Day 27, Mat129: Calculus I

Last time

Next time

Today:

Announcements:
- Your 3.3 homeworks are returned. Graded 10 and 28.
- Your quiz is returned.
- Let's have that next homework!
- Please put your rectangles away.
Last time: we worked more on section 3.5: how to graph a function.
- This section includes one new technique: the calculation of what are called "slant asymptotes" (which I've already mentioned in connection with the preceding section).
  This is just one type of asymptotic behavior that is sometimes useful. In fact, every rational function approaches a polynomial in its end behavior, so that we're interested here in those rational functions (and some others) which approach first degree (linear) polynomials as x gets large.
  E.g.,
  \[ f(x)=\frac{x^2-4x+17}{x-3} \]
  Even non-rational functions, such as
  \[ f(x)=3x-6 + \frac{sin(x)}{x} \]
  can approach linear functions.
- So here's the upshot on the graphing of functions, "the strategy":
  1. Domain -- find it.
  2. Symmetry -- hope for it!
    - even and odd functions;
    - functions that have symmetry about some displaced point;
    - periodicity.
    These help us reduce the amount of work we have to do
  3. Intercepts - crossings of the axes. The y-intercept is the point (0,f(0)) (if 0 is in the domain); the x-crossings are the roots of the equation f(x)=0: The special values of x that make this so.
  4. Asymptotes - vertical, horizontal, slant
  5. Intervals of increase or decrease - use the Increasing/Decreasing Test, based on the sign of the first derivative.
  6. Local maxima and minima - use the first or second derivative tests.
  7. Concavity and points of inflection - use the second derivative and the concavity test.
  8. Compute some points on the curve, especially any that are easy to calculate.
  9. Sketch the curve - sketch asymptotes as dashed lines; plot any known points on the curve (e.g. intercepts); finish by connecting the points in accord with the information above.
- Examples:
  - #50, p. 243
  - #58, p. 243
  - #23 (use completion of the square to get the TWO slant asymptotes)
Today we start a very important section 3.7 -- optimization problems. The best way to explain is to do some examples, but the point is that we're going to be looking for extrema of functions. There are thus two things to do:
1. Come up with the appropriate function, and then
2. find extrema in the ordinary ways.
As usual, I recommend my "UPCE" (oopsie!) strategy for solving story problems:
- Understand
- Plan
- Carry out
- Evaluate
(By the way: I say "my" strategy, but UPCE is basically the strategy of George Polya, who wrote How to Solve It -- a very famous book, and well worth the read. His "E" includes "Extend" -- can I solve more than this given problem?)
Examples:
- #15, p. 257
- #9
- #27 (what do you imagine the result might be?)
- #20

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