MA005: Calculus I

The graph of a numerical function $f$ consists of a plot of ordered pairs $(x, y)$ where $x$ is in the domain of $f$ and $y = f(x)$. A table of values of a numerical function consists of a list of some of the ordered pairs $(x, y)$ where $y = f(x)$. Fig. shows a graph of $f(x) = sin(x)$ for $–4 ≤ x ≤ 9$.

A function can be defined by a graph or by a table of values, and these types of definitions are common in applied fields. The outcome of an experiment will depend on the input, but the experimenter may not know the "rule" for predicting the outcome. In that case, the experimenter usually represents the experiment function as a table of measured outcome values verses input values or as a graph of the outcomes verses the inputs. The table and graph in Fig. 3 show the deflections obtained when weights are loaded at the end of a wooden stick. The graph in Fig. 4 shows the temperature of a hot cup of tea as a function of the time as it sits in a 68^o F room. In these experiments, the "rule" for the function is that $f(input) =$ actual outcome of the experiment.

Tables have the advantage of presenting the data explicitly, but it is often difficult to detect patterns simply from lists of numbers. Graphs tend to obscure some of the precision of the data, but patterns are much easier to detect visually - we can actually see what is happening with the numbers.