Unit 1 - Number and Algebra
Get Ready
Questions
a) Calculate the final value of a $1 investment which accumulates 100% interest over a single year.
b) Calculate the final value of a $1 investment which accumulates 100% interest p.a compounded quarterly.
c) Calculate the final value of a $1 investment which accumulates 100% interest p.a compounded monthly.
d) Calculate the final value of a $1 investment which accumulates 100% interest p.a compounded daily.
e) Calculate the final value of a $1 investment which accumulates 100% interest p.a compounded hourly.
f) Calculate the final value of a $1 investment which accumulates 100% interest p.a compounded every minute.
g) Calculate the final value of a $1 investment which accumulates 100% interest p.a compounded every second.
What do you notice about the value of your investment, as the frequency at which interest is paid increases?
Come up with a general formula for the calculating the final value of a $1 investment which accumulates 100% interest p.a compounded \(n\) times.
Solutions
Notes
The number \(e\) is an important mathematical constant, with significant applications in various fields, including calculus, complex analysis, finance, and statistics. The number \(e\) can be defined in several ways, such as through a limit or a series expansion. In finance, \(e\) appears in the formula for continuous compounding. Additionally, \(e\) plays a critical role in probability, statistics, and complex numbers.
\(e\) can be defined in several ways, but the most common is $$e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}$$It has an approximate value of $$e\approx 2.71828$$but much like \(\pi\) it’s decimal expansion continues randomly as it is an irrational number.
It is important enough that we have a special logarithm devoted to the base \(e\):$$\ln=\log_{e}x$$This means that $$\ln e^{x}=x$$and$$e^{\ln x}=x$$
Examples and Your Turns
Example
Find $$\ln e^{3}$$
Your Turn
Find $$\ln \sqrt{e}$$
Your Turn
Find $$e^{2\ln 5}$$
Investigation
On a spreadsheet use the formula =RAND() to create a random number between \(0\) and \(1\). You can also do this on your GDC if you know how.
Refresh the formula to get a new number.
Repeat this process until the sum is bigger than 1.
Record the number of numbers you needed for the sum to be greater than 1.
Repeat this process 10 times, so you have a list of 10 numbers (all integers as these are the number of numbers needed).
Calculate the average number of numbers needed for the sum to be bigger than 1.
Go to classes.interactive-maths.com/investigation-5.html.
Generate a large set of results automatically using this applet.
What do you think the result is getting closer to?
Express this result in words.
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.