One of the most important things to acquire in this course is the ability to qualitatively graph simple functions of a single variable.

Qualitative Graphing of Functions of a Single Variable . From experience, we know that the following items of information are important:

1. Intervals on which a function is increasing or decreasing [5.3]
2. Intervals on which a function is concave or convex [5.4]
3. points where the function crosses the x or y axis [3.12]
4. horizontal and vertical asymptotes [2.6]
5. points where f, f' or f'' fails to be continuous [3.1]

In this section, we consider item 1, intervals on which a function is increasing or decreasing.

Defininition: A function f is strictly increasing on an interval I if

f(x₁) < f(x₂) when x₁ < x₂

Defininition: A function f is strictly decreasing on an interval I if

f(x₁) > f(x₂) when x₁ < x₂

If the function is differentiable, we can characterize these intervals in the following way:

(a) If f'(x) > 0 for all x in (a,b) , then f is strictly increasing on (a,b).
(b) If f'(x) < 0 for all x in (a,b) , then f is strictly decreasing on (a,b).

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Previously, we saw that f'(c) = 0 was not, in itself, sufficient to show that x=c was a local maxima or minima. If we have some information about f'(x) in a neighborhood of x=c, we can say more

First Derivative Test. if

1. If f' changes sign from positive to negative at c, then f has a local maximum at x=c
2. If f' changes sign from negative to positive at c, then f has a local minimum at x=c
3. If f' does not change sign at c, then f does not have a local extremum at x=c

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In Section 5.5, we will address the other items in qualitatively graphing functions of a single variable .

Last modified Wed Nov 6 23:04:49 CST 1996