1Week 36

There are no tutorial classes in the first week, but you are strongly encouraged to test your skills by heading over to the BSS Mathematics brush-up course. This will help you master the necessary skills that are needed to get a good start with this course. The brush-up course consists of a series of quizzes, where you can test yourself. If you find something difficult, there are short explanatory notes that you can read to brush up.

When you've finished the brush-up course, there are a few more problems below that you can use to further practise your mathematics skills.

1.1 Readings for this week's lectures

Sections 6.1 – 6.7 in the textbook.

1.2 Readings for this week's exercises

The topics covered in the brush-up course roughly correspond to chapters 1 – 5 in the textbook.

1.3 Notes

If you are interested in the fascinating history of calculus, which features not only Newton and Leibniz, but goes all the way back to Archimedes, then I highly recommend the book Infinite Powers by Steven Strogatz.

1.3.1 Important rules

Here are some important rules, that you know from high school.

Fractions

$\begin{aligned} a \cdot \frac{b}{c} &= \frac{a \cdot b}{c} \qquad {}& \frac{a}{\frac{b}{c}} &= \frac{a \cdot c}{b} \qquad {}& \frac{\frac{a}{b}}{c} &= \frac{a}{b \cdot c} \\[4ex] \frac{\frac{a}{b}}{\frac{c}{d}} & =\frac{a \cdot d}{b \cdot c} {}& \frac{a}{b} \cdot \frac{c}{d} &=\frac{a \cdot c}{b \cdot d} \end{aligned}$

Identities for squares

$\begin{aligned} (a+b)^{2} &= a^{2}+b^{2}+2 ab \\ (a-b)^{2} &= a^{2}+b^{2}-2 ab \\ (a+b)(a-b) &= a^{2}-b^{2} \end{aligned}$

Powers

$\begin{aligned} a^{m} \cdot a^{n} &= a^{m+n} \qquad {}& \frac{a^{m}}{a^{n}} &= a^{m-n} \qquad {}& \left(a^{m}\right)^{n} &= a^{m \cdot n} \\[3ex] (a \cdot b)^{m} &= a^{m} \cdot b^{m} \qquad {}& \left(\frac{a}{b}\right)^{m} &= \frac{a^{m}}{b^{m}} \qquad {}& a^{0} &= 1 \\[3ex] a^{-m} &= \frac{1}{a^{m}} \qquad {}& \sqrt[m]{a} &= a^{\frac{1}{m}} \qquad {}& \sqrt[n]{a^{m}} &= a^{\frac{m}{n}} \\[3ex] \sqrt{a \cdot b} &= \sqrt{a} \cdot \sqrt{b} \qquad {}& \sqrt{\frac{a}{b}} &= \frac{\sqrt{a}}{\sqrt{b}} \qquad {}& \sqrt{a} &= a^{\frac{1}{2}} \end{aligned}$

Quadratic equations

A quadratic equation is an equation of the form

$ax^2 + bx + c = 0. \tag{1.1}$ The discriminant, $D$ , is defined as

$D = b^2 - 4ac. \tag{1.2}$ If $D > 0$ the equation has two solutions, given by

$x = \frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}. \tag{1.3}$ If $D = 0$ the equation has a single solution, given by:

$x = \frac{-b}{2 a}. \tag{1.4}$ If there are two solutions $x_1$ and $x_2$ the quadratic equation can be written

$a (x-x_1) (x-x_2) = 0 \tag{1.5}$ This is called factorisation of the quadratic equation, since it has been expressed as a product of two factors.

The natural exponential and logarithmic functions

Connection between the exponential and logarithmic functions

$y = \ln(x) \Leftrightarrow x = e^{y}$ The natural logarithm

$\begin{aligned} \ln(e) &= 1 \qquad {}& \ln(a \cdot b) &= \ln(a) + \ln(b) \\[3ex] \ln \left(\frac{a}{b}\right) &= \ln (a)-\ln (b) \qquad {}& \ln \left(a^{r}\right) &= r \cdot \ln (a) \end{aligned}$ The natural exponential function

$y = \exp(x) = e^x$

$e^x e^y = e^{x+y} \qquad \frac{e^x}{e^y} = e^{x-y}$

1.4 Exercises

Below, you will find a few exercises that build on skills that we expect you to have. If these are difficult, you may need to spend some time on the brush-up course first.

