top of page # 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

• 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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