Subsection Calculus as the science of functions
Calculus in its modern form, and as it will be presented to you at Northwestern, can be concisely described as the
science of functions. We will make this statement more precise in
DictumΒ 1.7. To do so we first recall the definition of a function and some related concepts.
Definition 1.1. Function.
Let \(X\) and \(Y\) be sets. A function \(f\) from \(X\) to \(Y\text{,}\) denoted
\begin{equation}
f\colon X\rightarrow Y\text{,}\tag{1.1}
\end{equation}
is a rule which, given any input \(x\in X\text{,}\) returns an output \(y\in Y\text{.}\) We write \(y=f(x)\) in this case, and call \(y\) the image of \(x\) under \(f\text{,}\) or the value of \(f\) at \(x\text{.}\) Similarly, we say \(f\) maps (or sends) the input \(x\) to the output \(y=f(x)\text{.}\)
The set
\(X\) is called the
domain of
\(f\text{;}\) the set
\(Y\) is called the
codomain of
\(f\text{.}\) The
range of
\(f\text{,}\) denoted
\(\range f\text{,}\) is the set of all outputs of
\(f\text{.}\) Using
set-builder notation, we have
\begin{equation}
\range f=\{y\in Y\colon y=f(x) \text{ for some } x\in X\}\text{.}\tag{1.2}
\end{equation}
As mentioned above, the textbook is less careful about dealing with the domain and codomain of a function. For example, you will find exercises in the book of the form
\begin{equation*}
\text{What is the domain of } f(x)=\frac{\sqrt{x-1}}{\sqrt{4-x^2}}\text{?}
\end{equation*}
To reconcile our approach with this particular issue, we make the following official declaration, or fiat.
Fiat 1.3. Implied domain.
If a function is
\(f\) introduced via a formula without specifying a domain or codomain, we will assume that the codomain is
\(\R\) and that the
implied domain \(D\) is the set of all real numbers for which the formula can be meaningfully evaluated.
Example 1.4. Implied domain.
Compute the implied domain
\(D\) of the function
\(f(x)=\sqrt{x-1}/\sqrt{4-x^2}\text{.}\)
Solution.
Since \(\sqrt{t}\) is defined only for \(t\geq 0\text{,}\) and since further a fraction \(r/s\) is defined only if \(s\ne 0\text{,}\) we can evaluate \(f(x)\) if and only if
\begin{align*}
x-1\amp\geq 0 \\
4-x^2 \amp > 0 \text{.}
\end{align*}
Solving this system of inequalities, we see that we must have \(x\geq 1\) and \(-2\leq x\leq 2\) and \(x\ne \pm 2\text{.}\) We conclude that
\begin{equation*}
D=[1,2)\text{.}
\end{equation*}
Example 1.6. Range versus codomain.
Define
\(f\colon \R\rightarrow \R\) as
\(f(x)=x^2\text{.}\) Compute
\(\range f\) and the codomain
\(Y\) of
\(f\text{.}\) Show that
\(\range f\subsetneq Y\text{.}\)
Solution.
The codomain of
\(f\) is
\(\R\) and the range of
\(f\) is
\([0,\infty)\) (the set of
nonnegative real numbers). Clearly
\([0,\infty)\subsetneq \R\text{,}\) thus
\(\range f\) is a proper subset of the codomain of
\(f\text{.}\)
With these notions in place we can finally describe the particular kind of functions that (single-variable) calculus is concerned with. We do so officially, as a
dictum, or saying.
Dictum 1.7. Calculus is the science of functions.
Single-variable calculus is the study of functions of the form
\(f\colon D\rightarrow \R\text{,}\) where
\(D\) is a subset of
\(\R\) (i.e.,
\(D\subseteq \R\)). This study is conducted in large part with the help of two fundamental tools: the
derivative and the
integral. The definitions of the derivative and integral rely in turn on an even more fundamental concept: the
limit.
In Math 220-1 (single-variable differential calculus), we introduce the limit and develop the theory of the derivative; in Math 220-2 (single-variable integral calculus), we study the integral. From now on, for the sake of convenience, we will assume that all functions considered in this course are of the particular form described in
DictumΒ 1.7.
Fiat 1.8. Calculus functions.
