# ML Wiki

Summer 2015

## Calculus: Single Variable (coursera)

### Differentiation

• Derivatives -Definition and interpretations of the derivative
• Differentiation rules -Rules for differentiating combinations of functions
• Linearization - First order Taylor approximations
• Higher derivatives -Definition and interpretation of higher derivatives
• Optimization -Classifying critical points and finding extrema
• Differentials -Implicit differentiation and related rates
• Differentiation as an operator -Using operators to compute other derivatives

### Integration

• Antidifferentiation -The indefinite integral and separable differential equations
• Exponential growth examples -More examples of exponential growth and decay
• More differential equations -Linear first order differential equations
• ODE Linearization -Solving harder differential equations
• Integration by Substitution -Substitution as an integration technique
• Integration by parts -Using the product rule as an integration technique
• Trigonometric substitution -Integration using trigonometric substitutions
• Partial fractions -Integration of rational functions using algebra
• Definite integrals -Definition and interpretation of the definite integral
• Fundamental Theorem of Integral Calculus -Connecting definite and indefinite integrals
• Improper integrals -Computing definite integrals when FTIC does not apply
• Trigonometric integrals -Products and powers of trigonometric functions
• Tables and computers -Using tables of integrals and mathematics software

### Applications

• Simple Areas -Finding the area of regions in the plane
• Complex Areas -Areas of more complex regions in the plane
• Volumes -Using the volume element to compute volume
• Volumes of revolution -Volumes from revolving a region about an axis
• Volumes in arbitrary dimension -Fourth dimension and beyond
• Arclength -Finding the length along a curve
• Surface area -Surface area of a solid of revolution
• Work -Computing work with integration
• Elements -Pressure, force, and other applications
• Averages -The average value of a function over an interval
• Centroids and centers of mass -Finding centroid and center of mass with integration
• Moments and gyrations -Moment of inertia and radius of gyration
• Fair probability -Uniform distribution
• Probability densities -Using the density function to compute probabilities
• Expectation and variance -Properties and interpretations of probability distributions

### Discretization

• Sequences -Discrete-input functions
• Differences -Derivatives of sequences
• Discrete Calculus -How to integrate discrete functions
• Numerical ODEs -Using sequences to solve ODEs
• Numerical integration -Using sequences to solve definite integrals
• Series -Infinite series as improper discrete integrals
• Convergence Tests -Comparison-type tests
• Convergence tests 2 -Geometric series-type tests
• Absolute Convergence and Conditional Convergence -Two types of series convergence
• Power Series -Interval and radius of convergence
• Taylor series redux -Details about Taylor series convergence
• Approximation and error -How to estimate an infinite series