Schedule

Functions

We start the course with a review about some topics from functions. A function is an object which, given an input from some set of objects, returns an output from another set of objects. For example, if I want to know how old someone will be in four years, then I input their (possibly decimal valued) age in years (from the set of years), and the function will output that age plus 4: f(x) = x + 4 (also in the set of years). In this course, we will focus on real-valued functions; that is, functions that taken in a single real value and which output a single real value (of which the above +4 ageing function is an example). We will discuss how to represent real valued functions, how to make new functions from template functions, and some special families of real-valued functions (linear, polynomial, trigonometric, exponential/logarithmic).

• Briggs 1.1, 1.2
• Function inside a function?

• Review Problems
• Challenge Problems

• Briggs 1.3
• The seduction of the exponential curve (2011 Forbes article)

• Briggs 1.4
• Trig for Games Interactive Tutorial

Limits

In this section, we take our first foray into problems involving dividing zero by zero, which form the foundation of the rest of calculus. In more concrete terms, we ask what happens to the output of a function when we can get as close as possible to a particular input for a function. This works even if the function isn't defined at the input we're approaching!

• Week 4 Review Sage Notebook
• Week 4 Extra Problems

Derivatives

Now we are able to introduce the first real workhorse of calculus: the derivative. This allows us to estimate the "tangent slopes" of functions. More concretely, it tells us how quickly functions are changing at different inputs. We will spend a lot of time building up the nuts and bolts on toy functions in this section before we move onto the applications.

• Briggs 2.7, 2.1, 3.1
• Desmos Delta/Epsilon Visualization Tool
• Usain Bolt's Speed

• Briggs 3.1, 3.2
• Shippensberg Derivative Secant Demo
• Desmos Derivative Secant Demo

• Briggs 3.3, 3.4
• Geogebra position/velocity/acceleration demo
• Quotient rule limit proof

• Extra Review Problems

• Playing with Exponential Tangents in Sage

• Briggs 3.8
• Implicit Differentiation of Loops Class Examples

• Extra Review Problems

Applications of The Derivative

Now that we are through the nuts and bolts of derivatives, we move onto some applications of derivatives. We will talk about how we can use derivatives to come up with accurate sketches of functions by understanding their "critical points" and "concavity." We will also see how to use some of these features to understand optimization problems (e.g. what's the largest volume enclosure I can create from a fixed amount of material?). We will end by finally being able to address the question of what infinity divided by infinity is.

• Briggs 3.11
• Conical tank filling app

• Briggs 3.11
• Falling Ladder App

• Briggs 4.3, 4.4
• Driving through inflection points

Introduction To Integration

We now look at the other major half of calculus, which is integration. We introduce this concept as the ability to find the "area under a complicated curve" as the limit of a sum of very small simple areas we know how to compute, which is more precisely referred to as a Riemann Sum, or a Definite Integral. In this way, we will be able to address what 0 times infinity is. Amazingly, we will see that integration turns out to be the "dual" of differentiation, in the sense that we can compute definite integrals by doing derivatives in reverse.