Calculus

9-12

Course Outline

 

  1. Limits

  • Introduction to Calculus – Limits

  • Finding limits from graphs

  • Continuity

  • Finding limits algebraically – direct substitution

  • Finding limits algebraically – when direct substitution is not possible

  • Infinite limits – vertical asymptotes

  • Limits at infinity – horizontal asymptotes

  • Intermediate value theorem

  • Squeeze theorem

  • Limit laws

 

  1. Derivatives

  • Definition of derivative

  • Estimating Derivatives from a table

  • Power rule

  • Slope and equation of tangent line

  • Chain rule

  • Derivative of trigonometric functions

  • Derivative of exponential functions

  • Product rule

  • Quotient rule

  • Implicit differentiation

  • Derivative of inverse trigonometric functions

  • Derivative of logarithmic functions

  • Higher order derivatives

 

  1. Derivative Applications

  • Position velocity acceleration

  • Critical number & maximum and minimum values

  • L’Hospital’s rule

  • Curve sketching

  • Optimization

  • Related rates

  • Rolle’s Theorem

  • Mean value theorem

  • Linear approximation

  • Quadratic approximation

  • Demand, revenue, cost & profit

  • Marginal revenue, and maximizing revenue & average revenue

  • Marginal cost, and minimizing cost & average cost

  • Marginal profit, and maximizing profit & average profit

  • Elasticity of demand

 

  1. Integrals

  • Antiderivatives

  • Riemann sum

  • Definite integral

  • Fundamental theorem of calculus

 

  1. Integration techniques

  • U-Substitution

  • Integration by parts

  • Integration using trigonometric identities

  • Trigonometric substitution

  • Integration of rational functions by partial fractions

  • Improper integrals

  • Numerical integration

 

  1. Integration Applications

  • Areas between curves

  • Volumes of solid with known cross-sections

  • Volumes of solids of revolution – Disk method

  • Volumes of solids of revolution – Shell method

  • Average value of a function

  • Arc length

  • Consumer and producer surplus

  • Continuous money flow

 

  1. Differential Equations

  • Order and solutions to differential equations

  • Separable equations

  • Modeling with differential equations

 

  1. Sequence and Series

  • Introduction to sequences

  • Monotonic and bounded sequences

  • Introduction to infinite series

  • Convergence and divergence of normal infinite series

  • Convergence & divergence of geometric series

  • Convergence & divergence of telescoping series

  • Divergence of harmonic series

  • p Series

  • Alternating series test

  • Divergence test

  • Comparison & limit comparison test

  • Integral test

  • Ratio test

  • Root test

  • Absolute & conditional convergence

  • Radius and interval of convergence with power series

  • Functions expressed as power series

  • Taylor series and Maclaurin series

  • Approximating functions with Taylor polynomial and error bounds

 

  1. Parametric Equations and Polar Coordinates

  • Defining curves with parametric equations

  • Tangent and concavity of parametric equations

  • Area of parametric equations

  • Arc length and surface area of parametric equations

  • Polar coordinates

  • Tangents of polar curves

  • Area of polar curves

  • Arc length of polar curves

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