The term extreme value is used to denote points where a function is as high or low as it gets.
For the level of blood sugar not to reach a dangerous amount among diabetics, continuous glucose monitors measure the current amount in the blood, and signals when the level gets too high.
The signal is sent to an insulin pump, that then injects a dose of this crucial hormone. It helps the body transfer glucose from the blood to the cells where it is used for fuel, and hence lowers the level in the blood.
If we think of the amount of blood sugar recorded by a continuous glucose monitor as a function of time, it determines where and what the function's extreme values will be. Essentially, it is sketching the graph.
Functions are essentially rules. They take some input and spit out some output. But this process does feel kinda abstract. I mean, what's actually going on?
Humans like visualising functions with graphs. It makes the function more tangible.
But we don't want to rely on graphing software to draw graphs. What if there's a power outage? How should you satisfy your basic need of looking at function graphs? You better be prepared.
There are, however, a few hacks for sketching graphs. Let's say we've computed the critical points of a graph, and we know whether the function increases or decreases before and after.
Now we can simply connect the dots. Literally.
If we have a function defined on the entirety of , then we can find minima, saddle points and maxima by a derivative test. Below we see a minimum point, saddle point and maximum in that order from the left.
The minimum and maximum are extremums. They are the highest or lowest point of a function in some region. We can also see that the tangent is flat at these points, which shows that the derivative must be zero there. That is, where is a minimum, saddle point or maximum.
Say we want to find the minimum of a function . The derivative is . Next we see that if , then . The point is a minimum point, which we can see if the graph below:
What is an extremum?
When it comes to finding the optimal value for something, the extremum is key. In everything from business to environmental analysis to study habits, we are interested in this.
These are all extrema:
In the graph above, we have:
a local maximum,
a local minimum,
a global maximum,
The graph has no global minimum: it keeps dropping as we walk along the horizontal axis towards .
There are three classes of extrema:
critical points, like and ,
end points, where the graph ends, and
singular points, where the derivative doesn't exist.
At a critical point the slope is zero. If we call the function , they are found by taking the derivative and setting it to zero:
Local extrema are found by setting the derivative to zero
To determine if this is a max or a min point, we can look at the function values at some points around , or the derivative itself in the vicinity. We can also, as we will see in the section on sketching graphs, use the second derivative.
Sometimes, the point we found is neither max nor min: if the function is increasing or decreasing all around the point. We call this local flattening a saddle point.
If the function is defined on an open interval, it happens that the local or global max or min doesn't exist. This graph shows an example where there is no local min:
Time to go extremum, for real, with an example:
The function is defined for all , so we will have no open end points to deal with.
Taking the derivative, we get:
The derivative is defined for all , so has no singular points.
The critical points are found by setting the derivative to zero:
So that's our critical point. Using it into , the function value is determined:
What is an asymptote?
For some functions, we can draw a straight line which follows the graph perfectly as or goes to . This line is called an asymptote.
The function above is . It has two asymptotes: the line and the axis. Look at the function: the first term causes the to go to infinity as approaches zero. Conversely, as gets big, the second term dominates, causing the function to behave like for big . We will formalize these intuitive ideas in a moment.
There are asymptotes of three kinds: vertical, horizontal and oblique. Basically, the line can be anywhere in the plane, as long as its straight.
They are a great help for sketching the graph, as we will see in the following section.
The three kinds of asymptotes require different approaches. We'll go through them one at a time.
The graph of has a vertical asymptote if:
Usually, this happens if is a rational expression whose denominator is zero at .
The definition of a horizontal asymptote is similar:
The graph of has a horizontal asymptote if:
The oblique asymptote is the black sheep of the trio:
The line , , is an oblique asymptote to if:
The expressions in the limits above just says: as we walk far down the axis, there is no difference between the function and the straight line .
- let's go asymptote hunting.
We go about our solution systematically, starting with vertical asymptotes.
Notice that if we put , the denominator is zero.
So there's a vertical asymptote at .
Next, what about horizontal asymptotes? Let's try the limit:
The first limit goes to infinity, the second disappear. So the condition for a horizontal asymptote is not met.
Lastly, what if we could find some oblique asymptote? Look back at the calculation from just a moment ago. The last step reveals some good stuff. Because we just showed that:
Aha! Moving the to the left side, we get:
So, is an asymptote!
So we have two asymptotes: one vertical, at , and one oblique, . The graph of the function can be seen below.
Concavity and convexity
For a function , tells us what the concavity of the function is at every point. That is: does it tend bend upward, or downward.
For an upward bend, forming a u-shape, the first derivative increase and so the second derivative is positive. If this is the case at some point, we say the function is convex there.
Similarly, where the function's second derivative is negative, making the first derivative decrease, it is said to be concave.
Let be twice differentiable at . Then is:
- Convex at if
- Concave at if
The points where a function changes from convex to concave, or the other way around, are known as inflection points. Knowing the concavity of a function, and finding its inflection points can help us sketch the graph.
We have previously gone over how to determine some characteristics of functions using critical points and asymptotes. Now, we will apply those concepts to draw the graphs of functions.
To do so, we start by making a sign table, containing the crucial features of a function we need in order to draw it. Initially, this will consist of an -axis followed by two rows for , and . The -axis will be marked with the values of we are interested in for the respective row, creating a column for every -value we are interested in and every interval between them.
As for finding those 's, we search for two things:
The values where the function has critical points
The values where the function has vertical asymptotes
For each such we find, we add a column to the sign table, with additional columns between them.
For , what we are interested in is the slope to the left and right of each critical point, as well as where is defined. Recall that the slope is zero at a critical point, and so the derivative's value around the point will tell us whether it is a local max, local min, or a saddle point.
When it comes to , we are interested in the concavity of the function to aid our sketching.
We insert this information about the function into our sign table as we yield it, and after we are done we should have everything we need to sketch it. To aid us in that process, we add one last row to the sign table where we state how tend to look at each point of interest, and in the intervals between them, based on the information we have collected.
The list of things we look before we get into the sketching part is long, and best remembered when seen in the form of an example.
Sketching a graph
Let us go through the steps of graphing the function:
Step 1: Find vertical asymptotes
With no denominator that can tend to as approach some value , the function has no vertical asymptotes that we need to consider in the sign table.
Step 2: Calculate and
Employing the power rule once, we find that:
Now, another application of the same rule, then we get:
Step 3: Examine
First, we find the critical points by setting the derivative to zero:
With little effort we see that the two critical points are and .
Now let's look at the derivative's value around those points. Our example is differentiable over the full range, so we can pick any points we like between and outside the critical points. For simplicity, we choose , , and :
Step 4: Examine
The first things we look for are points of inflection:
The expression is only zero where the polynomial has roots. Without showing the calculation here, those roots are and .
Next we are interested in the concavity on the intervals between and outside those points. Again, for simplicity, we choose some points , , and to evaluate at:
is convex for
is concave for
is convex for
Step 5: Examine
The first thing we would like to find here are the function's values at the critical points:
Next, we are after the intercepts for the - and -axis of the function respectively. By plugging in into the function, we find the intercept for the -axis to be . Then, by just looking at the function, we determine that its only root is , so there's no other intercept for the -axis.
Lastly, we would like to find potential non-vertical asymptotes:
horizontal asymptote at
no other asymptote
From sign table to sketch
The procedure has been long, but so worth it. After completing the sign table, it looks like this:
To draw an accurate graph of the function is now easy: