Day 18, Mat227

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Announcements:
- March is National Colorectal Awareness Month. I just thought that you'd like to know.... I'm trying to spread awareness!
- Your homework for 5.1 is due today.
- You have a homework for 5.2 due Wednesday.
- Your 6.1 homework will be due the Monday after spring break.
- We were scheduled to have another exam on 3/20 to celebrate the arrival of spring -- I'd like to push that one back to to 3/27. Any problems with that?
Section 6.1: Inverse functions
- While up in Canada recently, I had to convert between Fahrenheit and Celsius. In particular, I was given a temperature in Celsius (-20^oC) 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 ${F=g(C)}$ (Fahrenheit as a function of Celsius). Compute ${g(-20)}$ .
  ${F=g(C)}$
  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 ${g^{-1}}$ that converts F to C.
- Activity 2: given the graph of ${g({C})}$ , create the graph of the inverse function ( ${g^{-1}({F})}$ ).
  Now: would we 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?
  Mathematica summary of the temperature problem
- A function ${y{=}f({x})}$ is invertible if and only if each value of ${y}$ in the range is associated with a unique value of ${x}$ in the domain:
  
  ${f^{-1}(y)=x {\leftrightarrow} f(x){=}y}$
  Non-uniqueness is represented by a graph failing to pass the horizontal line test, as we saw last time in our example of the parabola.
- Why study inverse functions (and inverse problems, more generally)?
  - Pollution
  - Forensics
  - Ballistics
- Activity 3: Examine this figure:
  1. Is Temperature ${T}$ an invertible function of altitude ( ${h}$ )? Why (or why not)?
  2. Is Atmospheric Mass Density ${ \rho }$ an invertible function of altitude ( ${h}$ )? Why (or why not)?
  3. What does the inverse function measure, if it exists? What questions can it answer?
  4. How can you explain those values for mass density along the y axis?
- Examples: Let's run the odd exercises from 5-15, p. 390.
- Now, let's look at a particular function: : How do we
  - compute ${f^{-1}(-2)}$ ?
  - ${f^{-1}({0})}$ ?
  - find a formula for ${f^{-1}({x})}$ ?
- Now: is invertible? (Mathematica) How can we use
  - properties of this class of function to help us answer this question?
  - calculus to help us answer this question? (In particular, how we can use the derivative?)
- Let's derive the derivative of the inverse, using a neat trick: the inverse identity and the chain rule.
  (Notice that we're using ${x}$ as the independent variable, rather than ${y}$ . We could use any variable name here, of course, even "nose". It's just because we generally prefer the use of ${x}$ as an independent variable name. Sigh.)
  - 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).
  - How about the derivative of the inverse of ${g(x){=}x^3+x-2}$ ?
- Examples (more from p. 390): use the fact that
  - #3
  - #31
  - #24
Section 6.2 summary

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