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Saturday, January 31, 2009

Calculus Video Lectures + Bonus Basic Math

Hi all. This month I bring to you a bunch of calculus math video courses. As a bonus I have included some on fuzzy logic and really basic math such as trigonometry, algebra and geometry.


Calculus at San Francisco State University (SFSU, professor Dr. Goetz)


Course description:
The central object of the study in calculus is the concept of a function. Functions are used to describe the real world around us. Calculus introduces two fundamental concepts which enable us to describe and investigate functions. These are the derivative and the integral. The derivative describes how a function changes at a particular time. The integral carries information about the history of a function. Both, the derivative and the integral are defined using limits. Calculus 1 covers: Limits, Continuity, Asymptotes, The tangent problem, Rates of Change, Derivatives (including trigonometric and transcendental derivatives), Graphs, and their shapes, Optimizations, Riemann Sums, Integrals including two parts of the Fundamental Theorem of Calculus.

Course topics:
Introduction to limits. Limit Laws. Continuity. Long term behavior and asymptotes. Rates of change. Derivative. Derivatives of Polynomials and the Exponential Function. Product and Quotient Rules. Derivatives of Trigonometric Functions. The Chain Rule. Implicit Differentiation. Derivatives of Logarithmic Functions. Introduction to Differential equations and Exponential Growth and Decay. Minimum and Maximum Values. Shapes of a Graph. Local Extrema and Inflection Points. L`Hospital's Rule. Graphs. Optimization. Antiderivatives. Estimates of areas under graphs. Definite Integrals. The Area Function. Definite Integrals and Antiderivatives. Applications of the Fundamental Theorem of Calculus. Practice Problems.



The Calculus Lifesaver: All the Tools You Need to Excel at Calculus

These are Princeton introductory calculus courses MAT103 and MAT104 by Adrian Banner.


Course topics:
Functions. Trigonometric functions. Limits: theory and polynomial examples. Continuity, differentiability, trig limits. Trigonometric limits. How to solve differentiation problems. Trig derivatives. Simple harmonic motion. Implicit differentiation, related rates. Exponentials. Inverse functions. Inverse trig functions. Linearization. Extrema. Rolle/Mean Value Theorem. Critical points and the second derivatives. Classifying critical points, sketching graphs. Optimization, Lopital's Rule. Sigma notation. Riemann sums. Integration. Introduction to the Fundamental Theorems. Integration by substitution. Substitution revisited. Integration by parts. Integrals involving trig limits. Partial fractions. Trig integrals. Trig substitutions. Summary of integration techniques. Volumes of revolution, arc lengths, and surface areas. Improper integrals. Sequences and series. Introduction to power law. Estimation using Taylor series. The remainder/error term. Radius of convergence of power series. Introduction to complex numbers. Complex numbers, separable first-order differential equations. First order linear differential equations. First/second order constant coefficient linear differential equations.



MIT's Single Variable Calculus (18.01, professor David Jerison)


Course description:
The basic objective of Calculus is to relate small-scale (differential) quantities to large-scale (integrated) quantities. This is accomplished by means of the Fundamental Theorem of Calculus. Students should demonstrate an understanding of the integral as a cumulative sum, of the derivative as a rate of change, and of the inverse relationship between integration and differentiation.

Course topics:
Derivatives, slope, velocity, rate of change. Limits, continuity - Trigonometric limits. Derivatives of products, quotients, sine, cosine. Chain rule - Higher derivatives. Implicit differentiation, inverses. Exponential and log - Logarithmic differentiation; hyperbolic functions. Exam 1 review.


MIT's Multivariable Calculus (18.02, professor Denis Auroux)


Course description:
This course covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.

Course topics:
Vectors and matrices: Dot product. Determinants. Cross product. Matrices. Inverse matrices. Square systems. Equations of planes. Parametric equations for lines and curves. Acceleration - Kepler's second law. Partial derivatives: Level curves. Partial derivatives. Tangent plane approximation. Max-min problems. Least squares. Second derivative test. Boundaries and infinity. Differentials. Chain rule. Gradient; Directional derivative. Tangent plane. Lagrange multipliers. Non-independent variables. Partial differential equations. Double integrals and line integrals in the plane: Double integrals. Double integrals in polar coordinates. Applications. Change of variables. Vector fields and line integrals in the plane. Path independence and conservative fields. Gradient fields and potential functions. Green's theorem. Flux. Normal form of Green's theorem. Simply connected regions. Triple integrals and surface integrals in 3-space: Triple integrals in rectangular and cylindrical coordinates. Spherical coordinates. Surface area. Vector fields in 3D. Surface integrals and flux. Divergence theorem. Divergence theorem: applications and proof. Line integrals in space, curl, exactness and potentials. Stokes' theorem. Maxwell's equations. Final review.


