This course is designed to introduce you to the study of calculus. You will learn concrete applications of how calculus is used and, more importantly, why it works. Calculus is not a new discipline; it has been around since the days of Archimedes. However, Isaac Newton and Gottfried Leibniz, two seventeenth-century European mathematicians concurrently working on the same intellectual discovery hundreds of miles apart, were responsible for developing the field as we know it today. This brings us to our first question, what is today's calculus? In its simplest terms, calculus is the study of functions, rates of change, and continuity. While you may have cultivated a basic understanding of functions in previous math courses, in this course you will come to a more advanced understanding of their complexity, learning to take a closer look at their behaviors and nuances. In this course, we will address three major topics: limits, derivatives, and integrals, as well as study their respective foundations and applications. By the end of this course, you will have a solid understanding of the behavior of functions and graphs. Whether you are entirely new to calculus or just looking for a refresher on a particular topic, this course has something to offer, balancing computational proficiency with conceptual depth.
- Course Introduction
- Unit 1: Analytic Geometry
Most of the material in this unit will be review. However, the notions of points, lines, circles, distance, and functions will be central in everything that follows. Lines are basic geometric objects which will be of great importance in the study of differential calculus, particularly in the study of tangent lines and linear approximations.
We will also take a look at the practical uses of mathematical functions. This course will use mathematical models, or structures, that predict practical situations in order to describe and study a number of real-life problems and situations. They are essential to the development of every major business and every scientific field in the modern world.
Completing this unit should take you approximately 9 hours.
- 1.1: Lines
- 1.2: Distance between Two Points, Circles
- 1.3: Functions
- 1.4: Shifts and Dilations
- Unit 2: Instantaneous Rate of Change: The Derivative
In this unit, you will study the instantaneous rate of change of a function. Motivated by this concept, you will develop the notion of limits, continuity, and the derivative. The limit asks the question, "What does the function do as the independent variable becomes closer and closer to a certain value?" In simpler terms, the limit is the natural tendency of a function. The limit is incredibly important due to its relationship to the derivative, the integral, and countless other key mathematical concepts. A strong understanding of the limit is essential to success in the field of mathematics.
A derivative is a description of how a function changes as its input varies. In the case of a straight line, this derivative, or slope, is the same at every point, which is why we can describe the slope of an entire function with one number when it is linear. You will learn that we can do the same for nonlinear functions. The slope, however, will not be constant; it will change as the independent variable changes.
Completing this unit should take you approximately 16 hours.
- 2.1: The Slope of a Function
- 2.2: An Example
- 2.3: Limits
- 2.3.1: The Definition and Properties of Limits
- 2.3.2: The Squeeze Theorem
- 2.4: The Derivative Function
- 2.5: Adjectives for Functions
- 2.5.1: Continuous Functions
- 2.5.2: Differentiable Functions
- 2.5.3: The Intermediate Value Theorem
- Unit 3: Rules for Finding Derivates
Computing a derivative requires the computation of a limit. Because limit computations can be rather involved, we like to minimize the amount of work we have to do in practice. In this unit, we build up some rules for differentiation which will speed up our calculations of derivatives. In particular, you will see how to differentiate the sum, difference, product, quotient, and composition of two (or more) functions. You will also learn rules for differentiating power functions (including polynomial and root functions).
Completing this unit should take you approximately 12 hours.
- 3.1: The Power Rule
- 3.2: Linearity of the Derivative
- 3.3: The Product Rule
- 3.4: The Quotient Rule
- 3.5: The Chain Rule
- Unit 4: Transcendental Functions
In this unit, you will investigate the derivatives of trigonometric, inverse trigonometric, exponential, and logarithmic functions. Along the way, you will develop a technique of differentiation called implicit differentiation. Aside from allowing you to compute derivatives of inverse function, implicit differentiation will also be important in studying related rates problems later on.
Completing this unit should take you approximately 17 hours.
- 4.1: Trigonometric Functions
- 4.2: The Derivative of Sine
- 4.3: A Hard Limit
- 4.4: The Derivative of Sine, Continued
- 4.5: Derivatives of the Trigonometric Functions
- 4.6: Exponential and Logarithmic Functions
- 4.7: Derivatives of the Exponential and Logarithmic Functions
- 4.8: Implicit Differentiation
- 4.9: Inverse Trigonometric Functions
- 4.10: Limits Revisited
- 4.11: Hyperbolic Functions
- Unit 5: Curve Sketching
This unit will ask you to apply a little critical thinking to the topics this course has covered thus far. To properly sketch a curve, you must analyze the function and its first and second derivatives in order to obtain information about how the function behaves, taking into account its intercepts, asymptotes (vertical and horizontal), maximum values, minimum values, points of inflection, and the respective intervals between each of the above. After collecting this information, you will need to piece it all together in order to sketch an approximation to the graph of the original function.
Completing this unit should take you approximately 10 hours.
- 5.1: Maxima and Minima
- 5.2: The First Derivative Test
- 5.3: The Second Derivative Test
- 5.4: Concavity and Inflection Points
- 5.5: Asymptotes and Other Things to Look For
- Unit 6: Applications of the Derivative
With a sufficient amount of sophisticated machinery under your belt, you will now start to look at how differentiation can be used to solve problems in various applied settings. Optimization is an important notion in fields like biology, economics, and physics when we want to know when growth is maximized, for example.
In addition to providing methods to solve problems directly, the derivative can also be used to find approximate solutions to problems. You will explore two such methods in this section: Newton's method and the method of differentials.
Completing this unit should take you approximately 13 hours.
- 6.1: Optimization
- 6.2: Related Rates
- 6.3: Newton's Method
- 6.4: Linear Approximations
- 6.5: The Mean Value Theorem
- Unit 7: Integration
In this unit of the course, you will learn about integral calculus, a subfield of calculus that studies the area formed under the curve of a function. Though not necessarily intuitive, this concept is closely related to the derivative, which you will revisit in this unit.
Completing this unit should take you approximately 21 hours.
- 7.1: Motivation
- 7.2: The Fundamental Theorem of Calculus
- 7.3: Some Properties of Integrals
- 7.4: Integration by Substitution
- 7.5: Integration of Transcendental Functions
- 7.5.1: Exponential Functions
- 7.5.2: Natural Logarithmic Functions
- 7.5.3: Hyperbolic Functions
- Unit 8: Applications of Integration
In this unit, we will take a first look at how integration can and has been used to solve various types of problems. Now that you have conceptualized the relationship between integration and areas and distances, you are ready to take a closer look at various applications; these range from basic geometric identities to more advanced situations in physics and engineering.
Completing this unit should take you approximately 19 hours.
- 8.1: The Area between Curves
- 8.2: Volumes of Solids
- 8.3: Volume of Solids of Revolution
- 8.3.1: Disks and Washers
- 8.3.2: Cylindrical Shells
- 8.4: Lengths of Curves
- 8.5: Surface Areas of Solids
- 8.6: Average Value of Functions
- 8.7: Physical Applications
- 8.7.1: Distance
- 8.7.2: Mass and Density
- 8.7.3: Moments
- 8.7.4: Work