- "Class Agreements"
- Update on the STEM FORCE crew:
- They'll be by tomorrow morning to introduce themselves and their program.
- They'll have a morning session on Thursdays, 8:00 - 9:15 a.m., MEP 344, just for you.
- They'll have evening sessions at 7:30-9:00 PM every Wednesday in GH 230. This will be for students in various STEM courses (MAT 119, MAT 229, INF 120, CHE 120 and 121, CHE 310 and 311, PHY 210 and 211, and PHY 220 and 221), so it will not be just for you. But you can go then and get help.
- Both of these times will be especially convenient when it comes time for the tests.
- They've been made "teaching assistants" on our BlackBoard site, and I've given them free rein to use that site to their advantage.
- First of all, a quiz opportunity!
- I mentioned our canonical problem of an "inverse problem" last
time: while I was up in Canada recently, I had to convert between
Fahrenheit and Celsius. In particular, I was given a temperature in
Celsius (-20oC) and wanted to hear it in oF.
-
Activity 1: given a sheet of graph paper, the fact that the
relationship is linear, and using the two points
independent dependent oC oF 0 32 100 212 create a nice graph of
(Fahrenheit as a function of Celsius). Compute
.
Then write the linear function g that converts C to F.
As soon as we answer that question, we'll probably want to know: how do we convert between Celsius and Fahrenheit? That's the inverse function!
Write the linear function
that converts F to C.
-
Activity 2: given the graph of
, create the graph of the inverse function (
).
Now: would want to carry out stage three of the three-step process of computing an inverse function? That is, would you want to exchange the variable names of the dependent and independent variables?
- Why study inverse functions (and inverse problems, more generally)?
- Pollution
- Forensics
- Ballistics
- Activity 3: Examine this figure:
- Is Temperature
an invertible function of altitude (
)? Why (or why not)?
- Is Atmospheric Mass Density
an invertible function of altitude (
- What does the inverse measure, if it exists? What questions can it answer?
- How can you explain those values for mass density along the y axis?
- Is Temperature
- Examples: Let's run the odd exercises from 3-15, p. 390.
- Now, let's look at a particular function:
- How do we compute
?
-
?
- Find a formula for
?
- How do we compute
- Now: is
invertible? How can we use
- properties of this class of function to help us answer this question?
- calculus to help us answer this question?
- Let's derive the derivative of the inverse, using a neat trick:
the inverse identity
and the chain rule.
- Whether a function is invertible or not, one can propose to find the derivative of its inverse using this process. Use the process to find the derivative of the "inverse of sine" (although that clearly makes no sense at this point, since sine is not one-to-one).