Parametric Functions and Rates of Change

For an ordinary function you know how to measure change: the average rate of change on an interval, the slope of the secant line. A parametric function has two coordinates changing at once, so there are three rates worth asking about: how fast $x$ changes, how fast $y$ changes, and how steep the path is.

Two average rates, one for each coordinate

Take a parametric function $(x(t), y(t))$ and an interval $[t_1, t_2]$. Each coordinate is a function of $t$, so each has its own average rate of change:

These are the slopes of secant lines on the graphs of $x(t)$ and $y(t)$ against $t$. If $x$ is in meters and $t$ in seconds, the rate of $x$ is in meters per second. A positive rate of $x$ means the point ended up farther right; a negative rate of $y$ means it ended up lower.

The slope of the path

To measure how steep the path itself is between the two points, compare the change in $y$ with the change in $x$:

This is the average rate of change of $y$ with respect to $x$ on that piece of the path. Dividing the top and bottom by $t_2 - t_1$ shows that it is also the rate of $y$ divided by the rate of $x$.

Reading the sign

A slope of $0$ does not mean the point stood still. On $(t^2 + 1, 4t - t^2)$ from $t = 1$ to $t = 3$, the point goes from $(2, 3)$ to $(10, 3)$: the segment is horizontal, yet the point moved $8$ units right and rose and fell along the way.

How the rates themselves change

Over equal intervals of $t$, compare successive average rates. In the skier's log the rates of $x$ on $[0, 2]$, $[2, 4]$, $[4, 6]$ are $3$, $5$ and $7$ meters per second: they increase, so the rate of change of $x$ is increasing, and the skier covers more horizontal ground in each 2 seconds. The segment slopes $-\frac{2}{3}$, $-\frac{4}{5}$, $-\frac{6}{7}$ decrease, so the path gets steeper downhill from one stretch to the next.

Shorter intervals give a closer picture. The slope of a segment over a small interval of $t$ estimates how steep the path is near those points, and the rates of $x$ and $y$ over a small interval estimate how fast each coordinate is changing there.