Calculus II PLTL Fall 2014 Worksheet 4 These problems are to be done without the use of a calculator unless otherwise specified 1 Round Robin Reviewing the Product Quotient and Chain Rule for taking derivatives is useful since various integration methods require the use of these rules during the solving process Take the derivative of each function below and simplify a a t t 1 t2 1 b b u 1 u 2 u 2 c c v tan v 1 v 1 d 1 d w 1 sin w 1 ln w 3 2 Scribe The half life of C 14 is considered to be 5730 years a The decay equation for a radioactive element is y y0 e kt where y0 is the amount of radioactive material present at time zero and dy dt ky k 0 Explain how the half life of an element is related to the value of k b Suppose that of the original amount of C 14 in a human bone uncovered in Kenya only 10 remains today How long ago did death occur c A skull found in Kenya is reputed to be 1 800 000 years old Show that the percentage of C 14 remaining now would be negligible and so in this instance dating by means of C 14 would be meaningless 3 Pairs Use the method of substitution to calculate the following integrals R 1 a x ln x dx R sec2 x b tan x dx R x3 c x4 1 dx R ln x d dx x ln2 x 1 e What do all of the integrals have in common Make up a similar problem that is an integral that will share the same property as these 4 Round Robin Chris s car engine runs at 100 C On a day when the outside temperature is 21 C he turns off the ignition and notes that five minutes later the engine has cooled to 70 a Determine the engine s cooling constant k b What is the formula for y t the temperature of the engine at time t c When will the engine cool to 40 5 Scribe A spherical tank of radius 10 meters with a small hole at the top is filled with water How much work against gravity is done pumping the water out through the hole Note The density of water is 1000 kg m3
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