Online courses directory (381)

Identifying and graphing circles, ellipses, parabolas, and hyperbolas. Introduction to Conic Sections. Recognizing conic sections. Conic Sections: Intro to Circles. Graphing circles. Equation of a circle in factored form. Completing the square to write equation in standard form of a circle. Equation of a circle in non-factored form. Graphing circles 2. Conic Sections: Intro to Ellipses. Equation of an ellipse. Foci of an Ellipse. Shifting and scaling parabolas. Parabola intuition example 1. Parabola intuition 1. Parabola Focus and Directrix 1. Focus and Directrix of a Parabola 2. Parabola intuition 2. Conic Sections: Intro to Hyperbolas. Conic Sections: Hyperbolas 2. Conic Sections: Hyperbolas 3. Equation of a hyperbola. Foci of a Hyperbola. Proof: Hyperbola Foci. Identifying an ellipse from equation. Identifying a hyperbola from an equation. Identifying circles and parabolas from equations. Hyperbola and parabola examples. Introduction to Conic Sections. Recognizing conic sections. Conic Sections: Intro to Circles. Graphing circles. Equation of a circle in factored form. Completing the square to write equation in standard form of a circle. Equation of a circle in non-factored form. Graphing circles 2. Conic Sections: Intro to Ellipses. Equation of an ellipse. Foci of an Ellipse. Shifting and scaling parabolas. Parabola intuition example 1. Parabola intuition 1. Parabola Focus and Directrix 1. Focus and Directrix of a Parabola 2. Parabola intuition 2. Conic Sections: Intro to Hyperbolas. Conic Sections: Hyperbolas 2. Conic Sections: Hyperbolas 3. Equation of a hyperbola. Foci of a Hyperbola. Proof: Hyperbola Foci. Identifying an ellipse from equation. Identifying a hyperbola from an equation. Identifying circles and parabolas from equations. Hyperbola and parabola examples.

Your heart sits in the middle of your chest and pumps blood from about 4 weeks after conception until the day that you die. It never stops, and over your lifetime it will pump ~175 million liters of blood. To visualize that, imagine the amount of water that falls over Niagara falls in a few minutes. Remarkable! This little pump is the size of your clenched fist and in an adult can weigh about 300 grams. Watch these videos to learn more about how the heart works, blood flow in arteries and veins, and blood pressure. Meet the heart!. Layers of the Heart. Flow through the Heart. Two Circulations in the Body. Lub Dub. Heart Quiz. Circulatory System and the Heart. Arteries vs. Veins - What's the difference?. Layers of a blood vessel. Arteries, Arterioles, Venules, and Veins. Three Types of Capillaries. Pre-Capillary Sphincters. What is blood pressure?. Learn how a stethoscope can help determine blood pressure. Resistance in a tube. Adding up resistance in series and in parallel. Adding up resistance problem. Flow and Perfusion. Putting it all together: Pressure, flow, and resistance. Blood pressure changes over time. Regulation of blood pressure with baroreceptors. Parts of a nephron. General overview of the RAAS system - Cells and hormones. Renin production in the kidneys. Activating Angiotensin 2. Angiotensin 2 raises blood pressure. Aldosterone raises blood pressure and lowers potassium. Aldosterone removes acid from the blood. ADH secretion. ADH effects on blood pressure. Aldosterone and ADH. Meet the heart!. Layers of the Heart. Flow through the Heart. Two Circulations in the Body. Lub Dub. Heart Quiz. Circulatory System and the Heart. Arteries vs. Veins - What's the difference?. Layers of a blood vessel. Arteries, Arterioles, Venules, and Veins. Three Types of Capillaries. Pre-Capillary Sphincters. What is blood pressure?. Learn how a stethoscope can help determine blood pressure. Resistance in a tube. Adding up resistance in series and in parallel. Adding up resistance problem. Flow and Perfusion. Putting it all together: Pressure, flow, and resistance. Blood pressure changes over time. Regulation of blood pressure with baroreceptors. Parts of a nephron. General overview of the RAAS system - Cells and hormones. Renin production in the kidneys. Activating Angiotensin 2. Angiotensin 2 raises blood pressure. Aldosterone raises blood pressure and lowers potassium. Aldosterone removes acid from the blood. ADH secretion. ADH effects on blood pressure. Aldosterone and ADH.

