In dtkaplan/Znotes:

Thinking about skills/objectives

The idea is to divide along two directions:

1. The kind of action the student should be able to perform.
2. The topic to which (1) applies

Basic categories of action

• Graphical interpretation: Gi
• Graphics construction: Gc
• Symbolic computation: Sc
• Notation reading: Nr
• Numeric computation: Computer
• Hand computation: Hand
• Establish consistency: Confirm
• Conceptual relationship: Re
• Recognize and use vocabulary appropriately: Vo

Topics

Functions Fun

1. Notation for function creation: inputs, output, parameters, $\equiv$, makeFun()

. Notation conventions: scalars vs functions

. Evaluation of function

a. from formula
b. on computer

. Shapes of functions

a. slope, concavity, curvature, discontinuity, extrema, monotonic, inflection point

. Pattern-book functions

. Domain and range

. Parameters (basic modeling functions)

a. parameters with `makeFun()

. Input centering e.g $[x - x_0]$

. Input scaling

. Slope at a point

a. Directional slope at a point
b. Working with contours

. Piecewise functions

. Vector-valued functions (e.g. gradient)

. Table as a function, graph as a function [be able to evaluate, find slope, accumulate]

. Assembling functions

a. linear combination (coefficients, scalar multipliers)
a. composition
a. product

. Partial evaluation (e.g. in anti-derivatives or taking a slice)

. Functions from data

a. Splines and other interpolants
    i. constructing by software
    i. when is smoothness necessary (cubic spline) or not wanted (Bezier)
a. Fitting by eye for basic modeling functions
a. Least squares fitting
a. Evaluate function from a table (including multivariate)

. Exponentials

a. Interpretation as half-life, doubling-time
a. Complex exponentials and interpretation as damped (or growing) sinusoid
    i. determine frequency from eigenvalues

. Logarithms

a. Graphical uses
#. Graphics for functions
a. slice plot
a. contour plot
a. surface plot
a. vector field
a. graphical domain
a. paths and constraints

. Parametric description of path as a set of functions of $t$.

Optimization Opt

1. Vocabulary: argmax vs max

. Objective function

. Incommensurate objectives

. Constraints

a. Active or not
b. Equality and inequality
c. Shadow price (Lagrange multiplier)
d. constraint functions
e. parallel gradients for constraint function and objective function at argmax

. Numerical techniques

. Derivatives and critical points

. Gradient vector points uphill, not necessarily toward optimum

. Gradient ascent/descent algorithm

. Find argmax/min from gradient field

. Find argmax/min from contour plot

Data Dat

1. Organization of a data frame

. Point plot

. Estimating function parameters (linear and nonlinear for basic modeling functions)

a. straight-line
b. exponential
c. sinusoid
d. gaussian and sigmoid
e. distinguish between power-law and exponential
f. log axes

. Function as a table

. Residual between response var and output of a model function

Dimension analysis Dim

1. Fundamental dimensions

. Compound dimensions and dimensional arithmetic

a. invalid operations on dimensional quantities
b. Know dimension of basic physical quantities (acceleration, force, energy, power, pressure, area, volume, work=force x distance, energy = power x time, power = force x velocity)

. Determine dimensional consistency

. Unit conversion and flavors of 1

. Dimension of derivatives and anti-derivatives

. Power-law functions

Calculus operations Cal

1. Vocabulary

. With-respect-to input

. Notations: $\partial_t x$, $\dot{x}$, $\frac{dx}{dt}$, $\frac{\partial x}{\partial t}$

. Accumulation Acc

a. Symbolic anti-differentation
    i. pattern book functions
    i. basic modeling functions (simple chain rule)
a. Appropriate notation
a. Constant of integration (also form from `antiD()`)
a. Definite integral, net change
a. Numerical integration and anti-differentiation
a. "Area under a curve" and why it's signed area
a. Fundamental theorem
    i. relationship between differentiation and anti-differentiation
    i. confirm anti-derivative by differentiation
a. Estimate from graph
a. anti-differentiation: output is a function
a. integration: output is a quantity
a. `antiD()` in R and how to evaluate it for definite integral 
a. calculating work by integrating force over distance
a. calculate energy by integrating power over time
a. calculate center of mass, centroid, moment of inertia
a. find probability that random variable is in a range
a. discounting

. Differentiation Dif

a. Symbolic differentiation
    i. pattern book functions
    i. basic modeling functions (simple chain rule)
    i. chain rule
    i. product rule
    i. sum rule
    i. by computer
b. Numerical differentiation
c. Second and higher-order derivatives
d. Partial derivatives
    i. first order
    i. second-order, including mixed partials
e. Gradient vector
    i. relationship to contours
f. Order of smoothness
g. Relationship to velocity and acceleration
h. output is a function
i. identifying monotonicity, concavity, curvature, critical points
j. tangents: line, circle, polynomial

. Argmax finding

Other operations

1. Zero finding (and inversion, solution)

. Find zeros of linear functions by hand

. Simultaneous zeros (e.g. crossings of nullclines)

. Iteration

a. improvement
b. Newton method

Calculus theory Theory

1. Vocabulary

. Slope function

. Finite-difference approximation to derivative

a. limit definition of a derivative

. Euler accumulation

. Value/Existence for a limit

. Taylor polynomials

a. Pick center $x_0$
b. Memorize for sin(), cos(), exp()
c. Compute error for a given input, characterize error as a function of $x - x_0$.

. Local linear approximation

. Differentials

. Instantaneous vs average rate of change

Approximation Approx

1. Vocabulary

. Polynomials

a. linear, interaction, quadratic terms

. Linearization

. Low-order polynomial approximation

Linear algebra Lal

1. Vocabulary

. Acceleration, position, momentum, velocity are vector quantities

. Vectors are rootless, have just direction and magnitude

. Vector operations (add, scale, dot product)

. Matrices

. Target problem

. Eigenvectors and eigenvalues

. Subspace

a. Is a vector a member of a subspace? (numerically)
b. Projection
c. Residual

Dynamics Dyn

1. Dynamical system as set of first-order differential equations

. Numerical integration from initial condition

. Symbolic integration:

a. first-order linear
b. eigenvalues, characteristic values and stability
c. exponential ansatz
d. linear combinations of solutions are solutions in linear systems

. Flows

. Trajectories

. Time series

. Nullclines

. Unlimited and limited growth models, logistic growth

. Bifurcation (appearance and disappearance of fixed points in 1-D)

. Classic models: predator-prey, SIR

. Force-balance and equivalence to pair of first-order differential equations

. Identify physical quantities (resistance, restoring force) in force-balance

Probability Prob

1. meaning of random variable, event

. probability density vs relative density

. normalization of relative density

. expected value, variance, standard deviation,

Computer Comp

1. Translate math formula into R formula and vice versa

. Construct a function with parameters

a. set default values for parameters

. Graphics layers

Modeling concepts Mod

1. Different models for different purposes

. Frameworks:

a. low-order polynomials
    i. which terms are appropriate
    i. constructing from first principles
    i. constructing from data
    i. limitations for modeling (dog chasing squirrel)
a. differential equations

. Modeling cycle

dtkaplan/Znotes documentation built on Sept. 4, 2022, 10:21 a.m.