MAT 232 Calculus of Several Variables
2018-11-05-1 at 09h
  • Syllabus
  • Week 1
    • Dylan
    • Cartestian coordinates
    • Polar coordinates
    • Graphing in polar coordinates
    • Converting between polar and cartesian
    • Slop in polar coordinates
  • Week 2
    • Dylan
    • Area in polar coordinates
    • sin(k theta) for k even and odd
    • parametric equations
    • parametric curves
    • the cycloid
    • calculus with parametric curves
    • area with parametric curves
  • Week 3
    • Dylan
    • parametric curves
    • orientation of parametric curves
    • arclength
    • the coordinate system on R^3
    • the dot product
    • lines and planes
    • normal to a plane
    • parametric equations for lines in space
    • functions of several variables
    • domain of functions of several variables
    • cross sections
    • level sets
  • Week 4
    • Dylan
    • limits and continuity
    • continuity at a point
    • partial differentiation
    • partial derivatives
    • tangent planes
    • the multivariate chain rule
    • implicit partial differentiation
  • Week 5
    • directional derivatives
    • unit vectors
    • gradient vector
    • higher order partial derivatives
    • Clairaut's theorem
    • maxima and minima
    • critical points
    • Hessian matrix
    • multivariable second derivative test
    • reducing number of variables in optimization
  • Week 6
    • Lagrange multipliers
    • test review
MAT 135 Calculus
2018-11-05-1 at 09h
  • Week 1
    • Functions
    • Domain
    • Range
    • Simple graphs
    • Vertical line test
    • Piecewise functions
  • Week 2
    • Essential functions (line, parabolas, and trig)
    • sine and cosine
    • transforming graphs
    • horizontal / vertical shift
    • horizontal / vertical translation
    • horizontal / vertical stretch
    • horizontal / vertical compression
    • reflections about x-axis and y-axis
    • exponential functions
    • one-to-one functions
    • inverse functions
    • natural logarithm function
  • Week 3
    • inverses and logarithms
    • trigonometry
    • radians
    • inverse trig functions
    • sin(sin^{-1}(x))
    • unit circle diagram
  • Week 4
    • limits
    • one-sided limits
    • calculating limits visually
    • limit laws
    • squeeze theorem
    • continuity
    • left continuous
    • right continuous
    • intermediate value theorem
  • Week 5
    • showing solutions exist with the IVT
    • limits to infinity
    • asymptotes
    • horizontal asymptotes
    • derivatives and rates of change
    • secant lines
    • tangent lines
    • instantaneous rate of change
    • average rate of change
    • physics of falling objects
    • differentiability
    • cusps
    • derivative as a function
    • higher derivatives
  • Week 6
    • derivatives of polynomials and exponentials
    • the power rule
    • tangent lines
    • derivative of e^x
    • limits as derivatives
    • product and quotient rule
    • tangent lines passing through a given point
MAT B41 Techniques in Multivariate Calculus
2018-10-30-2 at 10h
  • Week 1
    • Kostya
    • Shrijan
    • vectors
    • lengths
    • angles
    • standard basis
    • i,j,k basis
    • parametric equation of lines
    • dot product
    • orthogonal
    • parallel
    • parametrix equation of a plane
    • normal form of a plane
  • Week 2
    • Shrijan
    • Eric
    • Kostya
    • matrix multiplication
    • transpose
    • inverse of a matrix
    • determinants
    • volume of parallelpipeds
    • co-factor expansion
    • determinant and row operations
    • det(AB) = det(A)det(B)
    • the cross product
  • Week 3
    • Kostya
    • vector geometry
    • geometry and dot products
    • limits
    • delta-epsilon limits
    • contour plots
    • continuity
    • the circle-box argument
    • open sets
    • open disks
    • partial derivatives
  • Week 4
    • directional derivative
    • differentiability
    • gradient vector
    • paths in space
    • total derivatives
    • the chain rule for R^n --> R
    • the chain rule for R^n --> R^n
  • Week 5
    • Thanusun
    • Taylor series
    • Euler's identity
  • Week 8
    • Tabeeb
