351a: Calculus Refresher

Feel of Mathematica
an introduction to using mathematica, plotting functions with mathematica.

1.01 Growth
rates of growth of linear functions/power functions/the exponential function, percentage growth rates, the global scale, interpolation of data sets.

1.02 Natural Logs and Exponentials
the natural base e and the natural logarithm, percentage growth of exponential functions: doubling time and half life, unnatural bases, exponential models, e and exponential data analysis, e and finance.

1.03 Instantaneous Growth Rates
instantaneous growth rates, instantaneous growth rate of x^k, sin[x], cos[x], Log[x] and exp[x]; average growth rate versus instantaneous growth rate.

1.04 Rules of the Derivative
derivatives, instantaneous growth rates, the chain rule, general rules for taking derivatives, using the logarithm to calculational advantage, dominance in growth rates

1.05 Using the Tools
finding maximum values and minimum values, good representative plots of functions, fitting data by curves.

1.06 The Differential Equations of Calculus
solving various differential equations like y'[x]=r y[x] (exponential growth and decay), y'[x]=r y[x](1 - y[x]/b) (the logistic equation), y'[x] = r y[x] + b, applications.

1.07 The Race Track Principle
the Race Track Principle, the Race Track Principle and differential equations, faking the plot of the solution of a differential equation, estimating roundoff error.

1.08 More Differential Equations
Euler's faker and Mathematica's faker, the predator-prey model.

1.09 Parametric Plotting
parametric plots in two dimensions: Circular parameters, parametric plots of curves in three dimensions, parametric plots of surfaces in three dimensions, derivatives for curves given parametrically: The cycloid, applications.

2.01Integrals for Measuring Area
the definite integral as a measure of signed area, three properties of integral, integration by approximation by trapezoids, areas suggested by data lists, functions which cannot be integrated over certain intervals.

2.02 Breaking the Code of the Integral: The Fundamental Formula
the fundamental theorem and the fundamental formula, measurements of distance and velocity via the fundamental formula, infinite integrals, the integral of the sum is the sum of the integralsmental, changing the order of the limits, velocity, acceleration and the fundamental formula, area between curves, approximating infinite integrals, the fundamental formula and its relation to differential equations, the indefinite integral.

2.03 Measurements
measurements based on slicing and accumulating: area and volume, measurements based on slicing and accumulating: density and mass, measurements based on approximating and accumulating: arc length, measurements based on the fundamental formula: accumulated growth, volumes of solids, linear dimension: volume and area.

2.04 Transforming Integrals
transforming integrals, measuring area under curves given parametrically, bell-shaped curves and Gauss's normal law, measuring area inside closed curves, polar plots and area measurements.
2.05 Integrals and the Gauss-Green Formula 2D integrals for volume measurements, the Gauss-Green formula, using a 2D integral to measure area.

2.0 More Tools and Measurements: Techniques for Calculating Integrals
separating the variables and integrating to get formulas for solutions of some differential equations, integration by part, using the complex exponential to calculate various integrals, the technique of calculating integrals by taking derivatives, using integration by parts to do integration by iteration

3.01 Splines
smooth splines, splining functions and polynomials.

3.02 Expansions in Powers of x
the expansion of a function f[x] in powers of x, some well known expansions, expansions for approximation, expansions by substitution, expansions by differentiation, expansions by integration.

3.03 Using Expansions
expansions in powers of (x - b) and approximations based on them, tangent lines and Newton's method, using expansions to help to calculate limits, expansions and the complex exponential function, square roots by Newton's method, using the complex exponential to generate trigonometric identities, using expansions to get precise estimates of integrals.

3.04 Taylor's Formula
Taylor's formula for the expansion of f[x] in powers of (x - b), four approximations based on Taylor's formula and how they can be used to estimate integrals, Taylor's formula in reverse, limits, approximations and fake plots of solutions of some differential equations.

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