Check out the flow of positive charge through the heart that happens every time your heart beats! Learn how the heart is able to keep a regular rate and rhythm throughout your life. Membrane Potentials - Part 1. Membrane Potentials - Part 2. Permeability and Membrane Potentials. Action Potentials in Pacemaker Cells. Action Potentials in Cardiac Myocytes. Reseting Cardiac Concentration Gradients. Electrical System of the Heart. Depolarization Waves Flowing through the Heart. A Race to Keep Pace!. Thinking about Heartbeats. New Perspective on the Heart.

Find out why having "stiff arteries" can increase blood pressure and cause uneven blood flow. It all comes back to the idea of "total energy"!. Compliance and Elastance. Bernoulli's equation of total energy. Stored elastic energy in large and middle sized arteries. Compliance - Decreased blood pressure. Compliance - Increased blood flow. Compliance and Elastance. Bernoulli's equation of total energy. Stored elastic energy in large and middle sized arteries. Compliance - Decreased blood pressure. Compliance - Increased blood flow.

Ever wonder about your lab values and what they mean? Lab values measure amounts of electrolytes or cells in your blood and occasionally tell you about how hormones and enzymes are working! Dive deeper and get a good understanding of concentrations as well!. Introduction to lab values and normal ranges. What's Inside of Blood?. Units for common medical lab values. What is an equivalent?. The mole and avogadro's number. Molarity vs. Molality. Molarity vs. Osmolarity. Calculate your own osmolarity. Molarity, Molality, Osmolarity, Osmolality, and Tonicity - What's the difference?. Tonicity - comparing 2 solutions. Introduction to lab values and normal ranges. What's Inside of Blood?. Units for common medical lab values. What is an equivalent?. The mole and avogadro's number. Molarity vs. Molality. Molarity vs. Osmolarity. Calculate your own osmolarity. Molarity, Molality, Osmolarity, Osmolality, and Tonicity - What's the difference?. Tonicity - comparing 2 solutions.

The art of this period is familiar, since the world of the Realists, Impressionists and Post-Impressionists is much like our own. More and more people lived in cities and worked in factories or shops for wages. Scientific and technological advances increased dramatically during this period and although there was dislocation and privation, standards of living increased sharply. In essence, modern mass culture was born. Artists responded sometimes by embracing these radical changes, and at other times by resisting them. Key here is understanding the authority of the various art academies in Europe, which controlled matters related to taste and art, and which were, to some extent, always connected to the government. A small number of artists rebelled against the strictures of the academy, and against the demand for art to tell clear stories for a middle class audience, and formed what we know as the

Thermo Can.

20th century solutions to new problems in Cryptography. The Fundamental Theorem of Arithmetic. Public Key Cryptography: what is it?. The Discrete Logarithm Problem. Diffie-Hellman Key Exchange. RSA Encryption: step 1. RSA Encryption: step 2. RSA Encryption: step 3. Euler's Totient Function. RSA Encryption: step 4. What should we learn next?.

