Definition 1.2.1. Accumulation of a variable quantity.
For variable quantity \(q\text{,}\) the accumulation between data points \(q_1\) and \(q_2\) is
\begin{equation*}
\change{q} = q_2 - q_1 \text{.}
\end{equation*}
Chapter 1 Variables and variation
Section 1.1 Variables, constants, and parameters
The use of symbols and alphabetic letters to represent numbers both known and unknown dates back millennia and was practised in numerous ancient cultures around the world.
We often use letters or symbols to represent quantities that maintain a constant value, which we (unsurprisingly) refer to as constants. Two familiar examples are the use of the Greek letter \(\pi\) to represent the common ratio of circumference to diameter that every circle exhibits and the use of the ordinary letter \(c\) to represent the speed of light in a vacuum.
While \(\pi\) and \(c\) are “known” constants (to the extent that the value of \(\pi\) can be “known”), an unknown value is sometimes referred to as a constant as well, and is assigned a letter like \(C\) or \(K\) to represent it, in the case the unknown value is known or assumed to be literally constant (unchanging) in its context. For example, a pair of parallel lines remain the same, fixed, perpendicular distance \(d\) apart along their lengths. While we have chosen to use a “variable” \(d\) to represent this value, it is unchanging in its context of two particular lines: no matter what perpendicular segment between the two lines is considered, the length \(d\) of that segment is always the same. However, it is an unknown constant as it is dependent on the particular pair of lines considered — a different pair of parallel lines might have a different constant distance \(d\) between them.
A quantity whose value can be controlled is often referred to as a parameter. It often represents a choice that can be made in creating or defining some mathematical object, and can be used to parametrize (or index) the entire collection of objects that can be created. For example, the collection of all lines in the Cartesian \(xy\)-plane that pass through the origin can be parametrized by their slopes — for each slope value \(m\text{,}\) we can create a line through the origin with equation
\begin{equation*}
y = m x \text{.}
\end{equation*}
For a particular such line the slope value \(m\) is constant, but a different slope creates a different line.
The letters \(x\) and \(y\) in the line-equation example above play a different role than the letter \(m\text{.}\) A line is a collection of points, and we think of these points as being at varying positions along the line, where \(x\) and \(y\) represent the coordinates of any one of these points. Any quantity that can vary will be represented by a variable.
Of course, how we categorize a quantity is based on point of view — a single line has a constant slope, it’s when we want to consider a family of lines with the same form of equation that slope becomes a parameter. How we refer to any letter that represents a quantity — as constant, parameter, or variable — indicates to the reader or listener a particular role that we wish that letter to play.
Section 1.2 Accumulation
When a quantity is variable, we often want to compare its values at different data points to identify trends. When the quantity grows larger, we think of it as accumulating, and when it becomes smaller, we think of it as diminishing or dissipating. However, we will use the same word accumulation for both situations, using positive and negative to distinguish between the two.
Notice the following.
\begin{align*}
q_2 \gt q_1 \amp \implies \change{q} \gt 0 \\
q_2 \lt q_1 \amp \implies \change{q} \lt 0
\end{align*}
So a positive accumulation indicates that the quantity has increased from the first to second data points, and a negative accumulation indicates that the quantity has decreased.
Example 1.2.2. Accumulation of rain.
You leave a one litre container out in the rain, and when the rain stops it is a quarter-full. A few days of sunny weather later and half of that has evaporated. So over the rainy period the accumulation in millilitres was approximately
\begin{align*}
\change{q} \amp = q_2 - q_1 \\
\amp = 250 - 0 \\
\amp = 250 \text{,}
\end{align*}
while over the sunny period the accumulation was approximately
\begin{align*}
\change{q} \amp = q_3 - q_2 \\
\amp = 125 - 250 \\
\amp = - 125 \text{.}
\end{align*}
Section 1.3 Rate of variation
It is often the case that two quantities are varying in tandem, and we wish to compare the amounts by which they vary.
Example 1.3.1. Volume varies by depth.
A graduated cylinder is a vessel for measuring a volume of liquid using a ruled scale printed vertically on its side. It works because at a constant diameter, the volume can be directly related to depth by
\begin{equation*}
V = \frac{\pi d^2 h}{4} \text{,}
\end{equation*}
where \(d\) is the diameter and \(h\) is the depth (or height) in the cylinder.
