Anna

Section 1

Define all the number systems.

Prove |Q| = |N|
Prove |R| > |N|

Prove Pythogoras’ Theorem in 2D and 3D.

Find the solutions of x^2 + px + q = 0

Section 2

Define sin(t), cos(t) and prove SOH CAH TOA.

Use calculus to write f’(x) in terms of f(x).

Now we have differentiation. Show that x^k / k! —D—> x^(k-1) / (k-1)!.

Given that f(x) and all its derivatives are continuous, find a function g(x) that behaves exactly the same as f(x) when x=0. i.e. g(0)=f(0), g’(0)=f’(0), …

State Taylor’s Theorem.

Use Taylor’s Theorem to give expansions for e^x, sin(x), cos(x).

Show e^x —D—> e^x.
Show S -> C -> -S -> -C -> …

State and prove Euler’s Identity.

Prove the double angle formulae for sin and cos in two different ways.

Section 3

Without using integration, show that integral 1/x^2 between 1 and infinity is finite, but integral 1/x between 1 and infinity is INFINITE.

State FTC and give sketch proof.

Now we have integration. Prove that x^k / k! —I—> x^(k+1) / (k+1)!.

This time using integration, show that integral 1/x^2 between 1 and infinity is finite, but integral 1/x between 1 and infinity is INFINITE.

Sketch e^x and ln(x).

Differentiate ln(x).

Give a linear approximation of f(x+h) in terms of f’(x). How does this connect with Taylor’s Theorem?

State and prove the chain rule.
State and prove the product rule.

Section 4

How do you construct the matrix for a rotation of 90° anticlockwise in 2D? How about theta radians? How about any 2D transform?

How would you solve three equations in three unknowns?

ax + by + cz = d
ex + fy + gz = h
ix + jy + kz = l

Section 5

What are the axioms for a group?

What extra axiom is needed for abelian groups?

Show, by creating the group table, that the symmetries on an equilateral triangle form a group.

Plot the five fifth-roots of unity on the complex plane, and show that they form a group under multiplication.

Find the matrix for a 120° rotation anticlockwise in 2D, and show that this matrix generates a group under matrix multiplication.

Section 6

Explain the Monty Hall problem.

Use induction to show 1^2 + 2^2 + … + n^2 = n(n+1)(2n+1) / 6

How many ways of choosing r objects out of n? Use induction to prove that this creates Pascal’s triangle.

Simplify (x^4 + 5x^3 - x^2 + 1) / (x^2 + 3x + 5) using polynomial division

Plot (x^2-16) / (x+1)(x+2)(x+3)

Section 7

Define an open set and a closed set.

A clopen set is a set that is both open and closed. Give examples of an open set, a closed set and a clopen set.

Show that in a connected open set, given any path from A to B, you can cover this path with finitely many epsilon-balls.

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Lastly, have I missed anything?