Mastering the basics of single variable calculus - A beginner's guide

In this course, we'll provide a comprehensive overview of single variable calculus, including key concepts like limits, derivatives, and integrals.

Table of contents

What is included in a calculus course?

The following 23 topics are typically included in a calculus course

1. Limits

Functions are not always defined for all inputs. For instance, division by zero is not allowed. To see how the function look close to such an input, we study the limit as the input goes toward this point.

2. Continuity

Integers are considered discrete numbers; they are distributed with gaps in-between them. The set of real numbers, however, is continuous, since between any two reals you will always find more of them. The same concept extend to functions, where a continuous function is one with no gaps.

3. Derivatives

The derivative of a function gives its instantaneous rate of change at any point. This is analogous to the slope of a line parallel to the function there, which, together with the concept of limits, formulate the definition of the derivative: $$f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$$

4. Differentiation rules

Finding derivatives is not always straight-forward using the definition. Luckily, the subject is well-researched, and multiple handy rules have been discovered, simplifying the task depending on the form of the function.

5. Mean value theorem

The mean value theorem is simple, yet powerful: Between any two points on a continuous function, the derivative will somewhere equal the slope of the line connecting the points.

6. Implicit derivatives

If y is not a function of x, but we still have a relation between the two through some equation, we can not formulate the derivative as a function of x. This is the case for a circle with radius $a$: $$x^2 + y^2 = a^2$$ A line tangent to this circle will have the slope $ -\frac{y}{x}$. This derivative is implicit, as opposed to explicit as a function of $x$ alone.

7. Inverse functions

A function is a way of processing numbers. You plug in some number and get out a new one. The inverse of a function is then a redo-process, which takes in the old output, and spits out the initial input.

8. Antiderivatives

As hinted by the name, antiderivatives are the reverses of derivatives. If a function $f(x)$ has the derivative $f'(x)$, $f(x)$ is an antiderivative of $f'(x)$. Given a function $f(x)$ may in turn have antiderivatives on its own, which are generally denoted as $F(x) + C$. The constant $C$ can always be added since taking the derivative removes any constants.

9. The natural logarithm

The natural logarithm $$\ln(x) = \log_{e}(x)$$ is the logarithmic function with Euler's number $e$ as the base. Logarithms and exponential functions have an inverse relationship to one another, and so the natural logarithm asks: What is the exponent $a$ that makes $e^{a} = x$.

10. Derivatives of logarithms and exponentials

A key property of the natural exponential function $e^{x}$ is that is constitutes its own derivative. This, along with some rules of exponentiation and differentiation can be used to find the derivative of any exponential function: $$\frac{d}{dx}a^{x} = a^{x}\ln{a}$$ In a similar fashion, the fact that the derivative of the natural logarithm is $1/x$ makes aids us in determining the derivative of the general logarithmic function: $$\frac{d}{dx}\log_{a}(x) = \frac{1}{x\ln(a)}$$

11. Inverse trigonometric functions

Trigonometric functions provide information about the ratio of the side lengths $a$, $b$ and $c$ in a right triangle, given an angle $x$. Now inverse trigonometric functions go the other way, and exposes the angles, given these ratios: $$\sin(x) = \frac{a}{c} \implies \arcsin\left(\frac{a}{c}\right) = x$$ $$\cos(x) = \frac{b}{c} \implies \arccos\left(\frac{b}{c}\right) = x$$ $$\tan(x) = \frac{a}{b} \implies \arctan\left(\frac{a}{b}\right) = x$$

12. Extreme values and sketching

Visualizing a function by sketching it can be of great help to understand the problem modelled by the function. Extreme values occur at points that are particularly interesting or helpful to paint the picture.

13. Optimization

Optimization techniques tend to involve a large portion of calculus. The core idea is what we can say about a function and its derivative where it takes on its maximum value. Since it will not increase regardless of what direction we change $x$ in, we find this point where the derivative is equal to zero.