When designing a graduated cylinder for manufacture, what diameter should be chosen so that the ruled markings can be printed as millimetres, with each centimetre of depth corresponding to \(10\) cubic centimetres of volume? A variation in height corresponds to a variation in volume, so that two different depths \(h_1\) and \(h_2\) with corresponding volumes \(V_1\) and \(V_2\text{,}\) we have
\begin{align*}
\change{V} \amp = V_2 - V_1 \\
\amp = \frac{\pi d^2 h_2}{4} - \frac{\pi d^2 h_1}{4} \\
\amp = \frac{\pi d^2}{4} (h_2 - h_1) \\
\amp = \frac{\pi d^2}{4} \change{h} \text{.}
\end{align*}
If we turn this into a ratio \(\inlineslope{V}{h}\text{,}\) we can directly see how a variation in depth leads to a variation in volume.
\begin{equation*}
\slope{V}{h} = \frac{\pi d^2}{4} \text{.}
\end{equation*}
We can think of this ratio as describing variation in volume per unit variation in depth. This is because if \(\change{h} = 1\text{,}\) then
\begin{equation*}
\slope{V}{h} = \frac{\change{V}}{1} = \change{V} \text{.}
\end{equation*}
So whenever the depth changes by \(1\) centimetre, the volume will change by
\begin{equation*}
\frac{\pi d^2}{4}
\end{equation*}
cubic centimetres. But if the depth changes by \(20\) centimetres, then the volume will change by
\begin{equation*}
20 \cdot \frac{\pi d^2}{4} = 5 \pi d^2
\end{equation*}
cubic centimetres.
A tall cylinder of diameter \(d\) is pictured, partially filled with a blue liquid. An additional volume of red liquid sits atop the blue liquid, and is labelled \(\change{V}\text{.}\) The additional height of the red liquid (from the top of the blue liquid to the top of the red liquid) is labelled as \(\change{h}\text{.}\)
We can now answer the original question: the requirement that each centimetre of change in depth correspond to \(10\) cubic centimetres change in volume can be stated as
\begin{equation*}
\slope{V}{h} = 10
\end{equation*}
if was assume that both the diameter \(d\) and the depth \(h\) are measured in centimetres. Solving, we obtain
\begin{align*}
\frac{\pi d^2}{4} \amp = 10 \\
d^2 \amp = \frac{40}{\pi} \\
d \amp = 2 \sqrt{\frac{10}{\pi}} \text{,}
\end{align*}
where we discard the negative solution since we want diameter to be a positive quantity. So the diameter should be approximately 3.57 cm.
Definition 1.3.3. Average rate of variation.
If \(q\) and \(t\) are two quantities that are varying in tandem, so that a variation in one of the quantities is always accompanied by a corresponding variation in the other, then a ratio
\begin{equation*}
\slope{q}{t} = \frac{q_2 - q_1}{t_2 - t_1}
\end{equation*}
is called an (average) rate of variation between data points \((t_1,q_1)\) and \((t_2,q_2)\text{.}\)
As suggested by the choice of placing \(t_1\) and \(t_2\) as the first coordinate in the data points, the quantity whose variation is used as the denominator in a rate ratio is usually considered to the be independent variable, while the other quantity is thought of as the dependent variable. That is, using the variables \(q\) and \(t\text{,}\) we think of variation in \(t\) as causing or “driving” the variation in \(q\text{.}\) We have chosen the variable \(t\) to represent the “general” independent variable instead of the more traditional variable \(x\) because we will usually consider time as the ultimate “independent variable,” and we want to invoke the intuition of a quantity varying over some period of time when we make general statements. However, in many situations the independent variable will represent something other than time. For example, in economics you might model variations in cost of production as the number of units being produced varies.
The following example from physics with time as the independent variable should be familiar to you.
Example 1.3.4. Average speed.
Suppose you drive 100 km in about an hour. If we let \(d\) represent distance from the starting point of the trip and \(t\) represent time, then
\begin{equation*}
\slope{d}{t} = \frac{100}{1} = 100
\end{equation*}
represents your average speed in kilometres per hour.
We will study average rate of variation further in later chapters.
Section 1.4 Constant variation
Normally we think of rate of variation as itself being a quantity that varies. For example, in Example 1.3.4, you likely didn’t leave your parking spot at 100 km⁄h and you likely didn’t crash into your destination at 100 km⁄h. Along the way, you would have sped up and slowed down at various points of the trip, perhaps even stopped completely at traffic lights. If we calculated an average rate of variation over the first five minutes of the trip, while you were travelling on city streets to get out onto the highway, the result would likely have been much lower than 100 km⁄h.
So a pair of quantities that vary in tandem so that the average rate of variation between them is always the same is a special situation.
