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# Calculus Graphs Study Guide (page 2)

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Updated on Oct 1, 2011

### Mathematical Notation Note

An apology must be made for mathematical notation here. An expression like (2,8) is ambiguous. Is this a single point with coordinates x = 2 and y = 8? Is this an interval consisting of all the points between 2 and 8? Only the context can make clear which is meant. If we read "at (2,8)," then this is a single point. If we read "on (2,8)," then it refers to an interval.

#### Example 2

Use the graph in Figure 2.6 to identify the domain of h, where it is increasing and decreasing, where it has local maxima and minima, where it is concave up and down, and where it has points of inflection.

#### Solution 2

The first thing to notice is that h has three breaks, or discontinuities. If we wanted to trace the graph of h with a continuous motion of a pencil, then we would have to lift up the pencil at x = –2, x = 2, and at x = 5. The little circle at (5,3) indicates a hole in the graph where a single point has been taken out. This means that x = 5 is not in the domain, just as x = –1 has no point above or below it. The situation at x = 2 is more interesting because x = 2 is in the domain, with the point (the shaded-in circle) at (2,–2) representing h(2) = –2. All of the points immediately before x = 2 have y-values close to y = 3, but then there is an abrupt jump down to x = 2. Such jumps look awkward on a graph, but occur often in real life, like the way the cost of postage leaps up as soon as a letter weighs more than one ounce.

Because of the discontinuities, we have to name each interval separately, as in: h increases on (–∞,–2), (–2,2), (2,5), and on (5,∞). As well, h is concave up on (–∞, –2), (2,5), and on (5,∞), and concave down on (–2,2).

There is a local minimum at (2,–2), because the point there is the lowest in its immediate vicinity, 1 < x < 3. There is no local maximum in that range because the y-values get really close to y = 3; there is no highest point in the range.

Similarly, a point of inflection can be seen at x = 2 but not at x = –2 because there can't be a point of inflection where there is no point!

The situation at x = – 2 is called an asymptote because the graph begins to flatten out like a straight line. The more we would continue to draw the graph off the top and bottom, the straighter this line would become. In this case, x = – 2 is a vertical asymptote because it approximates a vertical line at x = –2. Because the graph appears to flatten out like the straight horizontal line y = 0 (the x-axis) as the graph goes off to the left, this means that the graph of y = h(x) appears to have a horizontal asymptote at y= 0.

### Note

We can obtain all sorts of useful information from a graph, such as its maximal points, where it is increasing and decreasing, and so on. Calculus will enable us to get this information directly from the function. We will then be able to draw graphs intelligently, without having to calculate and plot thousands of points (the method graphing calculators use).

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