When you start looking at graphs of derivatives, you can easily lapse into thinking of them as regular functions — but they’re not. Fortunately, you can learn a lot about functions and their derivatives by looking at their graphs side by side and comparing their important features. For example, take the function, f (x) = 3x5 – 20x3.
You’re now going to travel along f from left to right, pausing to note its points of interest and also observing what’s happening to the graph of
at the same points. But first, check out the following (long) warning.
This is NOT the function! As you look at the graph of
in the figure, or the graph of any other derivative, you may need to slap yourself in the face every minute or so to remind yourself that “This is the derivative I’m looking at, not the function!” It’s easy to mistake graphs of derivatives for regular functions. You might, for instance, look at an interval that’s going up on the graph of a derivative and mistakenly conclude that the original function must also be going up in the same interval — an understandable mistake.
You know the first derivative is the same thing as slope. So when you see the graph of the first derivative going up, you may think, “Oh, the first derivative (the slope) is going up, and when the slope goes up that’s like going up a hill, so the original function must be rising.” This sounds reasonable because, loosely speaking, you can describe the front side of a hill as a slope that’s going up, increasing. But mathematically speaking, the front side of a hill has a positive slope, not necessarily an increasing slope. So, where a function is increasing, the graph of its derivative will be positive, but that derivative graph might be going up or down.
Say you’re going up a hill. As you approach the top of the hill, you’re still going up, but, in general, the slope (the steepness) is going down. It might be 3, then 2, then 1, and then, at the top of the hill, the slope is zero. So the slope is getting smaller or decreasing, even as you’re climbing the hill or increasing. In such an interval, the graph of the function is increasing, but the graph of its derivative is decreasing. Got that?
Okay, let’s get back to the f and its derivative in the figure. Beginning on the left and traveling toward the right, f increases until the local max at (–2, 64). It’s going up, so its slope is positive, but f is getting less and less steep so its slope is decreasing — the slope decreases until it becomes zero at the peak. This corresponds to the graph of
(the slope) which is positive (because it’s above the x-axis) but decreasing as it goes down to the point (2, 0). Let’s summarize your entire trip along f and
with the following list of rules.
An increasing interval on a function corresponds to an interval on the graph of its derivative that’s positive (or zero for a single point if the function has a horizontal inflection point). In other words, a function’s increasing interval corresponds to a part of the derivative graph that’s above the x-axis (or that touches the axis for a single point in the case of a horizontal inflection point). See intervals A and F in the figure.
A local max on the graph of a function (like (–2, 64) corresponds to a zero (an x-intercept) on an interval of the graph of its derivative that crosses the x-axis going down (like at (2, 0)).
On a derivative graph, you‘ve got an m-axis. When you’re looking at various points on the derivative graph, don’t forget that the y-coordinate of a point, like (2, 0), on a graph of a first derivative tells you the slope of the original function, not its height. Think of the y-axis on the first derivative graph as the slope-axis or the m-axis; you could think of general points on the first derivative graph as having coordinates (x, m).
A decreasing interval on a function corresponds to a negative interval on the graph of the derivative (or zero for a single point if the function has a horizontal inflection point). The negative interval on the derivative graph is below the x-axis (or in the case of a horizontal inflection point, the derivative graph touches the x-axis at a single point). See intervals B, C, D, and E in the figure (but consider them as a single section), where f goes down all the way from the local max at (–2, 64) to the local min at (2, –64) and where
is negative between (–2, 0) and (2, 0) except for at the point (0, 0) on
which corresponds to the horizontal inflection point on f.
A local min on the graph of a function corresponds to a zero (an x-intercept) on an interval of the graph of its derivative that crosses the x-axis going up (like at (2, 0)).
Now let’s take a second trip along f to consider its intervals of concavity and its inflection points. First, consider intervals A and B in the figure. The graph of f is concave down — which means the same thing as a decreasing slope — until it gets to the inflection point at about (–1.4, 39.6).
So, the graph of
decreases until it bottoms out at about (–1.4, –60). These coordinates tell you that the inflection point at –1.4 on f has a slope of –60. Note that the inflection point on f at (–1.4, 39.6) is the steepest point on that stretch of the function, but it has the smallest slope because its slope is a larger negative than the slope at any other nearby point.
Between (–1.4, 39.6) and the next inflection point at (0, 0), f is concave up, which means the same thing as an increasing slope. So the graph of
increases from about –1.4 to where it hits a local max at (0, 0). See interval C in the figure. Let’s take a break this trip for some more rules.
A concave down interval on the graph of a function corresponds to a decreasing interval on the graph of its derivative (intervals A, B, and D in the figure). And a concave up interval on the function corresponds to an increasing interval on the derivative (intervals C, E, and F).
An inflection point on a function (except for a vertical inflection point where the derivative is undefined) corresponds to a local extremum on the graph of its derivative. An inflection point of minimum slope (in its neighborhood) corresponds to a local min on the derivative graph; an inflection point of maximum slope (in its neighborhood) corresponds to a local max on the derivative graph.
Resuming your trip, after (0, 0), f is concave down till the inflection point at about (–1.4, 39.6) — this corresponds to the decreasing section of
from (0, 0) to its min at (1.4, –60) (interval D in the figure). Finally, f is concave up the rest of the way, which corresponds to the increasing section of
beginning at (1.4, –60) (intervals E and F in the figure).
Well, that pretty much brings you to the end of the road. Going back and forth between the graphs of a function and its derivative can be very trying at first. If your head starts to spin, take a break and come back to this stuff later.
Now, look again at the graph of the derivative,
in the figure and also at the sign graph for
in the next figure.
That sign graph, because it’s a second derivative sign graph, bears exactly (well, almost exactly) the same relationship to the graph of
as a first derivative sign graph bears to the graph of a regular function. In other words, negative intervals on the sign graph in the figure
show you where the graph of
is decreasing; positive intervals on the sign graph
show you where
is increasing. And points where the signs switch from positive to negative or vice versa
show you where
has local extrema.
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