    • critical points
    • the Hessian matrix
    • Clairaut's theorem
    • relative minima and maxima
    • cup, cap, and saddle
    • multivariable second derivative test
    • optimization
  • Week 9
    • Shrijan
    • Tabeeb
    • constrained optimization
    • open sets
    • closed sets
    • bounded sets
    • extreme value theorem
    • Lagrange multipliers
  • Week 10
    • Shrijan
    • Tabeeb
    • integration
    • fundamental theorem of calculus
    • Cavalieri's principle
    • Archimede's theorem: cone + semi-sphere = cylinder
    • double integrals
    • x-simple regions
    • y-simple regions
    • simple regions
    • area and integrals
    • Fubini's theorem
  • Week 11
    • Tabeeb
    • changing order of integration
    • upper and lower bound inequalities
    • triple integrals
    • applications of integrals
    • averages
    • mass and density
  • Week 12
    • change of variables
    • onto functions
    • one-to-one functions
    • domain and range
    • linear transformations
    • fundamental theorem of linear algebra
    • Jacobian matrix
    • polar coordinates
    • polar to cartesian
    • "r dr dtheta"
    • integrating in polar coordinates
MAT 133 Calculus and Linear Algebra for Commerce
2018-10-30-2 at 10h
  • Week 1
    • exponent laws
    • n'th roots
    • logarithms
    • exponential functions
  • Week 2
    • piecewise functions
    • the absolute value function
    • inequalities
    • sigma notation
    • geometric series
    • functions
    • composition of functions
    • domain and range
    • inverse functions
    • symmetry
    • even and odd functions
  • Week 3
    • financial math
    • simple compound interest
    • annual percentage rate
    • continuously compounded interest
    • effective continuous rate
    • annuities
    • future value
    • present value
  • compound interest worksheet
  • Week 4
    • future value of an annuity
    • present value of an annuity
  • Week 5
    • review of financial math
    • linear systems
    • types of linear systems
    • parameters
    • effective annual rate
    • matrix form
    • row echelon form
    • reduced row echelon form
    • the Gaussian algorithm
  • Week 6
    • matrix operations
    • column vectors
    • sum of matrics
    • transpose
    • linear combination
    • product of matrices
    • economic example of matrices
    • identity matrix
    • inverse matrix
    • matrix inversion algorithm
    • Ax = b
  • Week 7
    • one sided limits
    • infinite limits
    • existence of limits
    • continuity of polynomials
  • Week 8
    • limits to infinity
    • horizontal asymptotes
    • vertical asymptotes
    • horizontal asymptotes of rational functions
    • continuity
    • composition of continuous functions
    • one-sided continuity
    • continuous from the right
    • continuous from the left
    • types of discontinuities
    • removable discontinuity
    • jump discontinuity
    • essential discontinuity
  • Week 9
    • slope of lines
    • tangents
    • secant line
    • derivative
    • Leibniz notation
    • Newton notation
    • the product rule
    • the quotient rule
    • derivatives and competing companies
  • Week 10
    • types of dicontinuities
    • differentiable => continuous
    • failures of differentiability
    • vertical tangent lines
    • cusps
    • discontinuities
    • Leibniz notation
    • the chain rule
    • writing functions as compositions
    • tables of values
    • inverse functions and derivatives
    • logarithmic differentiation
  • Week 11
    • implicit differentiation
    • increasing and decreasing functions
    • critical points
    • concavity
    • concave up and down
    • points of inflection
  • Week 12
    • optimization
    • local maximum
    • global maxmimum
    • the first derivative test
    • the second derivative test
    • optimization algorithm
    • the curve sketching algorithm
  • Week 13
    • linear approximation
    • factorials
    • n'th order polynomial approximation
  • Week 14
    • definite integrals
    • approximating the area under a curve
    • integrable functions
    • summation formulas
    • Riemann sums
    • anti-derivatives