14. Taylor polynomials

Any mathematical function $f(x)$ can be written as a polynomial of this special form: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$ This so-called Taylor polynomial may be of infinite order, but a good approximation is often obtained by considering enough terms and letting the reference point $a$ be close to the point $x$ we are interested in. Terms involving higher order derivatives can then be disregarded without significant loss of accuracy.

15. Calculating limits

When the limit of a function is not immediately obvious, there are certain techniques we can employ to find it. It might for instance be the case that the function is a ratio of two expressions that both tend to infinity as $x$ grows large. In such a case we must consider which one of them grows faster compared to the other. One method to determine this is to compare the expressions' derivatives.

16. Integrals

Integrals, denoted by the iconic symbol $int$, are closely related to antiderivatives. It turns out that evaluating a function's antiderivative at two points and calculating the difference reveals fundamental information about the situation that the function describes.

17. Integration techniques

Just like there are short-cuts to finding a function's derivative, we can often employ standard methods to integrate a function depending on what type of function we are facing.

18. Applications and improper integrals

A somewhat counter-intuitive result from calculus is that even a function extending infinitely in one direction may confine an area of finite size. In light of this phenomenon, it is important not to jump to any conclusions when dealing with improper integrals; integrals involving infinities.

19. Series

A series $S$ is a sum of infinitely many terms: $$S = \sum_{n}^{\infty} a_n = a_1 + a_2 + a_3 + ...$$ Addition is probably the most basic and intuitive mathematical operation there is, and series often provide nice and simple ways to tackle hard problem.

20. Riemann sums and integral approximation

If you have a hard time understanding the seeming magic way of calculating areas using integrals, you are not alone. Not until integration was justified by the analogy to sums of infinitesimally narrow segments, so-called Riemann sums, did the majority of mathematicians start accepting the method.

21. Numerical methods

Calculating integrals analytically can be painful for complicated functions. In some cases, its not even possible due to the fact that not all functions have antiderivatives. When faced with such an issue, we are glad to have numerical methods at our service.

22. First order linear differential equations

In contrast to regular equations, were we set out to find the value of a variable $x$, differential equations explains a relationship between a function and its derivative, and what we are after is the original function. A first order linear differential equation can be written in the form: $$f'(x) + af(x) = 0$$

23. Second order linear differential equations

A second order differential equation is one containing not only the the derivative, of the function we want to find, but also the derivative of this derivative: $$f''(x) + af'(x) + bf(x) = 0$$

What is single variable calculus?

Single variable calculus is a branch of mathematics that deals with the study of functions and their rates of change. It involves the concepts of derivatives and integrals, which are used to study the behavior of functions and to solve problems in a wide range of fields, including physics, engineering, and economics.

Single variable calculus is typically divided into two subfields: differential calculus, which deals with the study of rates of change, and integral calculus, which deals with the study of the accumulation of quantities. Together, these two subfields form the basis of much of modern calculus.

FAQ

What is a limit?

A limit is a concept that describes the behavior of a function as its inputs get closer and closer to a specific value. The limit of a function at a particular point is the value that the function approaches as the inputs get closer and closer to that point.

What is the mean value theorem?

The mean value theorem states that for a curve stretching from one point to another, there will be at least one other point on the curve where its tangent line is parallel to the straight line between the endpoints.

What is the chain rule?

The chain rule is a fundamental concept in single variable calculus that is used to differentiate composite functions. A composite function is a function that is made up of two or more functions that are combined together. For example, if we have two functions $f(x)$ and $g(x)$, we can create a new function $h(x)$ that is the composite of $f(x)$ and $g(x)$ by defining $h(x)$ as $h(x) = f(g(x))$.

What is the fundamental theorem of calculus?

The Fundamental Theorem of Calculus is a fundamental result in single variable calculus that establishes the connection between the concepts of differentiation and integration. There are two parts to the theorem, both of which are important for understanding the relationship between these two concepts.

