Problem Six: 2000-2001
Calculus I (AB): A Beguiling Derivative
If 
,
what is the derivative of f when x = 1?
Difficulty Rating: 
Solution:
This function is a quotient. Therefore, you have
to use the quotient rule. In words, this means take the "bottom" (denominator)
times the derivative of the "top" (numerator) minus the top times the derivative
of the bottom, all divided by the bottom squared. Notice that you have
to use the product rule to find the derivative of the numerator (and the
chain rule as well), and you have to use the chain rule to find the derivative
of the denominator. If you simplify this much further, you are braver
than I. Remember that the free response questions on the AP test will not
require you to simplify.
That is one butt ugly derivative.
Calculus II (BC): Related Rates--Yankees
Playoff Version
(Artwork by Colleen...she won the graphics design contest for this
problem.)
Click here for Yankee Magic Sounds!
Difficulty Rating:
Solution:
Begin by drawing a picture of the situation.
The base paths from first to second and second to third are
perpendicular, creating a right triangle whose vertices are first base,
second base, and Derek's current position on his way to third. The rate
of change of x is 19, whereas the base path from first to second
will
not change during this problem; therefore, dy/dt = 0. In order
to find the length of d, use the Pythagorean Theorem. In addition,
you should use the Pythagorean Theorem to set up this related rates problem
and find the derivative with respect to t. Once you have derived,
plug in what you know and solve for dd/dt:
Make sure you label your units. As this is a rate in feet,
the rate of change is ft/sec.
