Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Find two or three classmates and a few feet of chalkboard Be sure to discuss how to solve the exercises how you get the solution is much more important than whether you get the solution If as a group you agree that you all understand a certain type of exercise move on to later problems You are not expected to solve all the exercises some are very hard Many of the exercises are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart these are marked with an Others are my own are from the mathematical folklore or are independently marked Here s a hint drawing pictures e g sketching graphs of functions will always make the problem easier Word Problems 1 When a raindrop falls it increases in size and so its mass m at time t is a function of t m m t The rate of growth of the mass is km t for some positive constant k Newton s Laws specify moreover that mv 0 gm where v v t is the velocity of the raindrop directed downward and g is the acceleration due to gravity Find an expression for the terminal velocity lim v t in terms of g and k t 2 When an object moves through a fluid the friction force on the object is generally a function of the velocity of the object When the object moves slowly a very good approximation is that the force is proportional to the velocity F av for some constant a When the object moves quickly a very good approximation is that the force is proportional to the square of the velocity F bv 2 Recall that an object of mass m moving under a force F experiences an acceleration change in velocity following the formula v 0 F m a Use physical intuition to explain why a 0 for any value of v and why b 0 for positive v and b 0 for negative v b Marbles moving through honey and car shock absorbers moving through oil are good examples of the viscous approximation F av Assume that an object with mass m starts out moving with velocity v0 slows down due to viscous friction and has no other forces acting on it Find its velocity as a function of time c Baseballs flying through the air are good examples of the turbulent approximation F bv 2 Assume that an object with mass m starts out moving with velocity v0 slows down due to turbulent friction and has no other forces acting on it Find its velocity as a function of time d Falling objects experience in addition to whatever friction forces surround them a constant downward acceleration g due to gravity Find differential equations describing the velocity of an object of mass m moving under gravity and a i viscous ii turbulent friction force e Solve the two differential equations from part d f The terminal velocity of a falling object is its limit lim v t Find the terminal velocity t of an object falling in the i viscous and ii turbulent regimes 1 3 The Pacific halibut fishery has been modeled by the differential equation dy y ky 1 dt K where y t is the biomass the total mass of the members of the population in at time t the carrying capacity is estimated to be K 8 107 kg and the relative growth rate is k 0 71 per year a Based only on the type of differential equation what can you say about the population of halibut b What are the equilibrium solutions Which solutions are stable and which are unstable If the pacific fishery starts with some positive amount of halibut and if the differential equation is perfectly satisfied what will be the long term population y limt y t Hint solve the problem in terms of the variables k and K and only substitute in numbers at the very end c If the biomass of halibut starts at 2 107 kg find the biomass a year later d If the biomass of halibut starts at 2 107 kg how long will it take for the biomass to double 4 a Consider the Pacific halibut fishery in the previous problem If fishers harvest a biomass of L halibut per year explain why dy y ky 1 L dt K is a reasonable model for the population of halibut For this problem don t substitute numbers until the very end work in terms of the unknown constants k K and L b What are the equilibrium populations Are they stable or unstable Explain qualitatively what will happen to the long term fish population c Explain how your answers to the previous question depend on the size of L In particular what amount of harvesting is too much in the sense that there is no sustainable population of halibut if fishers harvest that much fish per year P 5 Show that if P satisfies the logistic equation P 0 kP 1 K then d2 P P 2P k2 P 1 1 2 dt K K Deduce that a population grows fastest when it reaches half its carrying capacity 6 The doomsday equation for population growth is y 0 ky 1 c where k and c are positive constants Solve the doomsday equation with initial condition y y0 where y0 0 Show that there is a finite time t T doomsday such that lim y t t T 7 Find all functions f such that f 0 is continuous and Z x 2 f x 100 f t 2 f 0 t 2 dt 0 for all real x 8 Let f be a function with the property that f 0 1 f 0 0 1 and f a b f a f b for all real numbers a and b Show that f 0 x f x for all x and deduce that f x ex 2
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