Trig Review that was originally written for my Calculus I students. It it still geared mostly towards Calculus students with occasional comments on how a topic will be used in a Calculus class. However, anyone needing a review of some of the basic algebra, trig, exponential functions and logarithms should find the information of use. Not all the topics covered in an Algebra or Trig class are covered in this review. I've mostly covered topics that are of particular importance to students in a Calculus class. I have included a couple of topics that are not that important to a Calculus class, but students do seem to have trouble with on occasion. As time permits I will be adding more sections as well. The review is in the form of a problem set with the first solution containing detailed information on how to work that type of problem. Later solutions are usually not as detailed, but may contain more/new information as required. Complex Number Primer - This is a brief introduction to some of the basic ideas involved with Complex Numbers. The topics covered are a brief review of arithmetic with complex numbers, the complex conjugate, modulus, polar and exponential form and computing powers and roots of complex numbers. Note that this primer does assume that you've at least seen some complex numbers prior to reading. The purpose of this document is go a little beyond what most people see when the first are introduced to complex numbers in say a College Algebra class. Also, this document is in no way intended to be a complete picture of complex numbers nor do I cover all the concepts involved (that's a whole class in and of itself). Common Math Errors - As with the Algebra/Trig review this was originally written for my Calculus I class. However, only one of the five sections that I've given here directly addresses the topic of Calculus. The other four sections are more general errors or cover Algebra and Trig errors. There are a couple of calculus examples in the first four sections, but in all of these cases I've also tried to provide non Calculus examples as well. This portion of the site should be of interest to anyone looking for common math errors. If you aren't in a Calculus class or haven't taken Calculus you should just ignore the last section. How To Study Math - This is a short section with some advice on how to best study mathematics. [Contact Me] [Privacy Statement] [Site Help & FAQ] [Terms of Use] © 2003 - 2024 Paul Dawkins Page Last Modified : 10/9/2023