MTH6131 Chap.1 Functions and Continuously Compounded Interest
Functions and Continuously Compounded Interest
Why a calculus course opens with a bank balance
The first lecture chooses an example that anyone can state and that needs calculus to finish. A loan of principal P at rate r owes P(1 + r) at maturity. Roll the whole amount into a new loan on the same terms and the debt becomes P(1 + r)², and after n such loans it is P(1 + r) to the power n.
That is the compound interest model, and each step of it is ordinary arithmetic.
The interesting question comes next. Banks quote an annual rate and credit it in pieces, so with n credits a year for t years the balance is P(1 + r/n) to the power nt. What happens as n grows without bound?
The variable sits in the base and in the exponent at once, and the lecture resolves it by taking a logarithm and applying the rule for a zero over zero limit. The answer is the continuously compounded model, P times e to the power rt.
Three tools from later in the course, a limit, a logarithm and a derivative, are already needed on the first day, which is the case for teaching calculus before modelling.
The vocabulary of functions
The same lecture fixes the language used for the rest of the semester. A function from A to B assigns each input in the domain A exactly one output in the codomain B.
It is one to one when different inputs always give different outputs, onto when every element of the codomain is reached, and invertible exactly when it is both. Composition applies one function to the output of another, and the order matters: in general g of f and f of g are different functions even when both exist.
Even functions satisfy f(−x) = f(x) and odd functions satisfy f(−x) = −f(x); recognising either halves the work of a sketch later in the course.
What this chapter sets up for later
The number e reappears when logarithmic and exponential functions are differentiated, and the model itself returns at the end of the course as the differential equation dA/dt = rA, whose solution is the function found here as a limit.
Keeping that thread in view makes the final topic feel like a return rather than a new start.
What this chapter covers
- 01
The simple interest model and its repeated form
- 02
Splitting an annual rate into n credits
- 03
The limit of (1 + r/n) to the power nt
- 04
Domain, codomain and the one-output rule
- 05
One to one, onto and invertible functions
- 06
Composition and why its order matters
- 07
Even and odd functions
Worked example · free
Quarterly against continuous compounding
- 2Write the quarterly formula with the rate split into four and the number of periods counted.
- 2Write the continuous formula as the limit of the first.
- 2Compare the two values and justify the order.
Key terms
- Principal
- The amount lent or deposited at the start, on which interest is calculated.
- Compound Interest
- Interest that is added to the principal at the end of each period, so that later interest is earned on earlier interest as well.
- Continuous Compounding
- The limiting case of compounding as the number of periods per year grows without bound, which gives a balance of P times e to the power rt.
- Codomain
- The set a function maps into, which may be larger than the set of values actually reached.
- Bijective Function
- A function that is both one to one and onto, which is exactly the condition for it to have an inverse.
- Composite Function
- The function g of f obtained by applying f first and then g to its output.
Functions and Continuously Compounded Interest FAQ
Why does the course start with interest rather than with limits?
Because the interest example shows why limits are needed. Compounding more and more often leads to a quantity with the variable in both the base and the exponent, and settling what it approaches takes a logarithm and a derivative. The example therefore motivates the order of the whole semester: calculus first, then modelling.
Is y squared equals x a function of x?
No. A function must give exactly one output for each input, and here the input 4 is paired with both 2 and minus 2. The relation still describes a perfectly good curve, and later chapters find its slope by implicit differentiation even though it is not a function of x.
How do I show that a function is invertible?
Show that it is one to one, by assuming f(a) = f(b) and deducing a = b, and that it is onto its codomain, by showing every element is reached. The lecture states that a function is invertible exactly when it has both properties, and that its inverse is then unique.
Exam move
Work the compounding formula for one set of numbers at four frequencies and at the continuous limit, and keep the table: it is the quickest way to see the limit behind e before the formal definition arrives. Then practise composition by writing both orders for pairs of simple functions and evaluating each at zero, until checking the order becomes automatic.
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