Functions, Measurements, and Change

A position sensor records where a cart is at successive times. It can tell us how far the cart moved between two readings. A speedometer appears to answer a more demanding question: how fast the cart is moving at one instant. Dividing distance by elapsed time seems to require two instants. Calculus will make that tension precise rather than disguise it with a sufficiently small number.

For a deliberately simple model, let the cart’s position be $s(t)=t^2$ metres, where $t$ is measured in seconds and $t\geq0$. The coefficient of $t^2$ therefore has units metres per second squared. The formula is a model, not a claim about every cart. Between $t=2$ and $t=5$, the displacement is $25-4=21$ metres and the elapsed time is $3$ seconds. The average velocity is $7$ metres per second. It does not follow that the cart moved at that velocity throughout the interval.

Functions include their domains

A real-valued function assigns exactly one real output to each allowed input. Its domain is part of its specification. The formulas $x^2$ on $\mathbb R$ and $t^2$ on $[0,\infty)$ use the same algebra but represent different functions when domains matter.

Three restrictions recur throughout this book. A denominator cannot be zero. The argument of a real square root must be nonnegative. The argument of a real logarithm must be positive. For example,

$$f(x)=\frac{\sqrt{x+2}}{x-1} \quad\hbox{has domain}\quad [-2,1)\cup(1,\infty).$$

Writing $\sqrt{x+2}$ requires $x\geq-2$; division requires $x\ne1$. Both restrictions apply. A graphing program that silently omits the forbidden point has not removed the restriction.

Composition also carries restrictions. If $g(x)=\log x$ and $f(u)=1/(u-2)$, then $(f\circ g)(x)=1/(\log x-2)$ requires $x>0$ and $x\ne e^2$. We must check the input to each operation, in the order in which it is performed.

The difference quotient

For distinct inputs $a$ and $a+h$, the average rate of change is

$$\frac{f(a+h)-f(a)}h,\qquad h\ne0.$$

The numerator is the output change and the denominator the input change. On a graph, this is the slope of the line through $(a,f(a))$ and $(a+h,f(a+h))$, called a secant line. If the output is position and the input is time, its units are distance per time. If output is cost and input is production, the units are currency per item. A slope without its variable definitions can be mathematically correct and practically useless.

For $s(t)=t^2$, expand before simplifying:

$$\frac{s(a+h)-s(a)}h =\frac{a^2+2ah+h^2-a^2}h=2a+h\qquad(h\ne0).$$

At $a=2$, intervals of length $1$, $0.1$, and $0.01$ give average velocities $5$, $4.1$, and $4.01$. Backward intervals use negative $h$: $h=-0.1$ compares times $1.9$ and $2$, giving $3.9$. Both directions suggest $4$. The original quotient remains undefined at $h=0$, however. The simplified expression describes its behavior where the quotient is defined; it does not authorize division by zero.

Direction matters

Average velocity is displacement divided by elapsed time. Average speed is total distance travelled divided by elapsed time. A person who walks three metres east and three metres west in four seconds has zero average velocity and average speed $6/4=1.5$ metres per second. The endpoint positions alone determine displacement; they do not determine total distance.

For a graph that is a straight line $f(x)=mx+b$, every secant slope is $m$. For a curved graph the interval matters. Calculus asks whether there is a well-defined local rate as the interval contracts. It does not begin by assuming that every curve has one.

Algebraic preparation

The identity $(u+v)^2=u^2+2uv+v^2$ explains the cancellation above. Factoring and conjugates provide two more tools. Since $x^2-a^2=(x-a)(x+a)$, a quotient can simplify away from $x=a$. Since $(\sqrt{x}-\sqrt a)(\sqrt{x}+\sqrt a)=x-a$, multiplying by a conjugate can expose a hidden factor. In either case, preserve the original domain while manipulating the expression.

A useful check is to calculate the quotient at one ordinary permitted input before and after simplifying. Agreement at one input is not proof of an identity, but disagreement immediately exposes an error.

Exercises

1. Determine the real domain of $\sqrt{9-x^2}/(x-2)$.
2. For $p(t)=3t^2-2t+1$ metres, calculate average velocity from $t=1$ to $t=3$ seconds.
3. Simplify $[p(a+h)-p(a)]/h$ and state its restriction.
4. A tank contains $V(t)=100-4t$ litres for $0\leq t\leq25$ minutes. Interpret its average rate over any nonzero interval in this domain.
5. Explain why cancelling $x-1$ in $(x^2-1)/(x-1)$ does not define the original function at $1$.
6. A car returns to its starting position after an hour. What can be concluded about its average velocity and average speed?

Worked solutions

1. The square root requires $-3\leq x\leq3$ and the denominator excludes $2$, giving $[-3,2)\cup(2,3]$.
2. $p(1)=2$ and $p(3)=22$, so the average is $(22-2)/(3-1)=10$ metres per second.
3. Expansion gives $6a+3h-2$, valid for $h\ne0$ and permitted times. The terms $3a^2-2a+1$ cancel, leaving $h(6a+3h-2)$.
4. The rate is $-4$ litres per minute. The negative sign indicates loss of volume. The model stops when the tank empties; extrapolating to negative volume is outside its domain.
5. The two expressions agree for $x\ne1$. The expression $x+1$ has a value at $1$ that the original quotient lacks. Extending a function is a separate definition.
6. Average velocity is zero. Average speed is nonnegative and positive if any nonzero distance was travelled. Its value requires the total distance.