The Slope Problem

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Objectives
  • Define and understand the slope problem of Calculus
  • Examine ways we can estimate the instantaneous slope of a nonlinear function
  • Look at how and why we might want to interpret the slope of a function in a real life application
Lesson Description

One of the major calculus tasks is to solve "the slope problem". This lesson seeks to define the issue we are addressing when we study differential calculus, and introduces us to some estimation techniques that will eventually evolve into calculus principles.

 
Practice Problems

Practice problems and worksheet coming soon!

 

The Only Thing Constant is Change

The study of Calculus is generally divided into two operations: differentiating and integrating. Both operations allow for calculations of quantities that are constantly changing, but can be measured precisely at specific points. Our journey begins with differentiating for a few reasons, and differentiating (sometimes referred to as the process of finding or taking derivatives) is the measure of instantaneous rates of change. Thus, we are using derivatives to solve what I refer to as "The Slope Problem".In 2-dimensional analysis, nearly all rates of change can be visualized in a graph as the slope of a function. The only time in algebra in which we were able to rigorously study slope was during the study of linear equations and graphs, because any linear function has a constant slope.For example, the function depicted above seems to have a slope of approximately $1/2$.But what about all the stranger, curvier, functions that we studied in late Algebra and in Pre-Calculus, such as quadratics, higher polynomials, trig functions, or exponentials? All of these functions have their own properties and applications, but the one thing they have in common is that there is no one answer to the question "what is the slope of this function?"At some specific place on the graph, we can draw what we call a "tangent line" - or in other words a line that touches the function graph at exactly one point - but the slope of such a line will not be the same at varying locations.In the parabola above, we can see that the slope at a point may be very different from the slope at another point.

What Can We Do Now?

Without using the Calculus tools we are about to learn throughout the Calculus course, we can already use estimation techniques to get approximate slopes. Additionally, we can also discuss what slopes mean in terms of interpretation, in a word problem setting.EstimationLet's look at an example of a slope that we can "eyeball". Here's a graph of the function $f(x) = x^2 + 4x - 6$.At $x = -1$, we can see that the tangent line seems to exactly cross through the points $(-3,-13)$ and $(3,-1)$. Since we are discussing the slope of a straight line, all we have to do is use our classic slope formula from Algebra One:$$m = \frac{y_2 - y_1}{x_2 - x_1}$$$$m = \frac{(-1)-(-13)}{(3)-(-3)}$$$$m = \frac{12}{6} = 2$$Therefore, with good confidence, we can estimate the tangent slope of $f(x)$ at $x = -1$ to be $2$.We can use the counting method on any plot or graph in which we have good confidence in our artistry skills. However, the downfall of this approach, beside introducing the possibility of human error, is that you necessarily must graph the function to do this. When we dive into the Calculus based approaches to solving "The Slope Problem", we will be able to work purely algebraically.InterpretationThanks to repetitive practice in Algebra, we tend to think of slope conceptually as "rise over run". This doesn't have to change with Calculus, and in fact, can quickly get us an understanding of why slope is so useful in the real world.Let's take a look at the classic Physics application of slope - kinematics analysis. If we were to observe a moving object and graph it's position during some amount of time, we could get some very good information about not only its location at a given time, but also how fast it was moving.Here is a figure depicting location over time of a remote controlled car, measuring distance from the starting point on the $y$ axis, and elapsed time in seconds on the $x$ axis. We can see that at time $0$, the car is at position $0$, which aligns with the setup of this problem. After $60$ seconds, the car returns to its original starting place, as noted by the fact that the $y$ value is $0$. During this car's journey however, we can see a few things: that it traveled a maximum of $50$ meters from its starting position, and that it did not travel at a constant speed. How do we know about the speed? Since speed is measured in meters per second in this scenario, and since the graph is showing us distance in meters, then, at any point, the slope of the graph is the speed at that instant. Slope for this function is rise (meters) over run (seconds), and therefore the slope is measured in meters per second.There are many similar real-world situations that can be modeled this way, such as engineering or economics, and the slope of the function has whatever interpretation as you would expect, given the units you have. In this case, $y/x$ slope, in terms of units, was meters over seconds, and that's a measure of speed. We can see all sorts of useful facts with that interpretation of slope - the car was stopped between time $10$ and $20$ as evidenced by having a zero slope. The car was speeding up between time $20$ and $25$, as evidenced by the increasingly steeper slope over that period. This application and countless similar examples gives a small window as to the reason why the world at large cares about solving "The Slope Problem".