Khan Academy Video Lectures


Video descriptions:
Single man, Salman Khan, has made over 700 videos on all math topics: Calculus, Precalculus, Trigonometry, Algebra, Finance, Pre-algebra, Arithmetic, Geometry, Physics, SAT Preparation, Probability, Singapore Math, Linear Algebra, Differential Equations, Credit Crisis, Banking and Money.


Khan's Math Videos on Pre-Calculus


Precalculus video topics:
Introduction to Limits. Limit Examples. Squeeze Theorem. Sequences and Series. Permutations. Combinations. Binomial Theorem. Introduction to interest. Interest. Introduction to compound interest and e. Compound Interest and e. Exponential Growth.



Khan's Math Videos on Calculus


Calculus video topics:
Derivatives. Implicit Differentiation. Chain rule and implicit differentiation intuition. Maxima Minima Slope Intuition. Inflection point intuition. Monotonicity Theorem. Maximum and minimum values on an interval. Graphing using derivatives. Graphing with Calculus. Hairy inflection point problem. Optimization with Calculus. Introduction to rate-of-change problems. Ladder rate-of-change problem. Mean Value Theorem. The Indefinite Integral or Anti-derivative. Indefinite integrals. Introduction to definite integrals. Definite integrals. Area under curve. Definite integral with substitution. Introduction to differential equations. Solid of Revolution. Sequences and Series. Polynomial approximation of functions. Approximating functions with polynomials. Taylor Polynomials. Exponential Growth. Equation of a tangent line. AP Calculus BC Exams. The dot product. Dot vs. Cross Product. Calculating dot and cross products with unit vector notation. Partial Derivatives. Gradient. Gradient of a scalar field. Divergence. Curl. Double Integral. Triple Integrals.



Khan's Math Videos in Differential Equations


Differential equations video topics:
Introduction to differential equations. Separable Differential Equations. Exact Equations Intuition. Exact Equations Examples. Integrating factors. First order homegenous equations. First order homogenous equations. Second 2nd Order Linear Homogeneous Differential Equations. Complex roots of the characteristic equations. Repeated roots of the characteristic equation. Undetermined Coefficients. Laplace Transform. Laplace Transform to solve an equation. More Laplace Transform tools.



Khan's Math Videos on Trigonometry


Trigonometry video topics:
Basic Trigonometry. Radians and degrees. Using Trig Functions. The unit circle definition of trigonometric function. Unit Circle Definition of Trig Functions. Graph of the sine function. Graphs of trig functions. Graphing trig functions. Determining the equation of a trigonometric function. Trigonometric Identities. Proof: sin(a+b) = (cos a)(sin b) + (sin a)(cos b). Proof: cos(a+b) = (cos a)(cos b)-(sin a)(sin b). Trigonometry word problems. Law of cosines. Navigation Word Problem. Proof: Law of Sines. Ferris Wheel Trig Problem. Fun Trig Problems.



Khan's Math Videos on Pre-Algebra


Prealgebra video topics:
Order of operations. Adding/Subtracting negative numbers. Multiplying and dividing negative numbers. Adding and subtracting fractions. Multiplying fractions. Dividing fractions. Exponents. Negative Exponent Intuition. Exponent Rules. Simplifying radicals. Introduction to Logarithms. Unit conversion. Speed translation. Introduction to logarithm properties.



Khan's Math Videos on Algebra


Algebra video topics:
Linear Equations. Solving Inequalities. Graphing lines. Slope and Y-intercept. Equation of a line. Averages. Integer sums. Taking percentages. Growing by a percentage. Systems of equations. Introduction to Ratios. Age word problems. Multiplying expressions. Solving a quadratic by factoring. I and Imaginary numbers. Complex Numbers. Introduction to the quadratic equation. Completing the square. Quadratic Formula. Quadratic Inequalities. Introduction to functions. Domain of a function. Proof: log a + log b = log ab. Algebraic Long Division.



