A
microscopic bug is traveling along the graph of x2 + 4y2
= 25. (Assume distances are measured in centimeters, so the ellipse
is 10 cm wide and 5 cm high, and times are measured in seconds.) At time
t=0,
the bug is at the point (3,-2), and traveling so that its x-coordinate
is increasing at the rate of 1/16 cm/sec. How fast is its y-coordinate
changing at that instant?
There
are lots of reasonable strategies for attacking this question. Write down
at least one before you read the suggestions below.
1. Solve
for y in terms of x.
(The calculator
will give you more than one choice; make sure you use the right one.)
Then compute
the derivative of y with respect
to x when x = 3.
That gives
you the slope of the line tangent to the ellipse at the point (3,-2).
Then if
you know how fast the x-coordinate is changing, common sense (or the chain
rule) will tell you how fast the y-coordinate is changing.
2.
Use implicit differentiation to find the slope
of the tangent line, and then proceed as above.
3. Use a
linear approximation for the bug's x-coordinate: x = 3 + t/16.
Substitute
that expression for x in the equation of the ellipse, then solve
for y and compute dy/dt directly.
(You'll
still get two choices when you solve for y.)
Does it
matter that the x-coordinate is only approximately 3 + t/16?
4.
You can substitute x = 3 + t/16 and use implicit
differentiation to find dy/dt.
5. (My favorite.)
Use linear approximations for both the bug's x-coordinate and its y-coordinate:
Since we
don't know dy/dt, use some unknown slope, like m.
Then x
= 3 + t/16 and
y = -2 + m*t.
Substitute
these expressions for x and y in the equation for the ellipse
and solve for m.
(It's quickest
to differentiate the equation with respect
to t first.)
When you
solve for m, does the answer have t's in it?
What value
of t is the important one?
Question:
Do
you get the same answers if the bug is traveling along the tangent line
to the ellipse instead of along the ellipse itself? Does that make sense?
This type of problem is called a related rates problem, because the rate of change of x and the rate of change of y are related.
The traditional solution with pencil and paper requires that you differentiate the equation x2 + 4y2 = 25 with respect to t, keeping in mind that x and y both depend on t. You'll get an equation involving x, y, dx/dt, and dy/dt. Then substitute in numbers for the values you know.
This is exactly
the same as method #5 above, except in #5 we substitute
values before we differentiate instead
of afterwards (mostly because the calculator will do that more gracefully
than substituting afterwards.)
Exercises: 1. Apply each of these strategies to the bug problem.
2. Apply (more than one of) these strategies to related rates problems in the textbook. Make sure only two quantities in each problem are changing.