Example 1.4.1. Varying quantities at a constant rate.
Suppose \(q\) and \(t\) are quantities so that \(q\) varies as \(t\) varies, and it is observed that
\begin{equation*}
\slope{q}{t} = \frac{1}{2}
\end{equation*}
seems to always be true between every pair of data points. Then we can predict the quantity \(q\) at any level of quantity \(t\) using only a single known data point. For example, suppose we know an “initial” data point where \(q = 1\) when \(t = 0\text{.}\) Then using \(q_1 = 1\) and \(t_1 = 0\) in any average rate calculation will tell us the second data point. For example, if we use \(t = 4\) for the second data point, we can calculate the corresponding \(q\)-value as follows.
\begin{align*}
\slope{q}{t} \amp = \frac{1}{2} \\
\frac{q_2 - 1}{t_2 - 0} \amp = \frac{1}{2} \\
\frac{q_2 - 1}{4 - 0} \amp = \frac{1}{2} \\
q_2 - 1 \amp = \frac{4}{2} \\
q_2 \amp = 3
\end{align*}
We can use a similar calculation to determine the \(q\)-value corresponding to any value of \(t\text{.}\) In Figure 1.4.2, we have plotted some example data points obtained in this way on a set of \(tq\)-axes. From these data points, we observe a clear linear relationship between \(q\) and \(t\text{.}\)
Data points are plotted on a set of \(tq\)-axes, at coordinates \((-1,0.5), (0,1), (1,1.5), (2,2), (3,2.5), (4,3) \text{.}\) These points appear in a linear pattern, and a line is drawn through them.
Pattern 1.4.3. Constant rate means linear relationship.
If \(q\) and \(t\) are two quantities that are varying in tandem so that
\begin{equation*}
\slope{q}{t}
\end{equation*}
is constant, then \(q\) and \(t\) are linearly related by
\begin{equation*}
q = r t + q_0 \text{,}
\end{equation*}
where \(r\) is the constant value of \(\inlineslope{q}{t}\) and \(q_0\) is the “initial value” of quantity \(q\) corresponding to \(t = 0\text{.}\)
Graphically, when two quantities exhibit a constant rate of variation, we can plot the relationship between \(q\) and \(t\) on a set of \(tq\)-axes and obtain a line with equation as in Pattern 1.4.3, where the slope of the line is
\begin{equation*}
\slope{q}{t}
\end{equation*}
and the “\(q\)-intercept” is at the data point \((0,q_0)\text{.}\)
Example 1.4.4. Determining linear relation for a quantity with constant rate of variation.
Suppose \(q\) and \(t\) are quantities so that \(q\) exhibits a constant rate of variation relative to \(t\text{,}\) with
\begin{equation*}
\slope{q}{t} = -2
\end{equation*}
for all pairs of data points. Then using Pattern 1.4.3, we can say that
\begin{equation*}
q = -2 t + q_0 \text{.}
\end{equation*}
However, suppose we don’t know the data point at \(t = 0\text{,}\) but instead know that \(q = 1\) when \(t = 3\text{.}\) Then
\begin{gather*}
q = -2 t + q_0 \\
1 = -2 \cdot 3 + q_0 \\
1 = q_0 - 6 \\
q_0 = 7 \text{.}
\end{gather*}
With this information, we can plot the linear relationship between \(q\) and \(t\) on a set of \(tq\)-axes.
The line with equation \(q = -2 t + 7\) is drawn on a set of \(tq\)-axes. The line slopes downwards, crossing the vertical axis at \(q = 7\) and then crossing the horizontal axis at \(t = 3.5\text{.}\)
In Figure 1.4.5, we see that the plotted line crosses the \(q\)-axis at \(q_0 = 7\text{.}\) Interpreting
\begin{equation*}
\slope{q}{t} = -2
\end{equation*}
as the variation in \(q\) per unit variation in \(t\), we also see that the plotted line descends two units for each unit increase in \(t\text{.}\)
Compare the graphs in Figure 1.4.2 and Figure 1.4.5. In the first graph, the line goes up as it goes to the right, as we expect from its positive slope. In the second graph, the line goes down as it goes to the right, as we expect from its negative slope.
Pattern 1.4.6. Increasing/decreasing at a constant rate.
Suppose quantity \(q\) varies at a constant rate relative to quantity \(t\text{.}\) If that constant rate is positive, then values of \(q\) increase as \(t\) increases. If that constant rate is negative, then values of \(q\) decrease as \(t\) increases.
