Math 115: Calculus 1

Course Content

Broadly speaking, we will be reviewing precalculus content (e.g., functions), limits, differentiation, the applications of differentiation, and wrapping up with a study of the fundamentals of integration. In slightly more detail, the course is divide into the following four units. 

Unit 1: Review of Relevant Precalculus Content

This unit reviews the algebra, geometry, and precalculus content we'll need for learning single-variable calculus, in the context of only algebraic functions. (We'll return to transcendental functions in Unit 6.) We won't be reviewing all the content from those three areas, just the content most relevant to our calculus studies. This content includes a review of solving algebraic equations, various area and volume formulas from geometry, and content related to functions, including the definition of a function, its domain and range, how to graph functions, and various properties and transformations of functions. 

This unit will also introduce, reinforce, and emphasize mathematical modeling. This is the process of translating a real-world problem into mathematics, solving it, and interpreting your solution in the original real-world context that initiated the process. You no doubt have experience doing this in other mathematics courses (e.g., in math word problems). This unit will hone that skill and help you start to think more like an applied mathematician. This will, in turn, be useful for tackling your science courses and social science courses, many of which involve mathematical analyses and/or modeling. 

Unit 2: Limits

This unit begins our calculus adventure. It focuses its attention on the first pillar of calculus: limits. Every new concept in calculus is defined in terms of a limit, as we learned in the first chapter of Calculus Simplified (see, in particular, Figure 1.3 on pg. 4 and the discussion pertaining to it). So, we'll spend this unit learning all about what limits are, how we calculate them, and when they do and don't exist. We'll do all this only for algebraic functions. After we study transcendental functions in Unit 6 we'll study limits in that context (in Unit 7). 

Unit 3: Differentiation

This unit introduces the second calculus pillar -- the derivative -- and studies its definition, existence, interpretations, and various ways to calculate it.  We also discuss a variety of ways to understand and work with the derivative, including analytical (e.g., calculate the derivative from the equation of the function), graphical, and approximate (e.g., estimating the derivative from a table of values or a graph). Due to the origins and interpretations of the derivative, this unit is filled with real-world applications and connections to other mathematical concepts. (For example, we learn that the value of the derivative is the slope of a special line associated with the graph of the underlying function.)

Unit 4: Applications of Derivatives

In this unit we will focus on exploring the myriad applications of derivatives. The first lesson will showcase a real-word application of derivatives, but then we will pivot to applying derivatives to understand how to determine where a function has a maximum or minimum. The last lesson in the unit then circles back to real-world applications to discuss perhaps the most widely used application of derivatives: optimization. 

Unit 5: Integration

This unit marks a departure from the previous two units, which were focused on differentiation. In this unit we will begin our study of integration. As we will see, we will draw on  much of the knowledge and theory we built up in previous units, due in part to the fact that -- as we will discuss -- calculating integrals boils down to questions about derivatives. 

This being a Calculus 1 course, we will only scratch the surface of integration. In particular, we will only discuss one integration technique -- u-substitution. The follow-up to this course -- Calculus 2 -- traditionally discusses the plethora of other integration techniques and applications of integration. 

Unit 6: Transcendental Functions

This short unit is intended to review the precalculus-level treatment of trigonometric, exponential, and logarithmic functions. The goal is to get us comfortable with the basics and details about how these functions are defined, what their properties are, and the particular rules and relationships associated with them (e.g., the Rules of Logarithms). 

At the same time, we will see that in some contexts -- like in the context of introducing e, Euler's number -- we will draw on calculus to recast some of what we have already learned about trigonometric, exponential, and logarithmic functions. This is all to say that although the main focus of this unit is to review precalculus-level content, we will also at times apply our newly acquired calculus knowledge to enhance what we are learning about those functions.

Unit 7: The Calculus of Transcendental Functions

This is the final unit in the course. The focus in this unit is on applying the calculus concepts we have learned in the course to trigonometric, exponential, and logarithmic functions. We will therefore explore limits, differentiation, and integration in the context of those function families.

This unit is also in many ways a capstone unit. Because we have deferred the discussion of the calculus of those transcendental functions until the end of the course, while we are engaged in those discussions we will also be reviewing those calculus concepts, thereby helping to solidify our understanding of them. Furthermore, by seeing those concepts applied in new contexts, we will gain a greater appreciation of both what makes trigonometric, exponential, and logarithmic functions so much more interesting than algebraic functions and how calculus interacts with such functions.