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Unformatted text preview: Math 016A Final Mar., 2009 Name: ID Signature Show all your work to get full credits! No Calculators allowed. 1. Evaluate the following limits and derivatives: (a)[16pts] lim x → 2 √ x 2 3 x + 4 √ 2 x 2 lim x →∞ 3 x 2 x + 2 (b)[16pts] d dx x 2 x + 1 ¶ d dx ‡ 2 x √ 16 x 2 · 1 2. (20pts) Suppose that a function y = f ( x ) satisfies the equation x 3 + xy 2 + tan( x + y ) = 2 x. Use implicit differentiation to compute the derivative dy dx . Furthermore, write down the equation of the tangent line to the above curve at point (0 , 0) . 2 3. (24pts)A tank of water in the shape of a cone is leaking water at a constant rate of 2 cubic ft/m. The base radius of the tank is 2 ft and the height of the tank is 5 ft. (a) At what rate is the depth of the water in the tank changing when the depth of the water is 2.5 ft? (b) At what rate is the radius of the top of the water in the tank changing when the depth of the water is 2.5 ft? (Your answer will involve the number π .) 2 ft 5 ft 3 4. Optimization problems.4....
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 Spring '08
 DUANEKOUBA
 Calculus, Derivative, Limits, Continuous function, Limit of a function, largest possible volume

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