Workshop 10 1. Find all critical numbers of each function (on the given interval, if an interval is indicated). Then use the Second Derivaitve Test to classify the critical numbers. (a) f (x) = sin2 x + cos x, on the interval (0, π) (b) g(x) = e−x − 4e−2x 2. Let f (x) = x3/2 on the domain [0, 9]. (a) Use www.desmos.com, or a graphing calculator to sketch the graph of f (x). (b) Suppose we want to find the point on the graph of f (x) that is closest to the point (4, 0). What quantity do we need to minimize to find this point? Write an appropriate objective function for this quantity. (Recall that the objective function is the function on which we perform the optimization procedure.) (c) Use the optimization procedure on the function you found in part (b) to find the point on the graph of f (x) that is closest to the point (4, 0). 3. Sketch the graph of a function f that has the properties listed below. Identify in your graph any local extrema, inflection points, and asymptotes. • The domain of f is (−∞, ∞) • f is continuous on its entire domain • f 0 (x) < 0 for −∞ < x < −2 • f is not differentiable at x = −2 • f 0 (x) > 0 for −2 < x < 3 • f is concave down for −2 < x < 5 • f 0 (3) = 0 • lim f (x) = 1 x→∞ 4. The Democratic People’s Dictatorship of East Pisces (DPDEP) — a major exporter of caviar — is trying to decide how many pounds of caviar to produce per year to maximize its revenue. (Revenue is the amount of money a seller takes in; it is the total amount of goods sold times the price of the goods.) If DPDEP produces no caviar at all, then caviar will sell for $10,000 per pound on the world market. (Other countries also produce caviar, so those countries would be able to sell their caviar for $10,000 per pound.) When DPDEP produces x pounds of caviar per year, the price of caviar drops by 0.1x2 dollars per pound. DPDEP can produce up to 350 pounds of caviar per year. How many pounds of caviar should it produce per year to maximize its revenue? Note: If you haven’t taken economics, the basic principle is this: When a large producer “floods the market” by producing a lot of some product, the price of that product drops.

Find All Critical Numbers Workshop 10

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