**Advice from former MAM1000W students**

- Part I – an ActSci student whose grades kept going down, until they figured out the secret.
- Part II and III – Tutorial partners and useful videos.
- Part IV – Webassign and coming back from tough times
- Part V – “the methodology of thinking and applying helps me to this day”

**First semester**

Some introductory notes about the course and random links

- Intro to first year mathematics
- Why study mathematics?
- Tutorials
- Revision
- Ways to represent a function
- The first week of lectures
- Preamble and introduction
- MAM1000 Bootcamp

**Introductory topics**

**MAM1019H notes**

**Proof methods**

- Mathematical induction
- Mathematical induction with an inequalityProof by induction winning explanation
- Proof by contradiction – part 1
- Proof by contradiction – part 2

**Functions, Continuity and limits**

- Domain of a composite function – part 1
- Domain of a composite function – part 2
- Plotting rational functions.
- Arbitrary functions as the sum of an odd and an even function
- Can we find the inverse of a function which is not one-to-one? – part 1
- Can we find the inverse of a function which is not one-to-one? – part 2
- Continuity – part 1
- Continuity – part 2The squeeze theorem

**Second semester**

**Methods of integration**

- The distance problem
- Some sum identities Antiderivatives and integration
- The definite integral
- Riemann sums to definite integral conversions
- The fundamental theorem of calculus part 1 (part i)
- The fundamental theorem of calculus part 1 (part ii)
- The fundamental theorem of calculus part 1 (part iii)
- The fundamental theorem of calculus part 2 (part i)
- The fundamental theorem of calculus part 2 (part ii)
- Integration by substitution introduction
- Integration by substitution review
- Integration by parts
- More integration by parts
- Definite integration by parts
- Trig integrals
- Integrals using trig substitutionsHow to choose a substitution
- Integration by partial fractions
- Integration by partial fractions part 2A general expression for partial fractions
- Improper integrals of the first kind
- Improper integrals of the second kind

**Applications of integration**

- Areas between curves
- Volumes and cylinders
- Volumes by cylindrical cross-sections
- Volumes by cylindrical shellsVisualising volumes of revolution
- More volumes of integration
- More visualisations
- Arc lengths
- Average value of a function

**Combinations, permutations and the binomial theorem**

- Combinations and permutations
- Ordering objects
- Choosing r objects from p
- The binomial expansion

**Taylor polynomials**

- Taylor polynomials part 1
- Taylor polynomials part 2
- Taylor polynomials part 3
- Taylor polynomials continued
- Taylor polynomials continued

**Complex numbers**

- Complex numbers intro
- Complex numbers – the complex plane
- Complex numbers – mod/arg form
- Complex numbers – mod/arg form and multiplication
- Complex numbers – trig and exponential forms
- Complex numbers – exponentials of complex numbers
- Complex numbers – trig functions of complex numbers
- Complex numbers – roots of complex numbers
- Complex numbers – polynomials of complex numbers
- Complex numbers summary

**Differential equations**

- Differential equations – population growth
- Differential equations – the logistic equation
- Differential equations – direction flows and Euler’s method
- Differential equations – checking direction fields
- Differential equations – separable differential equations
- Differential equations – first order linear differential equations
- Differential equations – second order linear homogeneous differential equations with constant coefficient

**3D geometry, vectors and linear algebra**

- 3D geometry
- Vectors
- The scalar, or dot product
- Scalar and vector projections
- The vector, or cross product
- Vector representations
- Determinants
- The triple scalar and vector products
- The vector equation of a line
- Linear algebra – systems of linear equations
- Linear algebra – matrices
- Linear algebra – Gauss reduction

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