Khan's Math Videos in Linear Algebra


Linear Algebra video topics:
Introduction to matrices. Matrix multiplication. Inverse Matrix. Inverting matrices. Matrices to solve a system of equations. Matrices to solve a vector combination problem. Singular Matrices. 3-variable linear equations. 3 equations with 3 unknowns.



Khan's Math Videos on Arithmetic


Arithmetic video topics:
Basic Addition. Addition. Basic Subtraction. Subtraction. Why borrowing works. Multiplication. Division. Adding Decimals. Subtracting decimals. Multiplying decimals. Dividing decimal. Greatest Common Divisor. Least Common Multiple. Equivalent fractions. Mixed numbers and improper fractions. Converting fractions to decimals.Percent and decimals. Ordering numeric expressions. Why borrowing works.



Khan's Math Videos in Probability


Probability video topics:
What is probability. Flipping a coin. Free throws. Probability of getting a certain number roll in Monopoly. Introduction to conditional probability. Touch on Bayes' Theorem. Introduction to Bayes' Theorem. Introduction to permutations. Introduction to combinations.



Khan's Math Videos in Geometry


Geometry video topics:
Introduction to angles. Angles. Angles of parallel lines. The Angle Game. Similar triangles. Introduction to the Pythagorean Theorem. 45-45-90 Triangles. Intro to 30-60-90 Triangles.



Khan's Math Videos on California Standards Test in Algebra


California standards test in algebra video topics:
CA Algebra: Number Properties and Absolute Value. Simplifying Expressions. Simple Logical Arguments. Graphing Inequalities. Slope and Y-intercept. Systems of Inequalities. Simplying Expressions. Factoring Quadratics. Completing the Square. Quadratic Equation. Quadratic Roots. Rational Expressions. Rational Expressions. Word Problems. More Word Problems. Functions.



Khan's Math Videos on California Standards Test in Geometry


California standards test in geometry video topics:
Deductive reasoning. Proof by Contradiction. More Proofs. Similar Triangles. Similar Triangles. More on congruent and similar triangles. Triangles and Parallelograms. Area, Pythagorean Theorem. Area, Circumference, Volume. Pythagorean Theorem, Area. Exterior Angles. Deducing Angle Measures. Pythagorean Theorem, Compass Constructions. Compass Construction. Basic Trigonometry. More Trig. Circle Area Chords Tangent. Secants and Translations.



Fuzzy Logic


Math video description:
The tutorial will introduce the basics of fuzzy logic for data analysis. Fuzzy Logic can be used to model and deal with imprecise information, such as inexact measurements or available expert knowledge in the form of verbal descriptions. We will first introduce the concepts of fuzzy sets, degrees of membership and fuzzy set operators. After discussions on fuzzy numbers and arithmetic operations using them, the focus will shift to fuzzy rules and how such systems of rules can be derived from available data.


Math video topics:
Fuzzy Logic: Motivation. Characteristic Functions: Crisp Sets. Characteristic Functions: Fuzzy Sets. Linguistic Variables and Values. Linguistic Values & Context. Types of Membership Functions. Fuzzy Membership Function: Basic Concepts. Operators on Fuzzy Sets - Page 1. Operators on Fuzzy Sets - Page 2. Min / Max-Norm. Product / Bounded-Sum. T-norms and S-norms. Fuzzy TNorms and SNorms. Fuzzy Norms: Issues? Imprecise Reasoning. Joint Constraint (support distribution). Conditional Constraint (possibility distribution). Joint vs. Conditional Constraint. Fuzzy Rules. Example: Mamdani Rule. Example: Takagi-Sugeno Rule. Fuzzy Rule System (Mamdani). Fuzzy Rule Systems. Defuzzification: Center of Gravity. Defuzzification: Other Methods. Fuzzy Inference (Mamdani). Fuzzy Rule System (Mamdani). Fuzzy Rule System (Takagi Sugeno). Construction of Fuzzy Rule Systems. Grid Based Algorithms. Free Fuzzy Rules. Formation of Free Fuzzy Rules. Other Types of Fuzzy Rule Learning Methods. Fuzzy Numbers: Motivation. Fuzzy Numbers. Operations on Fuzzy Numbers. Imprecise Functions (Fuzzy Graphs).



Have fun with these math lectures!


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