  • Week 15
    • anti-derivatives and area
    • initial value problems
    • displacement
    • velocity
    • speed
    • acceleration
    • the Fundamental Theorem of Calculus
    • sketch of proof of the Fundamental Theorem of Calculus
    • signed area
    • total area
  • Week 16
    • integration by substitution
    • substitution for definite integrals
    • integration by parts
    • integration by parts for definite integrals
  • Week 17
    • area and integrals
    • signed area
    • total area
    • area between curves
    • horizontal slices
    • vertical slices
    • review of integration techniques
  • Week 18
    • improper integrals
    • convergence
    • divergence
    • integrating unbounded functions
    • divergence tests
    • direct comparison
    • limit comparison
  • Week 19
    • differential equations
    • seperable equations
    • linear differential equations
    • integrating factors
    • second order differential equations
  • Week 20
    • partial derivatives
    • tangent planes
    • gradient vector
    • marginal productivity
    • competing and complementary goods
  • Week 21
    • higher order partial derivatives
    • Clairaut's theorem
    • Hessian matrix
    • multivariate chain rule
    • unconstrained optimization
    • critical points
    • two variable second derivative test
    • cup, cap, and saddle
  • Week 22
    • constrained optimization
    • reducing variables
    • Lagrange multipliers
    • Lagrange with multiple constraints
  • Week 23
    • volume and iterated integrals
    • cross sectional area
    • Fubini's theorem
    • density and mass
  • Week 24
    • functions f : R^n --> R^k
    • coordinate systems
    • chain rule
    • change of variables
    • Jacobian matrix
    • linear maps and change of area
    • exam review
MAT 134 Calculus
2018-10-26-5 at 19h
  • Week 1
    • inequalities
    • the real number line
    • solving inequalities
    • set builder notation
    • intervals
    • functions
    • absolute value function
    • domain and range
    • graphing lines and quadratics
    • vertical line test
    • piecewise functions
  • Week 2
    • transforming graphs
    • vertical shifts
    • horizontal shifts
    • vertical stretch / compression
    • horizontal stretch / compression
    • trigonometry
    • radians and degrees
    • the special triangles
    • graphing trigonometric functions
  • Week 3
    • exponential functions
    • Euler's number
    • logarithms
    • natural logarithm
    • exponential growth
    • radioactive decay
    • one-to-one functions
    • horizontal line test
    • inverse functions
    • inversion algorithm
    • inverse trigonometric functions
    • limits
    • limit laws
    • sandwich theorem
    • squeeze theorem
  • Week 4
    • one sided limits
    • right hand limit
    • left hand limit
    • existence of limits
    • trigonometric limits
    • geometric proof of limit sin(x)/x
    • continuity
    • intermediate value theorem
    • limits to infinity
    • horizontal asymptotes
    • vertical asymptotes
  • Week 5
    • horizontal asymptotes
    • vertical asymptotes
    • rate of change
    • speed and travel
    • average rate of change
    • instant rate of change
    • slope of a line
    • slope of a curve
    • derivative as a function
    • Leibniz and Newton notation
    • one sided derivatives
    • left derivative
    • right derivative
    • differentiable => continuous
  • Week 6
    • derivatives
    • rate of change
    • velocity, speed, and acceleration
    • Mendel's law of inheritance
    • differentiation rules
    • scaling
    • adding
    • constants
    • power rule
    • horizontal tangent lines
    • product rule
    • quotient rule
  • Week 7
    • chain rule
    • implicit differentiation
    • fundamental law of muscle contraction
  • Week 8
    • inverse functio
    • differentiating inverse functions
    • logarithmic differentiations
    • inverse trigonometric functions
    • related rates
  • ./2017-2018-134/week-09/mat-134-week-9.pdf
  • ./2017-2018-134/week-09/mat-134-week-9ab.pdf
  • Week 10
    • monotonic functions
    • increasing / decreasing