What is so important about studying functions, and what sort of properties of functions do we wish to examine? The short answer is that functions are one of the most fundamental tools for modeling empirical phenomena. Scientific inquiry will often begin by treating one empirical quantity \(q\) as being determined or dependent on some other empirical quantity \(x\) (or even multiple quantities \(x_1, x_2,\dots, x_n\) in a multivariable model). In this situation it is natural to model the quantity \(q\) as a function of these other quantities: i.e., we assume there is a function \(f\) satisfying \(q=f(x)\) (or \(q=f(x_1,x_2,\dots, x_n)\) in a multivariable model). Some examples:
-
Model the vertical velocity
\(v\) of a skydiver in free fall as a function of the time
\(t\) elapsed since jumping from the plane. In this case we assume
\(v=f(t)\) for some function
\(f\text{.}\)
-
Model the atmospheric temperature
\(T\) as a function of altitude
\(a\text{.}\) In this case we assume
\(T=f(a)\) for some function
\(f\text{.}\)
-
A marketing firm models the yearly sales
\(S\) of a certain product as a function of its price
\(p\) and the yearly amount
\(a\) the producer spends on advertising. In this case we assume
\(S=f(p,a)\) for some (multivariable) function
\(f\text{.}\)
Observe how in these modeling situations the defining characteristic of a function \(f\text{,}\) that each input \(x\) determines a fixed output \(f(x)\text{,}\) captures our intuitive idea of one quantity being dependent on another. This is what motivates the language of dependent and independent variables sometimes used in function theory: i.e., given a function \(y=f(x)\text{,}\) we often call \(y\) the dependent variable, and \(x\) the independent variable.
Continuing with the empirical modeling context, suppose we are modeling quantity \(q\) as a function \(q=f(s)\) of quantity \(s\) (i.e., quantity \(q\) is dependent on, or determined by quantity \(s\)). Here are natural questions about \(f\) we are interested in when studying the relation \(f\) between \(s\) and \(q=f(s)\text{.}\)
-
Maximum/minimum values.
Is there a maximum value of
\(q\) that can be attained? Is there a minumum value? In terms of
\(f\text{,}\) we are asking whether
\(\range f\) has a maximum and/or minimum value. If it does, we would also like to know which inputs
\(s\) give rise to these maximum and minimum values.
-
Asymptotic values/asymptotic behavior.
Perhaps
\(q\) does not reach a maximum value. Does it
approach some maximum value as
\(s\) varies, without ever attaining that value? Suppose quantity
\(s\) ranges naturally over the set
\([0,\infty)\) of all nonnegative numbers: how does the output
\(q=f(s)\) vary as
\(s\) gets infinitely large?
-
Increasing/decreasing.
How do the values
\(q=f(s)\) change as we vary
\(s\text{?}\) More specifically, does
\(q\) increase as we increase
\(s\text{?}\) Does
\(q\) decrease as we increase
\(s\text{?}\)
-
Average/instantaneous rates of change.
Fix a particular value \(s_0\) of our quantity \(q\text{,}\) and let \(q_0=f(s_0)\) be the value of \(q\) corresponding to that input. How does \(q_0\) change as we vary \(q\) by arbitrarily small amounts about \(q_0\text{.}\) More specifically, call
\begin{equation*}
R_{q_1}=\frac{q_1-q_0}{s_1-s_0}=\frac{f(q_1)-f(q_0)}{s_1-s_0}
\end{equation*}
the average rate of change with respect to \(s\) as we vary \(s\) from \(s_0\) to \(s_1\text{.}\) What happens to these average rates of change if we keep \(q_0\) fixed, and pick \(q_1\) to be arbitrarily close to \(q\text{.}\) Do they approach a well-defined instantaneous rate of change?
All the questions above can be phrased as questions about the function \(f\) that models the relation between \(q\) and \(s\text{,}\) and we will be able to answer all of these questions, using calculus, by the end of the quarter!
Historically, the development of calculus was most plagued by the challenge of coming up with rigorous mathematical definitions for the notions of
approaching a value to arbitrarily close degree without necessarily attaining it. Much nonsense was produced in this endeavor (Isaac Newtonβs
fluxions and
fluents, and Gottfried Wilhelm Leibnizβs
infinitesimals, for example), as well as a fair dose of vitriolic recriminations (George Berkeley accusing Newton and Leibniz of peddling in dubious βghosts of departed quantitiesβ).
These shaky foundations were not firmed up until the 19th century, when the modern definition of the
limit was developed by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass. Fittingly, it is with the limit that this course will begin in earnest!