Expected Background Knowledge

We are assuming that students entering this paper are relatively mathematically confident, having done well in NCEA Level 3 mathematics. To be precise: we’re hoping that you’ve received a ‘merit’ or higher on one of the three externally-assessed NCEA L3 mathematics standards, namely differentiation, integration, or complex numbers.

In other systems, this is roughly equivalent to a passing mark in CIE A2 mathematics, or a C or better in CIE AS mathematics, or a 3/7 or higher in IB mathematics.

Compsci 120 is a paper that assumes you are already comfortable with a number of mathematical concepts and conventions. Trying to take this course without a solid background in mathematics is a bad idea. If you do not have this background, you should take Maths 102! Maths 102 is a course ran every semester (including summer semesters) at the University of Auckland, and has no prerequisites. It is designed to give you the skills to succeed in CS 120!

By “comfortable with”, we mean that you should be able to see these calculations done in lecture without further explanation, and be able perform these calculations yourself on an exam without a calculator.

To help clarify some of the specific concepts we’re expecting students to understand (in case you’re coming from overseas, or it’s been a while since you finished high school), here is a particular set of skills that we are hoping you’ve acquired over your career. This list is not exhaustive, but is just meant to point out the most common stumbling blocks that less-prepared students encounter in Compsci 120. Again, if you do not feel comfortable with these calculations, please enrol in Maths 102!

Exponents. You should know what $a^b$ is for any two integers $a,b$ , and know how to work with exponents. For example, the following calculations are ones you should be comfortable with:

2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32

2^3 \cdot 2^4 = 2^{3+4} = 2^{7} = 128

(2^2)^3 = 2^{2\cdot 3} = 2^6 = 64

7^{-2} = \dfrac{1}{7^2} = \dfrac{1}{49}

10^0 = 1

a^b \cdot a^c = a^{b+c}

(a^b)^c = a^{bc}

a^{-b} = \dfrac{1}{a^b}

Logarithms. You should know what $\log_2(n)$ and $\log_{10}(n)$ mean, and be comfortable with calculations like the following:

\log_2(32) = \log_2\left( 2^5\right) = 5

\log_{10}(1) = 0

\log_2(2^n) = n

\log_2(a) = a

Quiz .

What is $\log_2(4^n) - \log_{10}\left(\dfrac{2^n5^n}{10^{2n}}\right)$ ?

Expanding polynomials. You should be able to expand products of polynomials and other expressions. For example, the following calculations are ones that you should be able to read and understand without further explanation:

(x+1)^3 = x^3 + 3x^2 + 3x + 1

(x -y)(x^n + x^{n-1}y + x^{n-2}y^2 + \ldots + xy^{n-1} + y^n) = x^{n+1} + x^n y + x^{n-1}y^2 + \ldots + xy^n - x^ny - x^{n-1}y^2 - \ldots - xy^n - y^{n-1} = x^{n+1} - y^{n+1}

Fractions. You should be able to combine ratios and fractions with various arithmetic operations. For instance, you should be comfortable with the following calculations:

\dfrac{1}{3}-\dfrac{4}{8} = \dfrac{1\cdot 8 +(-4)\cdot 3}{24}=-\dfrac{4}{24}

3+\dfrac{7}{113} = \dfrac{113 \cdot 3 +7}{113}=\dfrac{346}{113}

\dfrac{1}{1-x} = \dfrac{ 1 + -x + x}{1-x} = 1 + \dfrac{x}{1-x}

\left(-\dfrac{45}{11}\right) \cdot \left(-\dfrac{5}{6}\right) = \dfrac{(-45) \cdot (-5) }{ 11\cdot 6 }=\dfrac{225}{66}

\dfrac{a}{b} + \dfrac{d}{bc} = \dfrac{ac + d}{bc}

\dfrac{1}{\sqrt{x} - a} = \dfrac{1}{\sqrt{x} - a} \cdot \dfrac{\sqrt{x}+a}{\sqrt{x} + a} = \dfrac{\sqrt{x} + a}{x - a^2}

Solving equalities and inequalities. Given an equation or inequality in one or more variables, you should be able to rearrange it to “solve” for one variable in terms of the others. For example, the following processes should be ones you’re comfortable with:

3x - 4 = 12 \Rightarrow 3x = 16 \Rightarrow x = \frac{16}{3}

(a^2-1)b + 1 =a \Rightarrow (a^2-1)b = a-1 \Rightarrow b = \frac{a-1}{a^2-1} = \frac{1}{a+1}, \textrm{ if } a \neq \pm 1

-5x - 7 \leq 3 \Rightarrow -5x \leq 10 \Rightarrow x \geq -2

\frac{20}{x^2+1} > 2 \Rightarrow \frac{x^2+1}{20} < \frac{1}{2} \Rightarrow x^2+1 < 10 \Rightarrow |x| < 3

x^2 - 3x + 2 > 0 \Rightarrow (x-2)(x-1) > 0 \\ \Rightarrow (x-2) \textrm{ and } (x-1) \textrm{ are} \textrm{ both } < 0, \textrm{ or } (x-2) \textrm{ and } (x-1) \textrm{ are both } > 0\\ \Rightarrow (x < 2 \textrm{ and } x< 1), \textrm{ or } (x >2 \textrm{ and } x > 1)\\ \Rightarrow (x< 1) \textrm{ or } (x >2).

Substitution. Given a function $f(x)$ , you should be able to plug in values and expressions in to this function, and get the correct output. For example, you should be capable of the following:

\textrm{If } f(x) = x^2 + 1\quad \textrm{ then }\quad f(4) = 4^2+1 = 17, \quad \textrm{ and }\quad f(x+1) = (x+1)^2 + 1 = x^2 + 2x + 2.

\textrm{If } g(n) = \frac{n(n+1)}{2}\quad\textrm{ then }\quad g(n+1) = \frac{(n+1)(n+2)}{2}, \quad\textrm{ and }\quad g(2n) = \frac{(2n)(2n+1)}{2} = n(2n+1).

\textrm{If } h(x) = 1-x\quad\textrm{ then }\quad h(y) = 1-y, \quad\textrm{ and }\quad h(h(y)) = h(1-y) = 1- (1-y) = y.