    • critical points
    • first derivative test for local extrema
    • concavity
    • concave up / down
    • second derivative test for concavity
    • point of inflection
    • curve sketching
    • oblique asymptotes
  • Week 11
    • curve sketching
    • applied optimization
    • optimization algorithm
    • indeterminate forms
    • L'Hopital's rule
  • Week 12
    • applied optimization
    • antiderivatives
    • indefinite integrals
  • Week 13
    • area
    • upper and lowers sums
    • average value
  • Week 14
    • sigma notation
    • algebra and sigma notation
    • geometric series
    • Riemann sums
    • definite integrals
    • algebraic properties of definite integrals
    • geometry and area
  • Week 15
    • definite integrals
    • area and derivatives
    • area and anti-derivatives
    • fundamental theorem of calculus
    • total area
    • signed area
    • anti-differentiation techniques
    • substitution for indefinite integrals
  • Week 16
    • substitution
    • definite integrals and substitution
    • area between curves
    • vertical slices
    • horizontal slices
    • area algorithm
    • volume
    • solids of revolution
    • washer method
  • Week 17
    • parametric curves
    • arclength
    • physical work
    • force
    • Hooke's law
  • Week 18
    • integration techniques
    • trigonometric integrals
    • the inverse trig functions
    • integration by parts
    • cylic parts
    • iterated parts
    • parts and definite integrals
    • trigonometric integrals
    • itegrating sin^n(x)cos^k(x)
    • itegrating sec^n(x)tan^k(x)
  • Week 19
    • trigonometric substitution
    • partial fractions
  • Week 20
    • improper integrals
    • convergence and divergence
    • limiting to infinity
    • integrating near vertical asymptotes
    • comparison test
    • limit comparison
    • sequences
  • Week 21
    • convergence of sequences
    • divergence of sequences
    • squeeze theorem for sequences
    • continuous image of a sequence
    • bounded sequences
    • bounded above
    • bounded below
    • monotone
    • non-increasing sequence
    • non-decreasing sequence
    • monotone convergence theorem
    • infinite series
    • partial sums
    • algebra for series
    • n'th term divergence test
    • geometric series
    • divergence of the harmonic series
    • integral test
  • Week 22
    • comparison for series
    • recursive sequences
    • ratio test
    • root test
    • absolute convergence
  • Week 23
    • absolute convergence
    • conditional convergence
    • alternating series
    • convegence flow chart
    • power series
    • domain of convergence
    • Maclaurin series
    • Taylor series
  • Week 24
    • Maclaurin series
    • convergence of power series
    • interval of convergence
    • radius of convergence
MAT 223 Linear Algebra I
2018-10-26-5 at 08h
  • ./2017-223/week-01/mat-223-week-1.pdf
    • linear equations
    • augmented matrices
    • elementary row operations
    • row echelon form (REF)
    • reduced row echelon form (RREF)
    • Gaussian algorithm
    • solving linear systems
    • consistent and inconsistent systems
  • ./2017-223/week-01/mat-223-week-1c.pdf
    • parametrized solutions
    • parameters and rank
    • homogeneous systems
    • trivial solution
    • column vectors
    • basic solutions
  • Week 2
    • dimensions of a matrix
    • matrix sum
    • matrix product
    • transpose
    • column vector
    • row vector
    • n-dimensional euclidean space
    • dot product
  • Week 3
    • handout
    • matrix multiplication
    • non-commutativity of matrix multiplication
    • the plane
    • linear maps acting on the plane
    • matrix inversion
    • determinant
    • adjugate
    • matrix inversion algorithm
    • Fundamental Theorem of Linear Algebra
  • Week 4
    • handout
    • linear transformations
    • matrix of a linear map
    • geometry of linear maps
    • rotations
    • determinants and co-factors
    • row operations and determinants
    • det(AB)=det(A)det(B)
  • Week 5
    • cofactor expansion
    • matrix equations
    • complex numbers
    • complex plane
    • modulus
    • argument
