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Math 125 Carter Test 2 Fall 2009 General: Do all your work and write your answers inside the blue book. Do not write on the test. Write your name on only the outside of the blue book. There are problems on both sides of this page. Good luck! Remember that the Toxic Pets smash single “I’d Rather be Killing Communist Babies,” appeared in the credits of the movie “Eight Heads in a Duffle Bag,” but never was heard in the movie. 1. (10 points each) For each of the following, compute the derivative of y dy with respect to x. That is find y 0 , f 0 (x), or dx (Same things — different notation). Do not simplify your answer! (a) (b) (c) (d) f (x) = 1+sin (x) 1−sin (x) y = ecos (x 2 ) x2 + y 3 = 5 √ f (x) = t2 + 1 + arctan (t) 2. (10 points) Compute the equation of the line tangent to the curve at the particular point, (a, f (a)) where y = 4 ln (9x + 2) at a = 2. 3. (10 points) Sand falls into the shape of a conical pile at the rate of 2 m3 per minute. The proportion of the radius to the height is always 1/3. At what rate is the height increasing when the height is 5 meters? The volume of a cone is V = π3 r2 h where r denotes the radius and h denotes the height. 4. (10 points) A boat is pulled into a dock by a rope through a windlass that is 3 feet above the boat. The rope is being hauled in at 1 foot per second. At what rate is the boat approaching the dock when it is 4 feet from the dock? 1 5. (10 points each) Sketch the graphs of the functions given below. Indicate critical behavior and when feasible the inflection points. (a) (b) f (x) = (x − 1)(x − 2)(x − 5) f (x) = e−x 2 6. (10 points each) Determine the Maximum and minimum values of the functions on the given intervals. (a) f (x) = −x2 + 10x + 8 on [3, 6]. (b) f (θ) = cos (θ) + sin (θ) on [0, 2π]. 2