Luca: Home

Each topic has one of the following states:

• "lock", we haven't covered this yet
• "pen" , we've covered this but need more practice
• "fire", we've learned this well

Resources

• Calculator: numbat.dev

Algebra

• Fundamentals

  • Exponents and properties
  • Logarithms and properties
  • Interval notation
  • Exponential functions
  • fn basics, composite & inverse functions

• Root manipulation

  • Rationalize denominator
  • Extract roots (sqrt(24) = 2sqrt(6))

• Quadratic equations

  • Complete the square
  • Difference of squares
  • Quadratic formula (have it memorized!)
  • Factoring polynomials by decomposition
  • Complex roots of polynomials

• Bonus

  • Polynomial long division

Precalculus

• Informal limits
  • As x approaches ±infinity
  • As x approaches a given point
  • Limits from the left and from the right
• Sketch polynomials
  • End behavior
  • Root behavior
• Sketch rational functions
  • What is a rational function?
  • End behavior
  • Horizontal & vertical asymptotes
  • Removable discontinuities
• Polar coordinates
  • Convert to/from cartesian coordinates
  • Sketch functions
• Conic sections

Linear algebra

• Vectors

  • Vector decomposition (using trigonometry)
  • Vector sums
  • Dot product
  • Projection
  • Orthogonality
  • Norms
  • Recognize that ℂ is isomorphic to ℝ^2
  • Basis sets

• Complex numbers

  • Addition, multiplication
  • Conjugates
  • Division
  • Norms
  • Dot product

• Matrices (Bonus)

  • Matrices as linear transformations
  • Multiplication with vectors
  • Multiplication with matrices
  • Solving linear systems
  • Other applications

Trigonometry

• Basics

  • Trigonometric ratios: sin, cos, tan
  • Trigonometric inverses: asin, acos, atan
  • Memorize the unit circle
    • sin, cos, tan from angle in degrees
    • sin, cos, tan from angle in radians
    • Map radians to degrees
    • Map degrees to radians
  • Trigonometric identities (TODO: which ones?)

• Advanced

  • Other ratios: csc, sec, cot
  • Law of sines
  • Law of cosines

Geometry

• Fundamentals

  • Point, Line, Plane
    • Parallel lines (definition)
    • Perpendicular lines (definition)
    • Line = constant distance from two points
  • Polygon classification
    • Triangle, Square, Rectangle, Parallelogram,
    • Trapezoid, Isoceles trapezoid, Rhombus, Kite
    • Circle, Ellipse
    • Convex polygon (definition)
  • Similarity
    • What are translations, dilations, rotations, reflections
    • What properties to these transformations preserve?
    • These are the only such transformations in the plane!
  • Cavalieri's principle

• Angle-chasing

  • Terms
    • acute, obtuse, right
    • vertical angles, complementary, supplementary
  • Sum of interior angles
  • Sum of exterior angles
  • Complementary and supplementary pairs
  • Parallel transverse lines (Alternate exterior angles)

• Circles

  • Terms: sector, chord, tangent, arc
  • Radians (definition, conversion)
  • Circumference and area (...of arcs and sectors)
  • Interior/exterior angles
  • Properties
    • Tangent line is perpendicular to radius
    • Center angle is 2x chord angle (with proof)
    • Common chord => equal angles

• Triagles

  • Fundamentals
    • Triangle inequality (with proof)
    • Pythagorean theorem (with proof)
    • Special right triangles (30, 60, 90 and isoceles)
    • Pythagorean triples
  • Similarity
    • three forms of triangle simarity
    • three forms of triangle equality

• Compass and ruler constructions (problems)

• Handouts

  • Euclidean I
  • Triangles I

Combinatorics

• Fundamentals
  • Factorial
  • n permutations k
  • n choose k (problems )
  • Double-counting (problems )
• Pascal's triangle
  • Stars and bars
  • Sum of row is 2^n
  • Symmetry of n choose k
  • Binomial coefficient
  • Stars and bars (technique)

Probability

• Fundamentals

  • Probability as a combinatorial ratio (frequentist)
  • Simple combinatorial problems
  • Set theory fundamentals

• Advanced

  • Expected value
  • Monty hall problem
  • Independent and dependent events
  • Conditional probability
  • Bayes' rule & bayesian probability
  • Birthday paradox

Logic & Proofwriting

• Fundamentals
  • Terms: theorem, axiom, lemma, conjecture
  • Implication
  • Contrapositive
• Proof by contradiction
• Proof by induction
• Proof by construction
• Pidgeonhole principle