Calculus I & II (MATH-M 211) summary notes covering following topics:
- Review: Functions
- Functions and their graphs
- Transformations and combinations of graphs
- Trigonometric functions
- Limits and continuity
- Limit of a function and limit laws
- Precise definition of a limit
- One-sided limits
- Continuity
- Limits involving infinity and asymptotes
- Differentiation
- Tangent lines and the derivative at a point
- The derivative as a function
- Differentiation rules
- The derivative as a rate of change
- Derivatives of trigonometric functions
- The chain rule
- Implicit differentiation
- Related rates
- Linearization and differentials
- Applications of differentiation
- Extreme values of functions on closed interval
- The mean value theorem
- Monotonic functions and the first derivative test
- Concavity and curve sketching
- Applied optimization
- Newton’s method
- Antiderivatives
- Integration
- Areas and estimating with finite sums
- Sigma notation and limits of finite sums
- Definite integrals
- The fundamental theorem of calculus
- Indefinite integrals and the substitution method
- Definite integral substitutions and the area between curves
- Applications of definite integration
- Volumes using disks and washers
- Volumes using shells
- Arc length
- Areas of surfaces of revolution
- Work and fluid forces
- Moments and centers of mass
- Transcendental functions
- Inverse functions and their derivatives
- Natural logarithms
- Exponential functions
- Exponential change and separable differential equations
- Indeterminate forms and L’Hˆopital’s rule
- Inverse trigonometric functions
- Hyperbolic functions
- Relative rates of growth
- Techniques of integration
- Using basic integration formulae
- Integration by parts
- Trigonometric integrals
- Trigonometric substitutions
- Integration of rational functions by partial functions
- Integral tables and computer algebra systems
- Numerical integration
- Improper integrals
- Probability
- First-order differential equations
- Solutions, slope fields and Euler’s method
- First-order linear equations
- Applications
- Graphical solutions of autonomous equations
- Systems of equations and phase planes
- Infinite sequences and series
- Sequences
- Infinite series
- The integral test
- Comparison tests
- Absolute convergence, ratio and root tests
- Alternating series and conditional convergence
- Power series
- Taylor and Maclaurin series
- Convergence of Taylor series
- Applications of Taylor series
- Parametric equations and polar coordinates
- Parametrizations of plane curves
- Calculus with parametric curves
- Polar coordinates
- Graphing polar coordinate equations
- Areas and lengths in polar coordinates
- Conic sections
- Conics in polar coordinates
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