Calculus

Comes after: Precalculus, Trigonometry

The course

1. Limits and continuity

   20 lessons
   ∫Limits∫Calculus∫A Preview of Calculus∫Introduction to Calculus∫Review of Functions∫Basic Classes of Functions∫Trigonometric Functions∫Inverse Functions∫Exponential and Logarithmic Functions∫The notion of limit∫The Limit of a Function∫Finding Limits: Numerical and Graphical Approaches∫The Limit Laws∫Finding Limits: Properties of Limits∫The Precise Definition of a Limit∫Limit (mathematics)∫Continuity∫Continuous function∫Limits, continuity, and differentiability∫Limits at Infinity and Asymptotes
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2. Derivatives: the definition and the rules

   18 lessons
   ∫Derivatives∫How do we measure velocity?∫The derivative of a function at a point∫Defining the Derivative∫Derivative∫The derivative function∫The Derivative as a Function∫Interpreting, estimating, and using the derivative∫Derivatives as Rates of Change∫The second derivative∫Elementary derivative rules∫Differentiation rules∫The product and quotient rules∫Product rule∫Quotient rule∫The sine and cosine functions∫Derivatives of Trigonometric Functions∫Derivatives of other trigonometric functions
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3. The chain rule and implicit differentiation

   6 lessons
   ∫The chain rule∫Implicit differentiation∫The Chain Rule∫Derivatives of Exponential and Logarithmic Functions∫Derivatives of Inverse Functions∫Derivatives of functions given implicitly
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4. Applications of derivatives

   20 lessons
   ∫Maxima and minima∫Related rates and optimisation∫Related rates∫The tangent line approximation∫Linear Approximations and Differentials∫Linear approximation∫Using derivatives to identify extreme values∫Maxima and Minima∫Global optimization∫The Mean Value Theorem∫Mean value theorem∫Derivatives and the Shape of a Graph∫Using derivatives to describe families of functions∫Curve sketching∫Applied optimization∫Applied Optimization Problems∫Optimization (mathematics)∫Using derivatives to evaluate limits∫L'Hôpital's rule∫Newton’s Method
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5. Integrals and the fundamental theorem

   19 lessons
   ∫Integrals∫Definite integrals∫Antiderivatives∫Antiderivative∫Approximating Areas∫Riemann sums∫Riemann sum∫The Definite Integral∫Integral∫Determining distance traveled from velocity∫The Fundamental Theorem of Calculus∫Fundamental theorem of calculus∫The Second Fundamental Theorem of Calculus∫Constructing accurate graphs of antiderivatives∫Integration Formulas and the Net Change Theorem∫Substitution∫Integration by substitution∫Integrals Involving Exponential and Logarithmic Functions∫Integrals Resulting in Inverse Trigonometric Functions
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6. Techniques of integration

   11 lessons
   ∫Integration techniques: substitution, parts, partial fractions∫Integration by parts∫Trigonometric Integrals∫Trigonometric substitution∫Partial Fractions∫Partial fraction decomposition∫Other Strategies for Integration∫Other options for finding algebraic antiderivatives∫Numerical Integration∫Improper Integrals∫Improper integral
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7. Applications of integration

   18 lessons
   ∫Applications of integration: area, volume, arc length∫Areas between Curves∫Using definite integrals to find area and length∫Determining Volumes by Slicing∫Using definite integrals to find volume∫Volumes of Revolution: Cylindrical Shells∫Solid of revolution∫Arc Length of a Curve and Surface Area∫Arc length∫Physical Applications∫Physics applications: work, force, and pressure∫Moments and Centers of Mass∫Density, mass, and center of mass∫Integrals, Exponential Functions, and Logarithms∫Calculus of the Hyperbolic Functions∫Exponential Growth and Decay∫Exponential growth∫Logistic function
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8. Sequences and series

   21 lessons
   ∫Infinite series and convergence tests∫Series and sums∫Taylor series∫Sequences∫Infinite Series∫Geometric sums∫The Divergence and Integral Tests∫Comparison Tests∫Alternating Series∫Ratio and Root Tests∫Convergence tests∫Power Series and Functions∫Power series∫Properties of Power Series∫Extending local linearization∫Taylor polynomials∫Taylor and Maclaurin Series∫Maclaurin series∫Working with Taylor Series∫Finding and using Taylor series∫Quantifying the accuracy of approximations
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