This exercise is meant to give you an understanding of quadratic functions and the quadratic equation, by using graphical representations. Below is a plot of the function

$f(x) = ax^2 + bx + c.$

In the plot, you can change the values of $a, b$ and $c$ by pulling the three sliders, and observe how the plot of $f(x)$ changes. See if you can give answers to the following questions.

Explain, in words, how changing the values of $a$ , $b$ and $c$ changes the plot.
Find the values of $a, b$ and $c$ for which the curve of $f(x)$ is a stright line through the points $(0,1)$ and $(1,0)$ .
What is the graphical interpretation of the equation $f(x) = 0$ ?
Find a set of values for $a, b$ and $c$ for which the equation $f(x) = 0$ does not have any solutions.
Find a set of values for $a, b$ and $c$ for which the solution to the equation $f(x) = 0$ are $x = -2$ and $x = 2$ .
This may be easier of you factorise the quadratic, as shown in Eq. 1.5.

Fractions, solving equations

Arbitrage is when two identical products have two different prices on different markets. When this is the case, you can buy the product in one market and sell it on another market — with a profit! In finance, it is common to work with the no arbitrage assumption, meaning that arbitrage is not possible.

However, this assumption is not always true, for instance in sports betting. When you play odds, you bet money that a particular outcome will happen, and you get a larger amount back if you guessed the outcome correctly. In the opposite case you lose the amount you bet.

As an example, consider tennis, which is a sport with two outcomes. Roger Federer plays against Rafael Nadal. Either Federer wins (with the odds $x_1$ ) or Nadal wins (odds $x_2$ ). The table below shows the odds on these two outcomes from two different bookmakers: Bet365 and Danske Spil.

$\def\arraystretch{1.5} \begin{array}{lll} \textbf{Bookmaker} & \textbf{Federer wins} (x_1) & \textbf{Nadal wins} (x_2) \\ \hline \text{Bet365} & 1.5 & 2.5 \\ \text{Danske Spil} & 2 & 1.75 \end{array}$

The amount you win if you guess the right outcome is found by multiplying the amount bet with the odds. If $y_1$ is the amount bet on Federer winning, and $y_2$ is the amount bet on Nadal winning, then if Nadal wins the payoff will be $x_2 \cdot y_2$ .

In other words, if Nadal won the game and you bet EUR 100 on this outcome with Danske Spil the payoff would be

$100 \cdot 1.75 = 175 \quad \text{(i.e. a profit of EUR 75 ).}$

Federer wins. Tina bet EUR 100 at Bet365 that Federer would win. What is Tina's payoff?

A surebet is when you bet on two different outcomes and you are sure to make a profit, no matter what the outcome is. Often, but not always, you will place a bet for one outcome with one bookmaker and the other outcome with the other bookmaker.

The condition for a surebet is $\frac{1}{x_1} + \frac{1}{x_2} < 1$ .

To bet on Federer with Bet365 and Nadal, also with Bet365, is not a surebet since $\frac{1}{1.5} + \frac{1}{2.5} = \frac{1}{\frac{3}{2}} + \frac{1}{\frac{5}{2}} = \frac{2}{3} + \frac{2}{5} = \frac{10}{15} + \frac{6}{15} = \frac{16}{15} > 1$ .

Is there a possibility of a surebet with the combination of odds in the table above?

Tina has identified a surebet: betting on Federer with Danske Spil and on Nadal with Bet365. She has a budget of EUR 100 in total to bet, i.e., $y_1 + y_2 = 100$ .

If Tina wants an identical payoff no matter what the outcome is, how wil she need to place her bets, i.e., what are the values of $y_1$ and $y_2$ ? How big is the payoff?