    • Euler's identity
    • polar coordinates
  • Week 6
    • eigen-handout
    • eigenvalues
    • eigenvectors
    • characteristic polynomials
    • complex eigenvalues
    • repeated eigenvalues
    • basic eigenvectors
    • multiplicity of eigenvectors
    • diagonalization
  • Week 7
    • cartesian coordinate system
    • lines and planes
    • length
    • geometric vectors
    • parallelogram law
    • parametric equation of a line
    • intersection of lines
    • dot products
    • angles
    • orthogonality
    • projections
  • Week 8
    • lines and planes
    • vector form of a plane
    • parametric form of a plane
    • normal to a plane
    • plane containing two lines
    • distance to a plane
    • distance between two lines
  • Week 9
    • subspaces
    • image of a linear transformation
    • kernel of a linear transformation
    • span of a set of vectors
    • linear independence
    • dimension
    • bases
  • Week 10
    • cross product
    • orthogonality
    • orthonormal
    • Pythagorean theorem in an orthonormal basis
    • projection to an orthonormal bases
  • Week 11
    • exam practice
    • rank of matrices
    • column space
    • row space
    • properties of span
    • invariance of row space under row operations
  • Week 12
    • orthogonal complement
    • Gram-Schmidt orthogonalization algorithm
    • similar matrices
    • diagonalization
    • eigenspaces
MAT A29 Calculus I for the Life Sciences
2018-10-25-4 at 19h
  • Week 1
    • quadratic functions
    • parabolas
    • factoring
    • quadratic formula
  • Week 2
    • angles
    • radians and degrees
    • visual angles
    • special triangles
    • Pythagorean identity
    • unit circle
  • Week 3
    • limits
    • continuity
    • sin(x)/x
    • existence of limits
    • limit principles
    • continuity principles
    • piecewise functions
  • Week 4
    • average rate of change
    • instant rate of change
    • derivatives
    • scaling
    • the power rule
    • quotient rule
  • Week 5
    • increasing
    • decreasing
    • critical points
    • relative maxima and maxima
    • first derivative test
    • curve sketching table
    • concave up and down
    • second derivative test
    • point of inflection
    • limits of rational functions
    • vertical asymptotes
    • horizontal asymptotes
    • oblique asymptotes
  • Week 6
    • curve sketching
  • Week 7
    • optimization
    • end points
    • implicit differentiation
    • related rates
    • the sliding ladder problem
  • Week 8
    • exponential functions
    • population growth
    • logarithms
    • properties of logarithms
    • Ebbinghaus learning model
  • Week 9
    • solving logarithmic equations
    • exponential decay
    • radioactive decay
    • radio carbon dating
    • semi-log graphing
    • log-log graphing
  • Week 10
    • integration
    • anti-derivatives
    • integrating polynomials
    • summation notation
  • Week 11
    • Riemann sums
    • definite integrals
    • indefinite integrals
    • fundamental theorem of calculus
    • integration by substitution
  • Week 12
    • integration by parts
    • iterated integration by parts
    • partial fractions
    • cross-sectional area
    • volume
MAT A31 Calculus for the Mathematical Sciences
2018-10-25-4 at 16h
  • Week 1:
    • functions
    • one-to-one
    • onto
    • graphs
    • transformations
    • function composition
    • inverses
    • intervals
    • inequalities
    • absolute values
    • distances
  • Week 2:
    • Exponential functions
    • logarithms
    • the unit circle
    • radians
    • degrees
    • trigonometric identities
    • quantifiers
    • implication
    • logic
    • counter-examples
    • proofs with quantifiers
    • proofs with inequalities
    • proofs as essays
  • Week 3:
    • limits
    • limits to infinity
    • existence of limits
    • uniqueness of limits
    • vertical asymptotes
    • horizontal asymptotes
  • Week 4:
  • Week 5:
    • Bounds
    • upper bound
    • lower bound
    • least upper bound (lub)
    • greatest lower bound (glb)
    • completeness of the reals
    • continuitty
    • continuity of linear functions