What We Want To Do

Ultimately, the goal of early differential Calculus will be to take any one-variable function and be able to analyze it based on properties of its slope. Generally, we can take a function and find its derivative function to use for various tasks. Unlike the original function which gives output values for a given $x$ input, the function's derivative function will the give instantaneous slope at a given $x$ input.In other words, what we will be able to do is, given a specific function $f(x)$, we will be able to find a new function, which we will often call $f'(x)$, which spits out the slope of $f(x)$ at a given $x$ value, instead of a function value. In this way, we will be able to solve "The Slope Problem" explicitly and algebraically, and without needing a graphical approach as we saw above. We will use $f(x)$ when we need to answer questions about function values, and we will use $f'(x)$ when we need to answer questions about the slope of $f(x)$.$f(x) \longrightarrow$ The original function$f'(x) \longrightarrow$ The derivative function$f(4) \longrightarrow$ The value of $f(x)$ when $x = 4$$f'(4) \longrightarrow$ The slope of $f(x)$ when $x = 4$While much of our job at first will involve learning how to find a function's derivative function, there are several analysis tasks and techniques that teachers and professors expect us to know as well. These analyses utilize the derivative functions, and there are so many connections and concepts that you may or may not learn them all based on the course you are taking and what your instructor chooses to put on the course syllabus. For example, some professors highly value your ability to graph both a function and the function's derivative, and be able to draw conclusions between the two, but others may ignore this topic completely. While I cannot know what will and will not be on your specific syllabus, I can assure you that all bases are covered in the Calculus strand of the DNA of Math », so that you can tackle what you need to, and ignore non-covered topics (unless you're curious!).
You Should Know
On the subject of Differential Calculus, there is a myriad of small concepts and topics - most of them are important and commonplace, but some are not and are consequently often skipped in a course. However, every professor is different, and some care greatly about a small little topic that 10 other profs might not even consider worth the time. As you go through the Mr. Math Calculus curriculum, I will always point out what is common for all students to learn (which, not coincidentally is usually the stuff that paves the foundation for later Calculus topics), and I will point out things that I very rarely see on my students' syllabi.

Put It To The Test

This material will likely not be tested explicitly, though it is possible that your teacher expects you to be able to "eyeball" your own tangent lines and slopes. Therefore, all we will do here is two quick problems that ask us to do just that.
 
Example 1Sketch the following graph, and estimate the tangent slope at $x = 2$ by "eyeballing" and drawing in your own tangent line.$$f(x) = 2^x$$
Show solution
$\blacktriangleright$ Via plotting a few quick points (such as $x = -1, \, 0, \, 1, \, 2$), or by using a graphing calculator, we should be able to come up with something close to the graph below:Now let's add in a decent guess at the tangent line near $x = 2$:It looks like the tangent line passes really close to the points $(0.5,0)$ and $(2.5,5.5)$. Let's use those two points with the classic slope formula to get the estimate we need.$$m = \frac{y_2 - y_1}{x_2 - x_1}$$$$m = \frac{(5.5)-(0)}{(2.5)-(0.5)}$$$$m = \frac{5.5}{2} = 2.75$$So our estimate for the tangent slope at $x=2$ is approximately $2.75$ (just FYI, the real value is approximately $2.77$, so not too bad!).
 
Example 2Sketch the following graph, and estimate the tangent slope at $x = -3$ by "eyeballing" and drawing in your own tangent line.$$f(x) = x^3 + 6x^2 + 9x + 2$$
Show solution
$\blacktriangleright$ Using the rational roots theorem, you may be able to quickly wrestle out the fact that $x = -2$ is a root of this equation. Once you factor out the $(x+2)$ term, you have$$f(x) = x^3 + 6x^2 + 9x + 2$$$$f(x) = (x+2)\left(x^2 + 4x + 1\right)$$Solving for the roots of the remaining quadratic, or just getting the graph from a graphing utility right from the start, should lead you to a similar picture as below.Exactness isn't required but as a reminder, when asked to graph something, we should at least have a few points labeled, and the $x$ and $y$ intercepts need to be realistic. In any case, now that we have the graph of the function, let's look at $x=-3$ and draw in our best estimate tangent line.Incidentally, it seems that a tangent line at $x = -3$ will be perfectly horizontal. Not coincidentally, this is also a point on the graph of relative maximum height. We'll be seeing more about that related fact in the near future. For now, if we are happy with our horizontal tangent estimate (and we should be, it's correct), then all we have to say to finish this problem is that we estimate the tangent slope of $f(x)$ at $x = -3$ to be $0$.
 
Lesson Takeaways
  • Be able to define and understand "The Slope Problem" of Calculus, and digest the fact that first semester Calculus is dedicated entirely to finding solutions to this problem in various capacities.
  • Understand the ever-changing quality of slopes of curvy functions, and appreciate the difficulty in trying to quantify the exact slope at a point
  • Apply estimation techniques to "eyeball" your best estimate of a function's instantaneous tangent slope
  • Be able to interpret slope in the context of a word problem, and begin to see how a function graph gives us extra information via slope qualities, not just function values

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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