Recreational mathematics and inspirational videos by resident mathemusician Vi Hart. Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3]. Doodling in Math Class: Spirals, Fibonacci, and Being a Plant [2 of 3]. Doodling in Math: Spirals, Fibonacci, and Being a Plant [Part 3 of 3]. Doodling in Math Class: Binary Trees. Doodling in Math Class: Stars. Doodling in Math Class: Snakes + Graphs. Doodling in Math Class: Infinity Elephants. Doodling in Math: Sick Number Games. Doodling in Math Class: Squiggle Inception. Doodling in Math Class: Triangle Party. Mobius Story: Wind and Mr. Ug. Math Improv: Fruit by the Foot. Wau: The Most Amazing, Ancient, and Singular Number. Are Shakespeare's Plays Encoded within Pi?. What is up with Noises? (The Science and Mathematics of Sound, Frequency, and Pitch). Open Letter to Nickelodeon, Re: SpongeBob's Pineapple under the Sea. How To Snakes. Origami Proof of the Pythagorean Theorem. 9.999... reasons that .999... = 1. Pi Is (still) Wrong.. Fractal Fractions. Angle-a-trons. Binary Hand Dance. Re: Visual Multiplication and 48/2(9+3). The Gauss Christmath Special. Rhapsody on the Proof of Pi = 4. Doodle Music. What was up with Pythagoras?. A Song About A Circle Constant. Dialogue for 2. Doodling in Math Class: Connecting Dots. Hexaflexagons. Hexaflexagons 2. Hexaflexagon Safety Guide. Flex Mex. Optimal Potatoes. Green Bean Matherole. Borromean Onion Rings. Thanksgiving Turduckenen-duckenen. Snowflakes, Starflakes, and Swirlflakes. Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3]. Doodling in Math Class: Spirals, Fibonacci, and Being a Plant [2 of 3]. Doodling in Math: Spirals, Fibonacci, and Being a Plant [Part 3 of 3]. Doodling in Math Class: Binary Trees. Doodling in Math Class: Stars. Doodling in Math Class: Snakes + Graphs. Doodling in Math Class: Infinity Elephants. Doodling in Math: Sick Number Games. Doodling in Math Class: Squiggle Inception. Doodling in Math Class: Triangle Party. Mobius Story: Wind and Mr. Ug. Math Improv: Fruit by the Foot. Wau: The Most Amazing, Ancient, and Singular Number. Are Shakespeare's Plays Encoded within Pi?. What is up with Noises? (The Science and Mathematics of Sound, Frequency, and Pitch). Open Letter to Nickelodeon, Re: SpongeBob's Pineapple under the Sea. How To Snakes. Origami Proof of the Pythagorean Theorem. 9.999... reasons that .999... = 1. Pi Is (still) Wrong.. Fractal Fractions. Angle-a-trons. Binary Hand Dance. Re: Visual Multiplication and 48/2(9+3). The Gauss Christmath Special. Rhapsody on the Proof of Pi = 4. Doodle Music. What was up with Pythagoras?. A Song About A Circle Constant. Dialogue for 2. Doodling in Math Class: Connecting Dots. Hexaflexagons. Hexaflexagons 2. Hexaflexagon Safety Guide. Flex Mex. Optimal Potatoes. Green Bean Matherole. Borromean Onion Rings. Thanksgiving Turduckenen-duckenen. Snowflakes, Starflakes, and Swirlflakes.

Get a ring-side seat as a pathologist talks about colon cells under the microscope. Normal Colon Tissue. Hyperplasia in Colon Tissue. Colon Dysplasia. Cancerous Colon Tissue. Normal Colon Tissue. Hyperplasia in Colon Tissue. Colon Dysplasia. Cancerous Colon Tissue.

Sal does the 80 problems from the released questions from the California Standards Test for Geometry. Basic understanding of Algebra I necessary. Interesting Perimeter and Area Problems. Challenging Perimeter Problem. CA Geometry: deductive reasoning. CA Geometry: Proof by Contradiction. CA Geometry: More Proofs. CA Geometry: Similar Triangles 1. CA Geometry: Similar Triangles 2. CA Geometry: More on congruent and similar triangles. CA Geometry: Triangles and Parallelograms. CA Geometry: Area, Pythagorean Theorem. CA Geometry: Area, Circumference, Volume. CA Geometry: Pythagorean Theorem, Area. CA Geometry: Exterior Angles. CA Geometry: Deducing Angle Measures. CA Geometry: Pythagorean Theorem, Compass Constructions. CA Geometry: Compass Construction. CA Geometry: Basic Trigonometry. CA Geometry: More Trig. CA Geometry: Circle Area Chords Tangent. CA Geometry: Secants and Translations. Interesting Perimeter and Area Problems. Challenging Perimeter Problem. CA Geometry: deductive reasoning. CA Geometry: Proof by Contradiction. CA Geometry: More Proofs. CA Geometry: Similar Triangles 1. CA Geometry: Similar Triangles 2. CA Geometry: More on congruent and similar triangles. CA Geometry: Triangles and Parallelograms. CA Geometry: Area, Pythagorean Theorem. CA Geometry: Area, Circumference, Volume. CA Geometry: Pythagorean Theorem, Area. CA Geometry: Exterior Angles. CA Geometry: Deducing Angle Measures. CA Geometry: Pythagorean Theorem, Compass Constructions. CA Geometry: Compass Construction. CA Geometry: Basic Trigonometry. CA Geometry: More Trig. CA Geometry: Circle Area Chords Tangent. CA Geometry: Secants and Translations.

