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Transcendental Functions
• Exponential functions (review)
• Logarithmic functions (review)
• Inverse trigonometric functions
• Arcsin and Arccos
• Arctan
• Derivatives of inverse trigonometric functions
• Indeterminate forms and L'Hospital's rule
• Indeterminate ratios
• Indeterminate products
• Indeterminate differences
• Indeterminate powers
Techniques of Integration
• Definite and indefinite integrals (review)
• The Fundamental Theorem of Calculus (review)
• Integration by substitution (review)
• Integration by parts
• For indefinite integrals
• For definite integrals
• Integrals involving sin and cos
• Integrals involving ln, arcsin, arctan
• Integrals involving exponentials
• Reduction formulas
• Trigonometric integrals
• Products of powers of sin and cos
• Products of powers of tan and sec
• Products of sin and cos of multiples of the variable
• Trigonometric substitutions
• Completing the square
• Substituions involving sin and cos
• Substitutions involving tan
• Substitutions involving sec
• Integration of rational functions
• Long division of polynomials
• Decomposition in simple fractions
• Simple linear factors
• Repeated linear factors
• Rationalizing substitutions
• Improper integrals
• Infinite intervals
• Discontinuous integrands
• Comparison test for improper integrals
Differential Equations
• Differential equations
• Population growth
• General differential equations
• Solution of a differential equation
• Initial-value problems
• Modeling with differential equations
• Direction fields
• Trajectories
• Euler's method
• Separable equations
• Method for separable equations
• Implicit and explicit solutions
• Exponential growth and decay
• Population growth
• Newton's law of cooling
• Continuously compounded interest
Parametric Curves
• Graphs of functions as curves
• Arc length of graphs
• Area of surface of revolution
• Parametric curves
• Curves defined by parametric equations in cartesian coordinates
• Sketching a curve from parametric equations
• Calculus with parametric curves
• Tangents to parametric curves
• Area enclosed by a parametric curve
• Arclength of a parametric curve
• Polar coordinates
• Curves defined in polar coordinates
• Sketching a curve from parametric equations in polar coordinates
• Tangents to polar curves
• Areas in polar coordinates
• Arclength in polar coordinates
Sequences and Series
• Sequences
• Definition of infinite sequences
• Methods of defining a sequence
• Limit of a sequence. Convergence and divergence of sequences
• Increasing and decreasing sequences
• Bounded and unbounded sequences
• The Monotone Sequence Theorem
• Series
• Definition of infinite series
• Sequence of partial sums
• Convergence and divergence of series
• Convergence of geometric series
• Harmonic series
• Telescopic series
• Tests for convergence/divergence
• The Test for Divergence
• The Integral Test
• Convergence of p-series
• The Comparison Test
• The Limit Comparison Test
• The Alternating Series Test. Error estimates.
• Absolute convergence and conditional convergence
• The Ratio Test
• The Root Test
• Power Series
• Sum of a power series
• Representations of functions as power series
• Differentiation and integration of power series
• Multiplication and division of power series
• Maclaurin and Taylor series
• The binomial series
• Applications of Taylor polynomials
• Approximations
• Error estimates
• Complex numbers
• Algebraic form
• Polar form
• Geometric representation
• Operations with complex numbers 