Video Lecture

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We’ve covered all kinds of ways to interpret different levels and values of derivatives graphically. Typically we’ve done this by looking at a graph and deducing things about the derivatives to build intuition and ultimately get to a new result or understanding. Now it’s time to go the other direction! We will be using derivatives to create a detailed sketch of a function which we would otherwise have little hope of sketching with only precalculus knowledge. Before we begin though, there are two things to keep in mind.

First and foremost, this is really something best learned and seen via examples. For that reason there will be plenty of example videos available for this process. Secondly, it is easy to get lost in the gritty details of careful computations when actually doing an example and miss the bigger picture. That bigger picture is what we attempt to discuss in this segment.

By way of warning then, be aware that we will be deliberately omitting a lot of the careful calculations involved in this segment because we are specifically after the “big picture” right now. In essence the example videos will show how to do the calculations, and which calculations need to be done, but this segment is more about interpreting what the calculations are telling us and how to translate that into a graph. For this reason don’t feel alarmed if you aren’t sure how a calculation was done or where a specific equation came from, that will be covered in detail in the example videos. You want to focus on how we translate the result of the calculation into a graph.

So, our goal is to turn a function into a reasonably detailed sketch of a graph. To this end, let’s start with a function. Consider the following function: $f(x) = x^3 - x^2 - x + 1$. Using our derivative rules and factoring techniques we can get the following:

• $f(x) = x^3 - x^2 - x + 1 = (x+1)(x-1)^2$
• $f'(x) = 3x^2 -2x -1 = (3x+1)(x-1)$
• $f''(x) = 6x - 2$

Now we want to list the zeros for each. $f(x)$ has zeros of $1$ and $-1$, $f'(x)$ has zeros of $-\frac {1}{3}$ and $1$, and $f''(x)$ has $\frac {1}{3}$ as its only zero. So all together we have as zeros $-1$, $-\frac {1}{3}$, $\frac {1}{3}$, and $1$. Next we want to build sign charts for each of these. It is ideal to build a sign chart that has all the zeros for each function as we will be breaking these up into regions, but if you want to do them separately that works just as well - just make sure to keep straight where the overlaps are. If we opt to include all the zeros and do a single chart that has all three functions, we get the following:

From this we can determine if we have maximums, minimums, and/or inflection points, and what the $x$-values are for each of them. In particular, we can see that $f'(x)$ goes from positive to negative at $-\frac {1}{3}$, so that is the $x$ value of a local maximum. Similarly, $x=1$ is a local minimum since $f'(x)$ transitions from negative to positive. We can also determine that $\frac {1}{3}$ is an inflection point since the second derivative changes sign at that value. Finally, we can plug in all these $x$ values, along with $x=0$ (which gives us the y-intercept) to get the following list of points and why they are important:

• $(-1,0)$, $(1,0)$ are $x$-intercepts (where $f(x)$ is zero)
• $(0, 1)$ is the $y$-intercept (compute $f(0)$)
• $\left (-\frac {1}{3}, \frac {32}{27}\right )$ is a local maximum. (where $f'(x)$ goes from positive to negative sign)
• $\left (1,0\right )$ is a local minimum (where $f'(x)$ goes from negative to positive sign)
• $\left (\frac {1}{3}, \frac {16}{27}\right )$ is an inflection point (where $f''(x)$ changes sign)

Ok, now that we have all the calculation parts done, it’s time to translate this all into a graph!

First, we start with a blank canvas:

Next up, we want to add in the points we already know for certain. These are all those points we listed a moment ago:

As a suggestion, it helps to (lightly) sketch a little curve through points to represent what they are. For example, for the maximums and minimums we can draw little parabolic curves through the points. They don’t need to be perfect, or even good, they are just to show you how the curve is suppose to pass through the point in general. Let’s see what I mean with just the maximums and minimums first:

Now, these curves aren’t exactly accurate - just looking at how far down the curve on the maximum is going there is no way it will end up hitting the y-intercept correctly, but that’s not the point. The point is to show me how to go through the point correctly once I have everything set up and ready to go. Let’s add in a curve to represent the inflection point (going from concave down to concave up) and the correct way to pass through each of the x-intercepts.

Finally we want to record the end-term behavior by calculating the limit as $x$ approaches positive and negative infinity - which turn out to be negative infinity to the left and positive infinity to the right. Once we have this, we want to try and smoothly connect the dots, going through each one in the way we’ve sketched as close to naturally as we can. We should end up with something like this:

We can see that the sketch segments didn’t quite line up perfectly as we tried to draw the full graph smoothly through all the points, but that’s ok! We only put those sketched segments there in order to know how the graph was suppose to pass through the point; whether it was suppose to arch away from the point to make a local extrema, or pass straight through, or go through but change concavity, etc. Keep this in mind when doing your own sketches, and remember that this isn’t an art class - we should be precise and accurate where we can (like when we found the specific points for the maximum, minimum, inflection points, zeros, etc) but this is still a sketch... just a much more detailed sketch than you would expect from a student in, say, precalculus.

So, we saw how we can use the ideas from derivatives like concavity and finding local extrema, along with some concrete values like intercepts, to make a much more detailed sketch of a graph than we could before we had the tools from calculus at our disposal. Graphing might seem like a somewhat abstract and theoretical application, but consider how many times we’ve pulled useful information out of a graph so far in this class alone. Being able to have a reasonably accurate graph, especially one that includes things like local and absolute extrema and inflection points, can be a powerful tool when presenting data or determining trends, or even trying to decide on a course of action for a company. There is a reason visual aids are so popular in presentations - they help!