Pierre de Fermat
Exercises

An optimization problem: A gardener
has 120 ft of fencing to enclose two adjoining rectangular plots of land
as in the diagram below. What dimensions should the garden be to
maximize the enclosed area? [Hint: if x and y are the
unknown dimensions, write equations that represent the total length of
the fencing and the area of the garden; solve for y in the first
and substitute into the second to obtain a quadratic function in x
which you must maximize.]

Another optimization problem: A producer
of novelty items can sell 1000 items per week at a price of $5 per item,
and guesses that for every $.10 drop in the price, it can sell 100 more
items weekly. The company has fixed overhead costs of $1050 associated
with manufacture and distribution in addition to costs of $1.10 per unit
in materials. How many units should it produce and sell to maximize
its profit? [Hint: revenue = price per item times the number of units
produced (and sold), and profit = revenue minus cost.]

Find the equation of the tangent line
to the parabola which is the graph of y = x^{2} +
2x at the point (2, 8) by computing a derivative.

At what points on the graph of the
equation y = 2x^{3} + 3x^{2} 
6x + 1 is the tangent line horizontal?
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last modified 12/8/00
Copyright (c) 2000. Daniel E. Otero