Day 44, Mat128: Calculus A

Last time:

Next time:

Today:

Announcements
- Here are the notes from last time: related rates
- No quiz today: we're reviewing for the final.
- Your final (which is Wednesday, 12:20-2:20) is comprehensive, and will be balanced between
  - 1/4 new material
  - 1/4 Exam 3 material (key)
  - 1/4 Exam 2 material (key)
  - 1/4 Exam 1 material (key)
  Of course calculus builds on itself pretty well, so, in some sense I could just give you a bunch of story problems -- but I've got a feeling that you'd rebel against that!:)
  You may use your notes, although I'd create a short version of things you're worried about.
- Solutions for all of our recent assignments are available for you to use to check your work:
  - Worksheets:
  - Homeworks:
    - HW 3.3 (1-4 at the end)
    - HW 3.4 (all nine)
    - HW 3.5 (2-6 at the end -- with Mathematica)
  - Other Problems:
    - solution to AC3.4.9
- I'll be around next week for my regular office hours, and class times, in case you want to come by for some help on anything. Best to give me a heads-up, however, just to make sure that we connect.
  I'm also available on Zoom, if those times don't work for you.
Last time:
- We went over the last quiz.
- We covered section 3.5: Related rates, and did a couple of problems.
Today: We review
- Linear things -- especially point-slope form: \[ y-y_0=m(x-x_0) \hspace{1in} \text{ or } \hspace{1in} y=y_0+m(x-x_0) \]
- slope = \(\frac{rise}{run}.\)
- Secant lines and slopes and limits: average rate of change
- Indeterminate things
- Continuity becomes essential
- Limit definition of the derivative at a point \(x=a\): slope of the tangent line there, and instantaneous rate of change.
- Limit definition of the derivative at \(x\): a function, giving the tangent line everywhere possible.
- Limit laws behave pretty much as you'd hope:
  - limit of sums is sum of limits
  - limit of products is product of limits
  - limit of quotients is quotient of limits
- Limits of continuous functions are especially nice. Lots of functions are continuous on their domains: polynomials, rational functions, trig functions, exponential functions, power functions,....
- Local linearization (point-slope form -- none other will do!)
- We leaned on the limit definition to create rules that made differentiation easier:
  - Sum and difference rules
  - product rule (this rule does not behave as we would hope)
  - constant multiple rule rule (corollary of product rule)
  - quotient rule (this rule does not behave as we would hope)
  - power rule (proven using dominoes, but only for integer powers; works for all, however)
  - chain rule (the major powerhouse! Allows for recursive derivative-discovery)
  - Derivatives of
    - Trig functions
    - Exponential functions
    - Inverse functions
  - By the way: with derivatives of exponentials, their inverses the logs, and the chain rule, we can return to prove that the power rule works for all powers.
- The sign of the derivative tells us when its function is increasing or decreasing -- which also tells us when it hits a peak or a pit (where the derivative is zero or DNE). This is the heart of the first derivative test.
- Second derivatives tell us about acceleration, rate of change of the derivative: so they inform us about the concavity of the original function. Poster children: \(x^2\) and \(-x^2\).
- Sign of the second derivative tells us if a function is concave up (plus) or concave down (minus); this is the heart of the second derivative test: concave up (bowl) means minimum; concave down (umbrella) means maximum.
- Zeros of the second derivative help identify inflection points.
- Implicitly defined functions often have graphs failing the vertical line test, but we can still compute their derivatives at points along their curves, and so construct tangent lines and local linearizations.
- Extrema Value Theorem: if a function is continuous on a closed and bounded domain, it has both a global max and global min on that interval. In most real problems, we're looking for global extrema.
- An example is linear regression, where we are trying to find the best fit line through a cloud of data. The example we considered was that of Elvis, the dog known as the King of Calculus, and a model of the form \(y=mx\), where are job was to find the best-fitting slope \(m\) to Elvis's empirical data, to see how close he gets to his theoretical best-fit line. (He got an A/B....)
- Story problems. Use UPCE: understand (draw! doodle!), plan, carry out, evaluate. Make sure that you give them what they asked for. We rely on all the math that you've learned up to this point: Pythagorean theorem, the dirt formula (d=rt), formulas for familiar geometric objects (e.g. squares, circles, spheres, etc.), similar triangles, trigonometric relationships, etc.
  Last night was a story problem fest for me, as I did the problems from section 3.4, and 3.5, and posted those with links above.
- Related rates is the subject of section 3.5, and there we are trying to find out how the rate of one thing affects the rate of another. An example is how the rate of administration of a drip IV affects a heart rate, or the rate of fever reduction, or.... But we did more banal examples, involving gravel, or balloons...:)
Links:
- Do Dogs Know Calculus?

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Footnotes