MA005: Calculus I

The real numbers (consisting of all integers, fractions, rational and irrational numbers) can be represented as a line, called the real number line (Fig. 1). Once we have selected a starting location, called the origin, a positive direction (usually up or to the right), and unit of length, then every number can be located as a point on the number line. If we move from a point $x = a$ to point $x = b$ on the line (Fig. 2), then we will have moved an increment of $b – a$. This increment is denoted by the symbol $∆x$ ( read "delta x" ).

The Greek capital letter delta, ∆, will appear often in the future and will represent the "change" in something. If b is larger than a, then we will have moved in the positive direction, and $∆x = b – a$ will be positive. If b is smaller than a, then $∆x = b – a$ will be negative and we will have moved in the negative direction. Finally, if $∆x = b – a$ is zero, then $a=b$ and we did not move at all.

We can also use the ∆ notation and absolute values to write the distance that we have moved. On the number line, the distance from $x = a$ to $x = b$ is

dist(a,b) =  $\left \{ \begin {array} {lI} b-a \text { if } b \geq a \\ a-b \text { if } b < a \end {array} \right.$

or simply, dist(a,b) = $| b – a | = | ∆x | = \sqrt{(∆x)^2}$ .

The midpoint of the segment from $x = a$ to $x = b$ is the point $M = \dfrac{a + b}{2}$ on the number line.


Example 1: Find the length and midpoint of the interval from $x = –3$ to $x = 6$.

Solution: $Dist \; (–3,6) = | 6 – (–3) | = | 9 | = 9$. The midpoint is at $\dfrac{(–3) + (6)}{2} = 3/2$.

Practice 1: Find the length and midpoint of the interval from $x = –7$ to $x = –2$.

(Note: Solutions to Practice Problems are given at the end of each section, after the Problems).