Calculus I & II (MATH-M 211) summary notes

Calculus I & II (MATH-M 211) summary notes covering following topics:

1. Review: Functions
   • Functions and their graphs
   • Transformations and combinations of graphs
   • Trigonometric functions
2. Limits and continuity
   • Limit of a function and limit laws
   • Precise definition of a limit
   • One-sided limits
   • Continuity
   • Limits involving infinity and asymptotes
3. 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
4. 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
5. 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
6. 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
7. 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
8. 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
9. 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
10. 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
11. 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