- You have a homework due (2.5).
- There's a quiz at the end of the hour (chain rule, implicit differentiation)
- Please put your rectangles away.
- 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 (good) notation. Use sensible variable
names. Assign symbols to all quantities that are
functions of time (usually time will be our
independent variable).
- 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.
- Don't forget your units.
"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 (basic example, no chain rule yet!).
- #6 (yep -- chain rule!)
- #23 (Key: similar triangles -- but this is a challenging one!)
- #31 (ladders! Pythagoras....)
- #34 (straightforward, implicit differentiation with chain rule)
- #37 (law of cosines)
- #38 looks like fun....
I've talked a bit about how the approximation $\sin(x) \approx x$ when $x$ is small is important in physics. That is mentioned in this section, which is one reason why I wanted to have a look at 2.9:
More generally, we frequently use the linear function representing the tangent line at a point to make certain approximations.
That linear function is called the linearization of $f$ at $a$:
- For example, suppose you need to compute $\sqrt{15.96}$. Now the
easy way is to use your calculator, but the cowboy way would be to use
the linearization! Let's have a go at that one.
- We know the square root of 16, which is close: our answer should be near 4.
- Draw a picture! The picture will demonstrate two different
types of quantities that we discuss in this section:
differentials $dy$, and increments $\Delta y$.
Differentials versus increments
The increment is the true change in the function value; the differential approximates the true change: One way to remember the difference:- Differentials $dy$ follow the direction suggested by the derivative
- Increments $\Delta y$ are actual changes in the function value.
$dy \approx \Delta y$
We want the increment, but may settle for the easily computed differential.
- We use the tangent line for our approximation, and, when we're done, we can ask "is our answer an over- or an under-estimate?"
- We know the square root of 16, which is close: our answer should be near 4.
- Examples
- #2, p. 187
- #5
- #38