- Your quiz is returned.
- Some of you have the quotient rule backwards
- $u'(v(x))v'(x)$
- $f(x)=x^3$, $g(x)=x\sin(x)$
- $f(x)=\sin(x)$, $g(x)=x^3\cos(x)$
- You have some chain rule practice problems due to hand in today.
- 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:
- Read the problem carefully.
- Draw a diagram if possible.
- Introduce notation. Assign symbols to all quantities that
are functions of time.
- Express the given information and the required rate in
terms of derivatives.
- 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).
- Use the Chain Rule to differentiate both sides of the
equation with respect to t.
- 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.
- Read the problem carefully.
- Examples:
- #4, p. 180
- #6
- #13 (similar triangles)
- #31 (ladders! Pythagoras....)
- #34
- #37 (law of cosines)
- 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?
- Maxima and minima are both types of extrema. Our author
begins by distinguishing between global (or
absolute) extrema and local extrema:
- Local is the biggest thing in the neighborhood;
- Global is the biggest thing anywhere!
- Take a moment and sketch a continuous function on the
interval [-2,2]. Make it look kind of interesting -- maybe even
ugly.
- Does your function have a global maximum and a global minimum?
- Does your function have local extrema which are not global extrema?
- If your function does have extrema, how would you characterize the derivative of the function at those extrema?
- By the way, our author requires that the function be defined in an
open interval about a local extremum. So in this figure from p. 199,
the right endpoint is not considered a local maximum:
- 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?
- Extreme Value Theorem
- Fermat's Theorem
- Find the critical numbers for f (that is, those values of x at which the derivative is zero or doesn't exist).
- Find the values of f at the critical numbers of f in (a,b).
- Find the values of f at the endpoints of the interval (at a and b).
- The largest and smallest of the values from steps 1 and 2 are the absolute or global extrema.
- #31, p. 205 (and let's decide whether they're local maxes or mins)
- #45 (closed interval method)
- #63