Day 16, Mat227

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Announcements:
- Sorry, I've not gotten your homework graded yet.
- Whoops! Looks like your 5.1 homework assignment got lost in the great website catastrophe. I've recovered it: pp. 349--, 1, 4, 5, 7, 8, 13, 14, 18, 21, 23, 28, 44, and 47 (let's make that due Monday, 3/2)
  You also have a homework for 5.2, and we'll make that due next Wednesday, 3/4.
Last time we started section 5.2. We'll continue with several more problems from 5.2, and get ready to jump into chapter 6, on inverse functions (even if we don't jump today).
Section 5.2: the volume
Let's see how to compute volumes of some simple objects using integrals.
Before you may have memorized some of these formulas. Now you can derive them!

This is the power of calculus.

In every case for what follows, we begin with what I consider the most important formula for integral calculus:

${V=\int{dV}}$
To calculate any quantity ${V}$ , we simply divide it up into infinitesimal quantities ${dV}$ , and add an infinity of them up.
Of course the letter ${V}$ is not crucial -- but in this chapter we consider volumes, and so we use ${V}$ . In the early going, we were computing areas, so one might write the most important formula for integral calculus as

${A=\int{dA}}$
That could be used to compute Andrews as well -- an Andrew is a sum of an infinite number of infinitesimal chunks of Andrew.
If we were computing masses, we'd write

${M=\int{dM}}$
a mass is made up of an infinite numbers of infinitesimal masses.

Now for volumes we usually know the extent of the object, so we'll march along the x-axis from a to b, stop at x, and add in the little ${dV({x})}$ that is found there. So in each case what we need to do is find ${dV({x})}$

${V=\int_{a}^{b}dV(x)}$
- The cube of side length ${s}$ has a differential of volume equal to the differential width ${dx}$ times the area of the face, ${A({x})}$ :
  
  ${dV(x)=A(x)dx=s^2dx}$
  
  ${V=\int_{0}^{s}dV(x)}$
- The right circular cylinder of radius ${r}$ and height ${h}$ has differential volume
  
  ${dV(x)=A(x)dx=\pi{r}^2dx}$
  
  ${V=\int_{0}^{h}dV(x)}$
- The cone of base radius ${r}$ and height ${h}$ , laid along the x-axis as its central axis (and its tip at ${x=h}$ ) -- draw a picture! -- has differential volume
  
  ${dV(x)=A(x)dx=\pi\left(r(1-\frac{x}{h})\right)^2dx=\pi{r^2}(1-\frac{2x}{h}{+}\frac{x^2}{h^2})dx}$
  and then, as usual,
  
  ${V=\int_{0}^{h}dV(x)}$
  (do you see a pattern here? The only thing that changes is the differential -- and the name on the variable at the endpoint. It's all about the differential! And there are those who would drop them -- shame!)
- The pyramid of height ${h}$ , and square base ${s}$ .
  What would its differential look like? How should we go about computing this volume?
- The sphere of radius ${r}$
  
  ${dV(x)=A(x)dx=\pi\left(\sqrt{r^2-x^2}\right)^2dx=\pi(r^2-x^2)dx}$
  
  ${V{=}2\int_{0}^{r}dV(x)}$
  (the "2" is out in front because we're using symmetry: we calculate the volume of half the sphere in running from 0 to r, then double it).
- Examples:
  - #48, p. 361
  - #52
  - #4 and 5, p. 360
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
  Now you made nice graphs: criticize my graphs in the Mathematica summary.
- 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 3-15, 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? 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.
  - 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 ${g(x)=x^3+x-2}$ ?
Section 6.2 summary

Website maintained by Andy Long. Comments appreciated.

independent	dependent
^oC	^oF
0	32
100	212