Calculus II PLTLFall 2014Worksheet 4These problems are to be do ne without the use of a calculator unless otherwisespecified.1) (Round Robin) Reviewing the Product, Quo tient and Chain Rule for taking derivativesis useful since var ious integration methods require the use of these rules d uring the solvingprocess. 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) d(w ) = [(1 + sin(w))/(1 − ln(w))]132) (Scribe) The half-life of C14is considered to be 5730 years.(a) The decay equation for a radioactive element is y = y0e−kt, where y0is theamount of radioactive material present at time zero and dy/dt = −ky, k > 0 . Explain howthe half-life of an element is related to the value of k.(b) Suppose that of the original amount of C14in a human bone uncovered in Kenya,only 10% remains today. How lon g ago did death occur?(c) A skull found in Kenya is reputed to be 1, 800, 000 years old. Show that thepercentage of C14remaining now would be negligible, and so in this ins tan ce, dating bymeans of C14would be meaningless.3) (Pairs) Use the method of s ubstitution to calculate the following integrals.(a)R1x ln xdx(b)Rsec2xtan xdx(c)Rx3x4+1dx(d)Rln(x)x(ln2x+1)dx(e) What do all of the integrals h ave in common? Make up a similar p roblem, thatis, 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 outsidetemperature is 21◦C, he turns off the igni tion and notes that fi ve minutes later, the enginehas 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 sp herical tank of radius 10 m eters with a small hole at the top is filledwith water. How much work (against gravity) is do ne pumping the water out through thehole?Note: The density of water is 1000
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