Take an online pre-test prior to week 1 (flexible scheduling)
Functions are incredibly important building blocks to describe all sorts of interesting things in mathematics, science and engineering. In this module, we wrestle with 3 key questions:
- What's a function?
- What's the average rate of change?
- What's a linear function?
In particular, we're going to ask you to think about these ideas in the context of the Keeling Curve application which describes the concentration of CO2 in Earth's atmosphere.
Functions and Data Fitting
Data fitting is a really, really big deal in many disciplines. We will start this module by looking at the gold-medal winning long jump performances in the Olympics since 1900. And we'll finish this module by asking you to interact more with the application, Keeling Curve, Revisited. Here are the questions that this topic generates:
- How do we approximate the data with a linear function when we have lots and lots of data.
- How do we measure the "total error"?
- How do we know if our linear function is really a good fit to the data?
Function Transformations are very, very helpful things. They give us ways to create more complicated functions out of very basic ones. Our questions:
- How do we shift a function horizontally and vertically?
- How do we stretch, compress, and/or flip a graph vertically? Horizontally?
Quadratic functions have a constant average rate of change of the average rate of change. That is, their concavity is constant. In this module we ask several questions to understand the properties and graphs of quadratic functions.
- How do we determine the concavity of a quadratic function?
- How do we determine the minimum (or maximum) of a quadratic function?
- How do we determine the roots of a quadratic function?
- What's the difference between the standard, factored, and vertex form of a quadratic function?
Introduction to Exponential Functions
Grow, Grow Grow, Go, Go, Go! Exponential functions are great for describing things that grow rapidly. We're going to examine how Ben Franklin's gift to the cities of Philadelphia and Boston grew over time to get an understanding of exponential functions. We conclude by looking at Malthus' theories of population growth. Questions we will interact with:
- What's an exponential function? (Ok, this seems kind of basic. But it's necessary)
- How can we distinguish between an exponential function and a linear function?
Compound Interest and the Number e
Over. And over. And over again and again and again. In this module, we explore how compounding interest over and over impacts the growth of a a bank account, the economy, and many other things. When we talk about Compound Interest we ask questions like:
- What's the effective annual interest rate if we compound interest several times a year?
- What's the effective annual interest rate if we compound interest all the time? Like, every moment of every day, i.e. continuously?
- Where does the number e come from? How does it relate to compound interest?
If exponential functions are "go, go go!" kinds of functions that grow rapidly, then logarithmic functions are "slow, slow, slow" kinds of functions that grow slowly. That's because they are the inverse function of an exponential function. In this module we answer the following questions.
- What's the difference between log(x) and ln(x)?
- What properties do log functions have?
- How do log functions "re-scale" information?
- What's the domain of a log function?
- What does it mean for one function to be the inverse of another?
- How do we solve equations involving exponents?
Geometry and Periodic Motion
We connect the basic ideas of periodic motion with the geometry of a right triangle and the sine and cosine functions. This leads to the following questions:
- How does the geometry of a right triangle relate to sine or cosine of an angle in the right triangle?
- How are the amplitude and midline related to sine and cosine functions?
Measuring Angles with Arc length
We define a new way to measure angles by using arc length. This leads to the idea of Radians.
We're going to look at an application of trigonometry by looking at a car jump scene from Gone in 60 Seconds with Nicholas Cage. We're asking "could that really happen?" Here are some questions that we must address before determining whether or not the car jump scene is realistic.
- How do we decompose a force into the horizontal and vertical components using trigonometry?
- What are the equations of motion and how do they come into play for both the vertical and horizontal distance an object travels in the air?
Catch-up on any missed work