Introduction

Calculus is the study of change. It provides a framework for modeling systems in which there is change, and for solving problems related to change.

Calculus has two main branches: differential calculus and integral calculus.

• Differential calculus deals with the study of the rate at which quantities change
• Iintegral calculus deals with the accumulation of quantities.

Calculus is a fundamental tool in many fields, including physics, engineering, economics, and computer science.

Calculus believes that everything can be broken down into infinitesimally small pieces, and that by understanding the behavior of these small pieces, we can understand the behavior of the whole. Many problem in calculus revolve around turning a discrete concept into a continuous one, and then solving the continuous problem.

History

In the late 17th century, Sir Isaac Newton and Gottfried Wilhelm Leibniz independently developed the foundations of calculus. Newton used calculus to develop his laws of motion and theory of gravity, while Leibniz used it to develop his theory of infinitesimal calculus.

Newton used calculus to develop his laws of motion and theory of gravity, eventually leading to the development of classical mechanics and the law of universal gravitation, solving the largest problem of the time, the motion of the planets.

The law of universal gravitation states that every point mass attracts every other point mass by a force acting along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them. In other words:

where is the magnitude of the force between the two masses, and are the masses of the objects, is the distance between the centers of the masses, and is the gravitational constant.

Leibniz used calculus to develop his theory of infinitesimal calculus, which is the basis for modern calculus. Leibniz's notation is still used today, and his work laid the foundation for the development of the calculus of variations, differential equations, and many other fields.

General Path

Typically, before studying calculus, students should have a solid understanding of algebra, trigonometry, and geometry. These subjects provide the foundation for calculus and are essential for understanding the concepts and techniques of calculus, and are therefore prerequisites for studying calculus, or "pre-calculus".

Then, students typically study differential calculus, which deals with the study of the rate at which quantities change. This includes topics such as limits, derivatives, and applications of derivatives.

After studying differential calculus, students typically study integral calculus, which deals with the accumulation of quantities. This includes topics such as integrals, techniques of integration, and applications of integrals.

Finally, students may study multivariable calculus, which extends the concepts of differential and integral calculus to functions of several variables.

Calculus 1 and 2 are typically the first two courses in a calculus sequence, and cover the topics of differential and integral calculus, respectively. These courses are typically taken by students in their first or second year of college or university. AP Calculus AB and BC are somewhat equivalent to Calculus 1 and 2, respectively, though there's some differences in the content covered.

Calculus 3 is typically the third course in a calculus sequence, and covers the topics of multivariable calculus. This course is typically taken by students in their second or third year of college or university.

The notes in this section cover AP Calculus BC, but is written as Calculus 1 and 2, since there will eventually be a Calculus 3 section.

There are many topics beyond Calculus 3.

• Differential Equations are typically studied after Calculus 3, and deal with the study of equations involving derivatives. This includes topics such as first-order differential equations, second-order differential equations, and systems of differential equations. This is important especially in physics and engineering.
• Linear Algebra is typically studied after Calculus 3, and deals with the study of vectors, vector spaces, matrices, and linear transformations. This is important especially in physics, engineering, and computer science.
• Real Analysis is typically studied after Calculus 3, and goes back to the foundations of calculus, rigorously defining limits, derivatives, and integrals. This is important especially for students who plan to study pure mathematics or theoretical physics.
• Complex Analysis is typically studied after Real Analysis, and deals with the study of complex numbers and functions. This is important especially for students who, again, plan to study pure mathematics or theoretical physics.
• Partial Differential Equations are typically studied after Differential Equations, and deal with the study of equations involving partial derivatives.

There are also some obscure topics in calculus, such as:

• Functional Analysis, which is the study of infinite-dimensional vector spaces and operators on those spaces.
• Differential Geometry, which is the study of geometry using calculus.
• Measure Theory, which is the study of measures and integration.
• Fractional Calculus, which is the study of calculus involving fractional derivatives and integrals. Basically, things like , which seems stupid, but is actually useful in some fields.

Resources

The main resources I used to create these notes are:

• Khan Academy - Really good site. A lot of questions and notes are grabbed straight from there.
• 3b1b - Excellent animations, for conceptual understanding.

Other Things

• Calculus is a difficult subject for many students, and requires a lot of practice to master. It is important to work through many problems and examples to develop a deep understanding of the concepts and techniques of calculus.
• Emphasize conceptual understanding over rote memorization. Understanding the underlying concepts and principles of calculus will help you solve problems more effectively and efficiently.
• I avoid using the notation (called "Lagrange's notation") in these notes, and instead use (called "Leibniz notation"). This is because Lagrange's notation is ambiguous and can be confusing, especially when dealing with higher derivatives and variable naming. Leibniz notation is more precise and easier to work with, especially when dealing with multiple variables. Of course there's exceptions, but I'll explicitly mention them when they come up.