Four Ways to Represent a Function Lecture Notes	The central objects of study in calculus are functions. Here we introduce various ways to visualize them.
Essential Functions Lecture Notes	There are far too many functions to study them all. Here we introduce several important classes of function, the ones we will study in most depth.
New Functions from Old Functions Lecture Notes	Here we look at how functions can be altered and how it affects their graphs.
Exponential Functions Lecture Notes	Exponential functions are a deeply important class of functions, apprearing all throughout nature. Here we introduce them along with the natural base e.
Inverse Functions Lecture Notes	Under what circumstances does a function admit an inverse? Here we explore this question as well as asking what graphs of inverse functions look like.
The Tangent and Velocity Problem Lecture Notes	The classical definiton of average velocity is the distance traveled divided by time taken to do so. This doesn't make sense if we ask what velocity means at a single moment in time. Here we explore this deep question.
The Limit of a Function Lecture Notes	To make sense of velocity at a single moment we had to think about how functions behaved as the input approaches (but does not equal) a specific value. Here we formalize this idea with the concept of a limit, the central technical tool in all of calculus.
Calculating Limits Using Limit Laws Lecture Notes	With the concept of a limit definited (at least informally), we develop methods to compute them for a wide class of functions.
The Precise Definition of a Limit Lecture Notes	The informal approach to limits is all well and good, but if we are to study functions in more depth we need a much more precise defintion. Here we introduce the rigorous, so called epsilon-delta defintion of a limit.
Continuity Lecture Notes	A function is continuous at a point if the limit there can be calculated in the most straightforward way possible. We carefully introduce this concept and relate it back to our core functions.
Limits at Infinity; Horizontal Asymptotes Lecture Notes	We introduce a new type of limit, considering how functions behave as the input grows without bound. We consider how this affects the shape of the graph.
Derivatives and Rates of Change Lecture Notes	Using the langage of limits we formally introduce the fundamental concept of the derivative of a function at a point. Informally we can interpret this as a slope of a tangent line.
The Derivative as a Function Lecture Notes	Having calculated the slope of the tangent line at a point, we observe this gives rise to a new function, namely the derivative. This is the main object of study in differential calculus.
Derivatives of Polynomial, Power and Exponential Functions Lecture Notes	We compute derivatives of polynomial, power and exponential functions, three of our core classes.
The Product and Quotient Rules Lecture Notes	If we know the derivatives of two functions, is there a simple way to calculate the derivative of their product or quotient? The answer will turn out to be quite unexpected.
Derivatives of Trigonometric Functions Lecture Notes	We compute derivatives of trigonometric functions, one of our core classes.
The Chain Rule Lecture Notes	If we know the derivatives of two functions, is there a simple way to calculate the derivative of their composite? The answer will turn out to be quite elegent.
Implicit Differentiation Lecture Notes	If a function is only known to satisfy some equation (involving its input and output), is there some way to calculate its derivative?
Derivatives of Logarithmic Functions Lecture Notes	Here we calculate derivatives of logarithmic functions using implicit methods. We also introduce the useful technique of logarithmic differentiation.
Rates of Change in the Natural and Social Sciences Lecture Notes	We explore various ways the derivative manifests itself in nature.
Exponential Growth and Decay Lecture Notes	We study functions which satisfy natural growth/decay, and explore how they occur in nature.
Maximum and Minimum Values Lecture Notes	We use the derivative to understand extreme values of functions.
The Mean Value Theorem Lecture Notes	If I travel 100 miles in one hour, even if my speed is fluctuating, there has to be a moment when I'm travelling at exactly 100mph. This is the mean value theorem. Making this precise will give us deeper understanding of a differentiable functions.
How Derivatives Affect the Shape of a Graph Lecture Notes	We use the derivative to understand when a function is increasing or decreasing, along with its convavity. We apply these ideas to finding extreme values.
l'Hospital's Rule Lecture Notes	This gives us a useful tool for computing certain types of limit. (Interesting Fact: This result is actually due to Johann Bernoulli, who was paid by l'Hospital to give him his mathematical discoveries.)
Summary of Curve Sketching Lecture Notes	We put everything we've learned together to sketch the graphs of functions.
Optimization Problems Lecture Notes	We use our techniques for finding extreme values of functions to calculate optimal solutions to various physical problems.
Antiderivatives Lecture Notes	We consider the problem for finding functions with specific derivatives. We call these antiderivatives.
Areas Lecture Notes	We consider how to calcuate the area bounded by the graph of a positive function and the x-axis.
The Definite Integral Lecture Notes	We introduce the fundamental concept in integral calculus: the definite integeral.
The Fundamental Theorem of Calculus and Indefinite Integrals Lecture Notes	We introduce the fundamental theorem, a deep connection between differential and integral calculus.
The Substitution Rule Lecture Notes	Integration by substitution is a systematic way to reverse the chain rule. It's a fundamental technique used to find antiderivatives.
Areas Between Curves Lecture Notes	We use the definite integral and the fundamental theorem to calculate areas between curves.
Volume Lecture Notes	We calculate the volumes of solid shapes by integrating cross-sectional area functions.