Sometimes the purpose of taking a course becomes obfuscated over its duration; it can become hard to appreciate the plot of a story when there’s too much exposition. This narrative overview, in its brevity, serves to summarize the plot of this course and to clarify our purpose, for the sake of keeping both student and instructor focused on what’s important.
Limits & Continuity
In the beginning of the story of calculus there are some minor characters we must meet. These characters’ primary importance is to introduce us to the protagonists of calculus, differentiation and the integration, but we should know them in their own right first. The first of these minor characters is the limit.
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Concept
Understand what the notation \( \lim_{x \to x_0} f(x) \) means, what the notation \( \lim_{x \to x_0^+} f(x) \) and \( \lim_{x \to x_0^-} f(x) \) means, and what it means if \(x_0\) is replaced by \(\pm\infty\) in any of those expressions. -
Skill
Be able to compute \( \lim_{x \to x_0} f(x) \), either numerically or precisely, given either a formulaic or graphical presentation of a function.
The important idea behind introducing the limit is that we need language and notation for talking what what’s happening around a point \(x_0\), instead of directly at the point. While discussing the purpose of the course, we’ll gradually reveal the limit’s relationship to the derivative and how it can be used to calculate the slope of a line tangent to a curve at a given point.
The other minor character is the concept of continuity of a function. Explicitly we’ll say a function \(f\) is continuous at a point \(x_0\) if \(f(x_0) = \lim_{x \to x_0} f(x) \), and continuous on a domain if it’s continuous at all points in that domain. It’s more important though to understand what continuity means conceptually.
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Concept
Understand what it means for a function to be continuous at a point and continuous on some domain, and why many natural phenomenon morally should be modelled a continuous function.
We can only apply the techniques of calculus and define the derivative and definite integral of a function on some domain as a formula, if we take as an hypothesis that the function is continuous on that domain; continuity is a crucially necessary moving forward.
Differentiation
Since learning the language of limits and the idea of continuity of functions, we now have all we need to define the derivative of a function and will immediately discuss two important things:
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Concept
Know and understand the definition of the derivative of a real-valued function, defined as \[ \frac{\mathrm{d}}{\mathrm{d}x}f(x) = \lim\limits_{h \to 0} \frac{f(x+h) - f(x)}{h}\,, \] and how you can think of this derivative as a rate of change of the function’s output relative to the function’s input, or how you can think of it geometrically as a slope of a tangent line to the graph \(y = f(x)\) of the function. -
Skill
Be able to write a formula for the derivative of any function defined in terms of sums, products, quotients, or composites of algebraic and trigonometric functions.
The former of these is a concept you must internalize as you work through exercises, reading examples, and generally reading about calculus. The latter is a skill you can practice until you feel proficient, and luckily such exercises are plentiful online. Sometimes those exercises are accompanied by solutions, but if they’re not a symbolic computer algebra system (CAS) can calculate the derivative of a function defined by a formula. A word of caution though: the expectation is that you can compute derivatives. Let a computer be your aid, not your crutch — like the relationship you have with your graphing calculator.
After you’re comfortable taking derivatives of functions \(f\) — where the output \(y = f(x)\) may be thought of as a variable explicitly dependent on the independent input variable \(x,\) the variable you’re differentiating with respect to — you’ll need to extend this skill, and apply it to equations where the relationship between \(x,\)\(y\) and other variables is only implicit.
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Skill
Be able to implicitly differentiate any equation involving sums, products, quotients, or composites of algebraic and trigonometric functions with respect to any variable in that equation.
Just like with explicit differentiation, there are plenty of exercises and worked-out examples online, and you can compare your computations with the output of a CAS.
In class we’ll spend time thinking about the graphs of functions, and what more we can say about those graphs given our new knowledge of derivatives. This isn’t an important “skill” we’re learning though. The purpose of doing this is to buff your understanding of the geometric concept of a derivative, and give you a visual perspective on the relationship between a function and its derivative.
There are three important uses of differentiation we’ll talk about.
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Application
Be able to apply your differentiation skills to analyze dynamic situations, i.e. situations that aren’t static, that are changing over time. The ubiquitous class of problems that require this ability to solve are called related rates problems.
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Application
Be able to apply your differentiation skills to optimize some quantity, minimize or maximize some function, relative to some changing parameters. The ubiquitous class of problems that require this ability to solve are called optimization problems.
The first of these will require you
to be adept at implicit differentiation.
The second will simply be an extension of a skill that
you’ve already practiced before in math class:
calculating zeros of functions.
Both of these will require that you be comfortable
synthesizing mathematical equations and relationships
from a given situation.
And luckily both of these topics are well documented on the internet;
you way want to append the phrases calculus 1
or single variable
to your search phrases though.
