Pierre de Fermat

Exercises

1. 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.]

2. 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.]
3. Find the equation of the tangent line to the parabola which is the graph of y = x² + 2x at the point (2, 8) by computing a derivative.
4. At what points on the graph of the equation y = 2x³ + 3x² - 6x + 1 is the tangent line horizontal?

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