Single Variable Calculus Tags

Limits and continuity

• Finding limits using graphs
• Rules of limits - basic
• Evaluating limits - factoring
• Evaluating limits - rationalizing
• Evaluating limits - rational expressions
• Evaluating limits - trigonometric
• Squeeze theorem
• One-sided limits - concept of
• Continuity - concept of
• Continuity - classifying discontinuities
• Continuity - intermediate value theorem
• Infinite limits and vertical asymptotes
• Limits at infinity, horizontal and oblique asymptotes
• Estimating limits numerically
• Applications - instantaneous rate of change
• Applications - tangent lines and slopes
• Applications - finding all asymptotes
• Applications - other
• Definitions and existence

Differentiation

• Definition of the derivative
• Conceptual understanding of derivatives
• Derivatives of polynomials and power functions
• Product rule
• Quotient rule
• Derivatives of trigonometric functions
• Derivatives of exponential functions
• Derivatives of logarithmic functions
• Derivatives of inverse functions
• Derivatives of inverse trigonometric functions
• Hyperbolic functions
• Derivatives of hyperbolic functions
• Chain rule
• Higher-order derivatives
• Derivatives involving multiple rules
• Logarithmic differentiation
• Implicit differentiation

Applications of differentiation

• Mean value theorem
• Rates of change
• Increasing/decreasing functions and local extrema
• Concavity and points of inflection
• Global extrema
• Summary of curve sketching
• Optimization
• Linear approximation and differentials
• Related rates
• Indeterminate forms and L'Hopital's rule
• Newton's method
• Elasticity of demand

Integrals

• Conceptual understanding of integration
• Antiderivatives
• Indefinite integrals
• Riemann sums
• Definite integrals
• Fundamental theorem of calculus
• Improper integrals

Techniques of integration

• Substitution
• Integration by parts
• Trigonometric integrals
• Hyperbolic functions
• Partial fractions
• Trigonometric substitution
• Tables of integrals
• Computer algebra system
• Mixed techniques
• Challenging integrals
• Approximation

Applications of integration

• Average value of a function
• Areas between curves
• Volumes by slices
• Volumes by disks
• Volumes by washers
• Volumes by cylindrical shells
• Volumes by multiple methods
• Arc length
• Surfaces of revolution
• Distance, velocity, acceleration
• Work
• Hydrostatic pressure
• Center of gravity
• Other physics and engineering applications
• Economics
• Biology
• Probability and statistics

Infinite sequences and series

• Limit of a sequence
• Series notation
• Partial sums
• Taylor polynomials
• Geometric
• Test for divergence
• Comparison tests
• Integral test
• Ratio test
• Root test
• Alternating series test
• Absolute and conditional convergence
• Strategy for testing series
• Interval of convergence of a power series
• Maclaurin series
• Taylor series
• Power series
• Representations of functions as series
• Applications of Taylor polynomials
• Fourier series

Parametric

• Curves
• Eliminating the parameter
• Tangents, velocity, and speed
• Higher order derivatives
• Arc length
• Area
• Volumes of revolution
• Surface area of surfaces of revolution

Polar

• Similar figures (Trigonometry, geometric and algebraic foundations for trigonometry, similar figures)
• Polar and rectangular coordinates (Trigonometry, polar coordinates and vectors polar and rectangular coordinates
• Curves (Trigonometry, polar coordinates, vectors, curves)
• Inequalities (Trigonometry, polar coordinates, vectors inequalities)
• Tangents
• Area
• Arc length
• Other applications