    • continuity of f(x) = x^2
    • intermediate value theorem
  • Week 6:
    • Uniqueness of limits
    • squeeze theorem
    • continuity of polynomials
  • Week 7:
    • Slope
    • the slope function
    • secant lines
    • tangent line
    • left derivative
    • right derivative
    • derivative
    • physics of falling objects
  • Week 8:
    • Scaling
    • adding
    • power rule
    • product rule
    • quotient rule
    • composition
    • chain rule
    • higher order derivatives
    • anti-derivatives
  • Week 9:
    • Logarithms
    • exponentials
    • logarithmic differentiation
    • differentiation of piecewise functions
    • power series for e^x
    • derivatives of inverse functions
  • Week 10:
    • maxima and minima
    • curve sketching
    • local extrema
    • global extrema
    • critical points
    • Fermat's theorem (extrema are critical)
    • Rolle's theorem
    • closed interval method
    • mean value theorem
    • equivalence of mean value theorem and Rolle
    • first derivative test
  • Week 11:
    • curve sketching algorithm
    • concavity
  • Week 12:
    • l'Hopital's rule
    • logarithms and limits
    • anti-derivatives
    • anti-differentiation
    • integration by substitution
MAT 246 Concepts in Abstract Mathematics
2018-10-25-4 at 11h
  • Pre-Course Evaluation
  • Week 1
    • natural numbers
    • divisibility
    • primes
    • the sieve of Eratosthenes
    • infinitude of primes
    • binomial coefficients
  • Week 2:
    • induction
    • sum of the first n naturals
    • n! > 2^n
    • closed form of the Fibonacci numbers
    • well ordering principle
    • sqrt(2) is irrational
    • strong induction
    • binary form of natural numbers
    • binomial coefficients
  • Week 3:
    • divisibility algorithm
    • greatest common divisors
    • fundamental theorem of arithmetic
    • primes and products
    • parity
    • modular arithmetic
    • casting out nines
    • divisibility by nine
  • Week 4:
    • greatest common divisors
    • Euclidean algorithm
    • Lamé's theorem (the runtime of the Euclidean algorithm)
    • linear diophantine equations
    • modular arithmetic and inverses
    • arithmetic modulo a prime
    • Wilson's theorem
  • Week 5:
    • cryptography
    • Caesar ciphers
    • ROT13 encoding
    • one time pad
    • RSA cryptosystem
    • rational numbers
    • rational roots test
    • irrational numbers
    • test overview
  • Week 6:
    • complex numbers
    • complex conjugate
    • Argand diagram
    • real part
    • imaginary part
    • modulus
    • argument
    • Euler's identity
    • trigonometry using Euler's identity
    • the quadratic formula
    • the Fundamental Theorem of Algebra
    • topological proof of the fundamental theorem of algebra
    • multiplicity of roots
    • polynomial long division
    • De Moivre's theorem
    • roots of unity
  • Week 7:
    • Complex numbers
    • polar form
    • rectangular form
    • large powers
    • roots of unity
    • roots of z
    • set theory
    • set builder notation
    • Russell's paradox
    • union
    • intersection
    • disjoint unions
    • functions
    • pigeon hole principle
    • injective
    • sujective
    • bijective
    • infinite sets
    • |N|=|Q|
    • Cantor diagonalization
  • Week 8:
    • cardinality
    • musical chairs
    • power sets
    • |X| < |P(X)|
    • Hilbert's hotel
    • countability
    • products of sets
    • countable union of countable sets
    • the continuum
    • decimal expansions
  • Week 9:
    • Cantor-Bernstein theorem
    • finite subsets
    • the labelling principle
    • Cantor diagonalization
    • Euler's phi function
    • test review
  • Week 10:
    • euclidean geometry
    • triangles
    • angles
    • congruence
    • isoceles
    • equilateral
    • side-side-side
    • side-angle-side
    • angle-side-angle
    • aparallel postulate
    • Poincaré disk model of hyperbolic geometry
  • Week 11:
    • construcatibility
    • fields
    • constructions and fields
    • square roots are constructible
    • towers of fields
    • surds
    • conjugation and roots
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