There is one last, rather fundamental, numerical/computational application of differentiation we’ll discuss.
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Application
Perform Newton’s method to numerically approximate the solution to an equation, and, more generally speaking, understand how lines can be used to approximate curves.
It’s more important that you understand the idea behind Newton’s method rather than be proficient at working it out manually; the goal here is familiarity, not fluency, so that one day you could implement it with a computer if you need to. If you’re practicing computer programming while taking this class though, now would be a fine time to implement it.
Integration Basics
Now that we know basically everything about differentiation, it’s time to turn our attention to the other protagonist in the story of calculus, integration. Doing this we must immediately surmount two conceptual hurdles: we must define and understand a definite integral, and we must flush out how integration is dual — in a sense a foil — to the operation of differentiation.
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Concept
Understand that the definite integral \(\int_a^b f(t) \;\mathrm{d}t\) has a “physical” interpretation as the total amount of something that \(f\) has accumulated as its input moves from \(a\) to \(b\), but also a geometric interpretation as the signed area of the region bound by the graph of \(y=f(t)\) and the \(t\)-axis between \(a\) and \(b\). Furthermore understand what the function \(\int_0^t f\) represents. -
Concept
Understand that a definite integral is defined as the limit of a Riemann sum — any kind of Riemann sum — as the number of summands approaches infinity. E.g. \[ \int\limits_a^b f(x) \,\mathrm{d}x \;\;=\;\; \lim\limits_{n \to \infty} \frac{1}{n} \sum_{i=0}^{n} f\!\left(a + \frac{b-a}{n}i\right) \] -
Concept
Have a working understanding of the Fundamental Theorem of Calculus, knowing abstractly that differentiation and integration are “inverse” operations that “undo” each other, and knowing concretely that this relationship is captured by these two equations: \[ \newcommand{\ddt}{\frac{\mathrm{d}}{\mathrm{d}t}} \ddt \int\limits_a^t f = f(t) \quad\text{ and }\quad \int\limits_a^b \ddt f(t) \;\mathrm{d}t = f(b) - f(a) \,. \]
Knowing these two parts of the Fundamental Theorem of Calculus is at the core of being fluent in the language of integral calculus. It’s important to both understand what \(\int\) means when encountered outside of math class, and to be able to comfortably work with \(\int\) in calculations. After internalizing these concepts, there is a skill that the second equation of the fundamental theorem of calculus demands we practice.
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Skill
Be able to compute an antiderivative of a function. I.e. given a formula for a function’s derivative \(\frac{\mathrm{d}}{\mathrm{d}t}f\), figure out a formula for \(f\). Also know that the indefinite integral of a function simple refers to the family of all its antiderivatives.
Unlike the analogous skill of computing derivatives, there is no reliable procedure to follow when computing antiderivatives. Instead, honing this skill involves some creativity. You will not master this skill in this class, but you’ll learn a few basic tools, and get some practice channeling that creativity. In the next course you’ll learn to further hone this skill.
The first of these basic tools is simply pattern recognition. E.g. you know that the derivative of \(\tan(x)\) is \(\sec^2(x)\), so you should recognize that you know \(\int \sec^2(x) \;\mathrm{d}x\). The next of these tools is that the power-rule works in reverse: \[ \int x^n \;\mathrm{d}x = \frac{1}{n+1}x^{n+1} + C \qquad\text{for } x\neq -1\,. \] And then we’ll talk about the method of substitution, a means of undoing the chain rule, which is encapsulated in by this formula: \[ \int f'\bigl(g(x)\bigr)\,g'(x) \;\mathrm{d}x = f\bigl(g(x)\bigr) + C \]
Once this skill is developed we’ll talk about some applications of integration that rely heavily on wielding that skill and understanding the core concepts of integration.
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Application
Be able to compute the area of any region in the coordinate plane bound by piece-wise differentiable curves. -
Application
Know that the volume of a solid of revolution can be thought of as the accumulation of areas of cross-sections along either an axis or a radius, and be able to compute this volume as an integral. -
Application
Compute work, the accumulation of force applied over the displacement of some mass. I.e. \[ \text{Work} = \int\limits_a^b \text{Force} \,\mathrm{d}x\,. \]
Each of these require the understanding of an integral as an accumulation of something. The first two are applications of integration to geometry and provide good practice of your spatial reasoning. The latter one exemplifies calculus’ utility within the study of physics, and science overall.
And that’s it! That’s the beginning of the story of calculus. In the sequel to this class we’ll discuss the calculus of exponential and logarithmic functions, dive deeper into techniques of evaluating indefinite integrals, learn to calculate the sum of infinitely many numbers, and finally we’ll culminate the story of single-variable calculus by learning that every infinitely differentiable function can be considered as the limit of a sequence of polynomial functions.