Learn about the cervical (neck) spine through an Xray!. Adequacy of the Lateral Cervical Spine X-Ray. Assessing Alignment of the Lateral Cervical Spine (neck) X-Ray. Cervical Spine Protection in Airway Management (not a substitute for formal training). Adequacy of the Lateral Cervical Spine X-Ray. Assessing Alignment of the Lateral Cervical Spine (neck) X-Ray. Cervical Spine Protection in Airway Management (not a substitute for formal training).

Overview of how to set up accounts for students under 13 and 13+, how to add a coach on Khan Academy, and how to separate students into class lists. For more information on getting started and technology needs, check out http://www.khanacademy.org/toolkit/getting-started. Creating Khan Academy accounts for users ages 13+. Creating Khan Academy accounts for users under age 13. Adding a coach and creating class lists on Khan Academy. How to delete a student or class list. How to add a coach on Khan Academy.

Learn some basics about the Health Care system in the US. Blue Cross and Blue Shield. Conversation About Drug Pricing. Health Care Costs in US vs Europe. Healthcare System Overview. Paying Doctors. Medicare Overview. Blue Cross and Blue Shield. Conversation About Drug Pricing. Health Care Costs in US vs Europe. Healthcare System Overview. Paying Doctors. Medicare Overview.

The Lungs and Pulmonary System. Red blood cells. Circulatory System and the Heart. Hemoglobin. Anatomy of a Neuron. Sodium Potassium Pump. Correction to Sodium and Potassium Pump Video. Electrotonic and Action Potentials. Saltatory Conduction in Neurons. Neuronal Synapses (Chemical). Myosin and Actin. Tropomyosin and troponin and their role in regulating muscle contraction. Role of the Sarcoplasmic Reticulum in Muscle Cells. Anatomy of a muscle cell. The Kidney and Nephron. Secondary Active Transport in the Nephron. The Lungs and Pulmonary System. Red blood cells. Circulatory System and the Heart. Hemoglobin. Anatomy of a Neuron. Sodium Potassium Pump. Correction to Sodium and Potassium Pump Video. Electrotonic and Action Potentials. Saltatory Conduction in Neurons. Neuronal Synapses (Chemical). Myosin and Actin. Tropomyosin and troponin and their role in regulating muscle contraction. Role of the Sarcoplasmic Reticulum in Muscle Cells. Anatomy of a muscle cell. The Kidney and Nephron. Secondary Active Transport in the Nephron.

Khan Academy Vision. Sal Khan on Digital and Physical Learning. Khan Academy Vision. Sal Khan on Digital and Physical Learning.

Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra. Introduction to matrices. Matrix multiplication (part 1). Matrix multiplication (part 2). Idea Behind Inverting a 2x2 Matrix. Inverting matrices (part 2). Inverting Matrices (part 3). Matrices to solve a system of equations. Matrices to solve a vector combination problem. Singular Matrices. 3-variable linear equations (part 1). Solving 3 Equations with 3 Unknowns. Introduction to Vectors. Vector Examples. Parametric Representations of Lines. Linear Combinations and Span. Introduction to Linear Independence. More on linear independence. Span and Linear Independence Example. Linear Subspaces. Basis of a Subspace. Vector Dot Product and Vector Length. Proving Vector Dot Product Properties. Proof of the Cauchy-Schwarz Inequality. Vector Triangle Inequality. Defining the angle between vectors. Defining a plane in R3 with a point and normal vector. Cross Product Introduction. Proof: Relationship between cross product and sin of angle. Dot and Cross Product Comparison/Intuition. Matrices: Reduced Row Echelon Form 1. Matrices: Reduced Row Echelon Form 2. Matrices: Reduced Row Echelon Form 3. Matrix Vector Products. Introduction to the Null Space of a Matrix. Null Space 2: Calculating the null space of a matrix. Null Space 3: Relation to Linear Independence. Column Space of a Matrix. Null Space and Column Space Basis. Visualizing a Column Space as a Plane in R3. Proof: Any subspace basis has same number of elements. Dimension of the Null Space or Nullity. Dimension of the Column Space or Rank. Showing relation between basis cols and pivot cols. Showing that the candidate basis does span C(A). A more formal understanding of functions. Vector Transformations. Linear Transformations. Matrix Vector Products as Linear Transformations. Linear Transformations as Matrix Vector Products. Image of a subset under a transformation. im(T): Image of a Transformation. Preimage of a set. Preimage and Kernel Example. Sums and Scalar Multiples of Linear Transformations. More on Matrix Addition and Scalar Multiplication. Linear Transformation Examples: Scaling and Reflections. Linear Transformation Examples: Rotations in R2. Rotation in R3 around the X-axis. Unit Vectors. Introduction to Projections. Expressing a Projection on to a line as a Matrix Vector prod. Compositions of Linear Transformations 1. Compositions of Linear Transformations 2. Matrix Product Examples. Matrix Product Associativity. Distributive Property of Matrix Products. Introduction to the inverse of a function. Proof: Invertibility implies a unique solution to f(x)=y. Surjective (onto) and Injective (one-to-one) functions. Relating invertibility to being onto and one-to-one. Determining whether a transformation is onto. Exploring the solution set of Ax=b. Matrix condition for one-to-one trans. Simplifying conditions for invertibility. Showing that Inverses are Linear. Deriving a method for determining inverses. Example of Finding Matrix Inverse. Formula for 2x2 inverse. 3x3 Determinant. nxn Determinant. Determinants along other rows/cols. Rule of Sarrus of Determinants. Determinant when row multiplied by scalar. (correction) scalar multiplication of row. Determinant when row is added. Duplicate Row Determinant. Determinant after row operations. Upper Triangular Determinant. Simpler 4x4 determinant. Determinant and area of a parallelogram. Determinant as Scaling Factor. Transpose of a Matrix. Determinant of Transpose. Transpose of a Matrix Product. Transposes of sums and inverses. Transpose of a Vector. Rowspace and Left Nullspace. Visualizations of Left Nullspace and Rowspace. Orthogonal Complements. Rank(A) = Rank(transpose of A). dim(V) + dim(orthogonal complement of V)=n. Representing vectors in Rn using subspace members. Orthogonal Complement of the Orthogonal Complement. Orthogonal Complement of the Nullspace. Unique rowspace solution to Ax=b. Rowspace Solution to Ax=b example. Showing that A-transpose x A is invertible. Projections onto Subspaces. Visualizing a projection onto a plane. A Projection onto a Subspace is a Linear Transforma. Subspace Projection Matrix Example. Another Example of a Projection Matrix. Projection is closest vector in subspace. Least Squares Approximation. Least Squares Examples. Another Least Squares Example. Coordinates with Respect to a Basis. Change of Basis Matrix. Invertible Change of Basis Matrix. Transformation Matrix with Respect to a Basis. Alternate Basis Transformation Matrix Example. Alternate Basis Transformation Matrix Example Part 2. Changing coordinate systems to help find a transformation matrix. Introduction to Orthonormal Bases. Coordinates with respect to orthonormal bases. Projections onto subspaces with orthonormal bases. Finding projection onto subspace with orthonormal basis example. Example using orthogonal change-of-basis matrix to find transformation matrix. Orthogonal matrices preserve angles and lengths. The Gram-Schmidt Process. Gram-Schmidt Process Example. Gram-Schmidt example with 3 basis vectors. Introduction to Eigenvalues and Eigenvectors. Proof of formula for determining Eigenvalues. Example solving for the eigenvalues of a 2x2 matrix. Finding Eigenvectors and Eigenspaces example. Eigenvalues of a 3x3 matrix. Eigenvectors and Eigenspaces for a 3x3 matrix. Showing that an eigenbasis makes for good coordinate systems. Vector Triple Product Expansion (very optional). Normal vector from plane equation. Point distance to plane. Distance Between Planes.

Fluids (part 1). Fluids (part 2). Fluids (part 3). Fluids (part 4). Fluids (part 5). Fluids (part 6). Fluids (part 7). Fluids (part 8). Fluids (part 9). Fluids (part 10). Fluids (part 11). Fluids (part 12). Fluids (part 1). Fluids (part 2). Fluids (part 3). Fluids (part 4). Fluids (part 5). Fluids (part 6). Fluids (part 7). Fluids (part 8). Fluids (part 9). Fluids (part 10). Fluids (part 11). Fluids (part 12).

