Day 19, Mat129: Calculus I

Last time:

Next time:

Today:

Announcements
- We'll have a short quiz at the end of the hour.
- We won't meet in the lab on Friday. I need to use that time to cover some more material.
- Homework returned: pp. 146--, graded #6, 23, 48
  - #6: Show the derivative of secant....
  - #23: make sure that your slope in a tangent line is a number, not a function of $x$
  - #48: some of you were very sneaky and used L'Hopital's rule. Not yet! At this point, we use an even more important idea about the sine function, and its behavior at $x=0$....
- You have some chain rule practice problems due to hand in on Friday.
- I've also assigned some IMath exercises -- keep up!
- Reminder MATHEMATICS & STATISTICS Major/Minor Lunch
  Friday 02/26/16
  11:30a.m.-1:30p.m.
  MEP 4th floor
  All Mathematics and Statistics Majors and Minors are Welcome!
Last time we wrapped up the chain rule, introduced Section 2.6: Implicit Differentiation (which we need for related rates problems), and we started related rates.
For another example of functions defined implicitly, think of the ideal gas law: $P$ is the pressure, $V$ is the volume, $T$ is the temperature, and $c$ is a positive constant: \[ P=c\ \frac{T}{V} \] So we can think of pressure as a function of temperature and volume. But we could just as easily write \[ V=c\ \frac{T}{P} \] Alternatively, we could just say that \[ PV-c T=0 \] The three are related, and each can be thought of as an implicit function of the others.
You can make any explicit equation $y=f(x)$ into an implicit equation just as easily: \[ y-f(x)=0 \] Then, if we differentiate with respect to $x$, we get \[ y'(x)-f'(x)=0 \] or \[ y'(x)=f'(x) \] (which we knew!:).
Today we're going to do a few more problems from Section 2.8: related rates before moving on to max/mins.
- In calculus you encounter the famous ladder problem. Example 2 on p. 177. We wish to establish the rate of one thing, relative to the rate of the other. These are so-called "related rates". So the ladder's top rung is falling vertically at one speed, as the feet are slipping out at another. Speeds are, in this case, rates of change of positions.
- The problems considered here are the fun ones, the story problems:
- Everyone loves story problems, right?! Let's go over the strategy, and then try some.
  I jokingly use the acronym "UPCE" (oopsie!) for the general problem solving strategy:
  - Understand
  - Plan
  - Carry out
  - Evaluate
- Here is our author's more detailed approach to these related rates problems:
  1. Read the problem carefully.
  2. Draw a diagram if possible.
  3. Introduce notation. Assign symbols to all quantities that are functions of time.
  4. Express the given information and the required rate in terms of derivatives.
  5. Write an equation that relates the various quantities of the problem. If necessary, use the geometry of the situation to eliminate one of the variables by substitution (as in Example 3).
  6. Use the Chain Rule to differentiate both sides of the equation with respect to t.
  7. Substitute the given information into the resulting equation and solve for the unknown rate.
  "Warning: a common error is to substitute the given numerical information (for quantities that vary with time) too early." (p. 179). Substitute only after the differentiation is complete.
- Examples:
  - #4, p. 180
  - #6
  - #13 (similar triangles)
  - #31 (ladders! Pythagoras....)
  - #34
  - #37 (law of cosines)
  - #38 looks like fun....
Today we begin talking about section 3.1: Maximum and Minimum Values.
- Our author begins by distinguishing between global extrema and local extrema:
  - What do you imagine is the difference?
  - Why do you suppose that we care?
- Take a moment and sketch a continuous function on the interval [-2,2]. Make it look kind of interesting -- maybe even ugly.
  Now, answer these questions:
  1. does your function have a global maximum and a global minimum?
  2. does your function have local extrema which are not global extrema?
  3. If your function does have extrema, how would you characterize the derivative of the function at those extrema?
- Now I want you to sketch a function defined on the interval [-2,2] that does not have any global extrema. What is true about your function?
- Now I want you to sketch a smooth function on [-2,2] that has a value of x=a where $f'(a)=0$ but no extremum there. What is true about your function?
By now we should have a good idea of what these two theorems say:
- Extreme Value Theorem
- Fermat's Theorem
We have hopefully deduced these. We might also have deduced the closed interval method, to find the extrema of a function on a closed interval [a,b]:
1. Find the critical numbers for f (that is, those values of x at which the derivative is zero or doesn't exist).
2. Find the values of f at the critical numbers of f in (a,b).
3. Find the values of f at the endpoints of the interval (at a and b).
4. The largest and smallest of the values from steps 1 and 2 are the absolute or global extrema.
Example application of extrema:
- The Chickens of Manitoba
- How was the flock egg production over time?
- How long should we let the flock live to maximize the average daily return?
Now let's look at some problems....
- #5, p. 204
- #31, p. 205 (and let's decide whether they're local maxes or mins)
- #45
- #63

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