The first part of the theorem states that the definite integral of a function over some interval can be computed by evaluating the function at the endpoints of the interval and taking the difference.

The second part of the theorem states that the indefinite integral of a function (also known as its antiderivative) can be found by evaluating a certain definite integral. Together, these two parts of the theorem provide a powerful tool for solving a wide range of problems in calculus.

What is calculus used for? - 6 practical use cases

1. Calculations and analysis of prices of goods

With the help of implicit derivatives, one can solve equations where ordinary differentiation falls flat. For example. in order to be able to predict the prices of goods, one must understand the relationship between many different variables that affect its price. In a market economy, the price of all goods is determined based on supply and demand, implicit derivatives are therefore a must for all stockbrokers!

2. Encryption

An encryption function takes a message as input, scrambles it, and spits out a scrambled message. To decrypt a message, the inverse function needs to be found which reverses the enciphering.

One of the most famous examples of encryption was the Enigma, used by the Germans during the Second World War to encrypt their messages. In Enigma, each letter was automatically reassigned a new letter, making the cipher harder to break.

The cryptologists eventually invented a machine for finding the settings of Enigma. The breaking of the Enigma code, which was crucial for the outcome of the war, meant constructing an inverse function.

3. Carbon-14 dating of organic matter

Carbon-14 is a form of carbon found in all living things. However, as an organism dies, this radioactive element starts to decay with time. Therefore, by measuring the amount of carbon-14 present in a dead object, radiocarbon dating tells us how long ago the organism died.

This decay is exponential, meaning that the rate of decrease depends on the current amount left. While the exponential function tells us how much carbon-14 is left at the time $t$, the natural logarithm answers the question: Given the amount of carbon-14 left, what is $t$?

4. Dosage of medicine

For the level of blood sugar not to reach a dangerous amount among diabetics, 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.

5. Cancer diagnosis

The field of medicine have made a lot of progress in recent years when it comes to cancer treatment. Although not yet perfect, the process of curing patients from the disease have in many ways been optimized.

With modern machine learning techniques, medical teams can use various types of imaging to scan a patients tissue to detect tumors.

A crucial part of the algorithms for computer vision used to diagnose patients from images is to maximizing the program's probability of finding cancer cells, while minimizing the risks of making erroneous predictions.

After diagnosis, another type of optimization comes into play as it is time to get rid of the tumor.

For successful radiation therapy, it is important to balance the amount of radiation to be effective for killing the malignant cells, while not exceeding an overall unhealthy level.

6. Predicting populations over time

Since 1970, humanity has wiped out more than 60% of all animal populations. But we started way earlier: examples of human-driven extinction date back more than a hundred thousand years.

For example, the arrival of humans to South America is the most probable reason why the animal called, the giant ground sloth, went extinct about eleven thousand years ago.

Differential equations let us calculate how many animals of a given population there will be at some point later in time. We only need to know how many we start with and how the amount changes.

Is single variable calculus hard?

Calculus in one variable is the course that is most similar to high school mathematics, which tend to make students confident. But be aware, many students do worse on the exam than they thought they would.

The reason why students do worse than expected is that they feel a false sense of security, as most of the material can be recognized from high school calculus. University calculus, however, tend to be much more demanding, both in theory and in problem solving. One could say it's the big reset for any student's mathematical journey.

The most difficult part of single variable calculus is typically considered to be the concept of limits. In order to understand calculus, it's essential to be able to grasp the idea of a limit, which is a fundamental concept that underlies many of the other ideas in calculus. A limit describes the behavior of a function as its inputs get closer and closer to a specific value, and understanding how to evaluate limits is crucial for being able to work with derivatives and integrals.

Other difficult concepts in single variable calculus include the chain rule, which is used to differentiate composite functions, and the Fundamental Theorem of Calculus, which connects the concepts of differentiation and integration.

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