Topics covered in a first year course in differential equations. Need to understand basic differentiation and integration from Calculus playlist before starting here. What is a differential equation. Separable Differential Equations. Separable differential equations 2. Exact Equations Intuition 1 (proofy). Exact Equations Intuition 2 (proofy). Exact Equations Example 1. Exact Equations Example 2. Exact Equations Example 3. Integrating factors 1. Integrating factors 2. First order homegenous equations. First order homogeneous equations 2. 2nd Order Linear Homogeneous Differential Equations 1. 2nd Order Linear Homogeneous Differential Equations 2. 2nd Order Linear Homogeneous Differential Equations 3. 2nd Order Linear Homogeneous Differential Equations 4. Complex roots of the characteristic equations 1. Complex roots of the characteristic equations 2. Complex roots of the characteristic equations 3. Repeated roots of the characteristic equation. Repeated roots of the characteristic equations part 2. Undetermined Coefficients 1. Undetermined Coefficients 2. Undetermined Coefficients 3. Undetermined Coefficients 4. Laplace Transform 1. Laplace Transform 2. Laplace Transform 3 (L{sin(at)}). Laplace Transform 4. Laplace Transform 5. Laplace Transform 6. Laplace Transform to solve an equation. Laplace Transform solves an equation 2. More Laplace Transform tools. Using the Laplace Transform to solve a nonhomogeneous eq. Laplace Transform of : L{t}. Laplace Transform of t^n: L{t^n}. Laplace Transform of the Unit Step Function. Inverse Laplace Examples. Laplace/Step Function Differential Equation. Dirac Delta Function. Laplace Transform of the Dirac Delta Function. Introduction to the Convolution. The Convolution and the Laplace Transform. Using the Convolution Theorem to Solve an Initial Value Prob.

Topics covered in a traditional college level introductory microeconomics course. Production Possibilities Frontier. Opportunity Cost. Increasing Opportunity Cost. Allocative Efficiency and Marginal Benefit. Economic Growth through Investment. Comparative Advantage Specialization and Gains from Trade. Comparative Advantage and Absolute Advantage. Law of Demand. Price of Related Products and Demand. Changes in Income, Population, or Preferences. Normal and Inferior Goods. Inferior Goods Clarification. Law of Supply. Factors Affecting Supply. Market Equilibrium. Changes in Market Equilibrium. Price Elasticity of Demand. More on Elasticity of Demand. Perfect Inelasticity and Perfect Elasticity of Demand. Constant Unit Elasticity. Total Revenue and Elasticity. More on Total Revenue and Elasticity. Cross Elasticity of Demand. Elasticity of Supply. Elasticity and Strange Percent Changes. Demand Curve as Marginal Benefit Curve. Consumer Surplus Introduction. Total Consumer Surplus as Area. Producer Surplus. Rent Control and Deadweight Loss. Minimum Wage and Price Floors. Taxation and Dead Weight Loss. Percentage Tax on Hamburgers. Taxes and Perfectly Inelastic Demand. Taxes and Perfectly Elastic Demand. Marginal Utility. Equalizing Marginal Utility per Dollar Spent. Deriving Demand Curve from Tweaking Marginal Utility per Dollar. Budget Line. Optimal Point on Budget Line. Types of Indifference Curves. Economic Profit vs Accounting Profit. Depreciation and Opportunity Cost of Capital. Fixed, Variable, and Marginal Cost.. Visualizing Average Costs and Marginal Costs as Slope. Marginal Cost and Average Total Cost. Marginal Revenue and Marginal Cost. Marginal Revenue Below Average Total Cost. Long Term Supply Curve and Economic Profit. Perfect Competition. Monopoly Basics. Review of Revenue and Cost Graphs for a Monopoly. Monopolist Optimizing Price (part 1)- Total Revenue.. Monopolist Optimizing Price (part 2)- Marginal Revenue. Monopolist Optimizing Price (part 3)- Dead Weight Loss.avi. Optional Calculus Proof to Show that MR has Twice Slope of Demand. Oligopolies and Monopolistic Competition. Monopolistic Competition and Economic Profit. Oligopolies, Duopolies, Collusion, and Cartels. Prisoners' Dilemma and Nash Equilibrium. More on Nash Equilibrium. Why Parties to Cartels Cheat. Game Theory of Cheating Firms. Negative Externalities. Taxes for Factoring in Negative Externalities. Positive Externalities. Tragedy of the Commons. First Degree Price Discrimination. A Firm's Marginal Product Revenue Curve. How Many People to Hire Given the MPR curve. Adding Demand Curves.