01 May 398. A potato, initially at room temperature (70 degrees), is placed in a hot oven (350
398. A potato, initially at room temperature (70 degrees), is placed in a hot oven (350
degrees) for thirty minutes. After being taken out of the oven, the potato sits undisturbed for
thirty more minutes on a plate in the same room (70 degrees). Let F(t) be the temperature
of the potato at time t during the 60-minute time interval 0 ? t ? 60. On separate axes,
draw plausible graphs of both F and F0
. Other than F(0), you are not expected to know
any specific values of F.
399.With the help of the Chain Rule and Power Rule, show that the derivative of the
function R(t) = 1
g(t) = [g(t)]1
is R0
(t) =
g0
(t)
[g(t)]2 . Now try to find a general Quotient Rule
that deals with functions of the form Q(t) = f(t)
g(t)
. A good way to start is to rewrite the
preceding as Q(t) = f(t) · 1
g(t)
. Express your answer in the form numerator(t)
denominator(t) .
400. Consider the circle given by (x
1)2 + (y + 3)2 = 25. The slope of the tangent line at
a point (x, y) on this curve is a function of both x and y. Find a formula for this function,
and use it to find the tangent slope at (4, 1), (1, 8),
and (3,
0).
45
401. Derivatives can sometimes be found with little e?ort.
The bowl shown in the figure is 3 feet deep, and is situated
so that its octagonal opening is parallel to the ground. Its
volume is 125 cubic feet. The water in it is being siphoned
out at a steady rate of 20 cubic feet per minute. After t
minutes, let V be the volume of water in the bowl, y be the
depth of the water, and A be the area of th
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