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Berkeley MATH 1B - Discussion Exercises

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Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Find two or three classmates and a few feet of chalkboard. Be sure to discuss how to solve theexercises — how you get the solution is much more important than whether you get the solution. Ifas 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 Berkeleyby James Stewart; these are marked with an §. Others are my own, are from the mathematicalfolklore, or are independently marked.Here’s a hint: drawing pictures — e.g. sketching graphs of functions — will always make theproblem easier.Word Problems1. § 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’sLaws 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 theterminal velocity limt→∞v(t) in terms of g and k.2. When an object moves through a fluid, the friction force on the object is generally a functionof the velocity of the object. When the object moves slowly, a very good approximation isthat the force is proportional to the velocity: F = av for some constant a. When the objectmoves quickly, a very good approximation is that the force is proportional to the square ofthe velocity: F = bv2. Recall that an object of mass m moving under a force F experiencesan acceleration — change in velocity — following the formula v0= F/m.(a) Use physical intuition to explain why a < 0 for any value of v , and why b < 0 for positivev and b > 0 for negative v.(b) Marbles moving through honey and car shock absorbers moving through oil are goodexamples of the viscous approximation F = av. Assume that an object with mass mstarts out moving with velocity v0, slows down due to viscous friction, and has no otherforces 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 =bv2. Assume that an object with mass m starts out moving with velocity v0, slows downdue to turbulent friction, and has no other forces acting on it. Find its velocity as afunction of time.(d) Falling objects experience, in addition to whatever friction forces surround them, aconstant downward acceleration g due to gravity. Find differential equations describingthe velocity of an object of mass m moving under gravity and a (i) viscous, (ii) turbulentfriction force.(e) Solve the two differential equations from part (d).(f) The terminal velocity of a falling object is its limit limt→∞v(t). Find the terminal velocityof an object falling in the (i) viscous and (ii) turbulent regimes.13. § The Pacific halibut fishery has been modeled by the differential equationdydt= ky1 −yKwhere y(t) is the biomass (the total mass of the members of the population) in at time t, thecarrying capacity is estimated to be K = 8 × 107kg, and the relative growth rate is k = 0.71per year.(a) Based only on the type of differential equation, what can you say about the populationof halibut?(b) What are the equilibrium solutions? Which solutions are stable, and which are un-stable? If the pacific fishery starts with some positive amount of halibut, and if thedifferential 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 onlysubstitute in numbers at the very end.(c) If the biomass of halibut starts at 2 × 107kg, find the biomass a year later.(d) If the biomass of halibut starts at 2 × 107kg, how long will it take for the biomass todouble?4. (a) Consider the Pacific halibut fishery in the previous problem. If fishers harvest a biomassof L halibut per year, explain whydydt= ky1 −yK− Lis a reasonable model for the population of halibut. For this problem, don’t substitutenumbers 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 qualita-tively 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 sustainablepopulation of halibut if fishers harvest that much fish per year?5. § Show that if P satisfies the logistic equation P0= kP (1 −PK), then:d2Pdt2= k2P1 −PK1 −2PKDeduce that a population grows fastest when it reaches half its carrying capacity.6. § The doomsday equation for population growth is y0= ky1+cwhere k and c are positiveconstants. Solve the doomsday equation with initial condition y = y0, where y0> 0. Showthat there is a finite time t = T (doomsday) such that limt→T−y(t) = ∞.7. § Find all functions f such that f0is continuous and[f(x)]2= 100 +Zx0[f(t)]2+ [f0(t)]2dtfor all real x.8. § Let f be a function with the property that f(0) = 1, f0(0) = 1, and f(a + b) = f(a)f (b) forall real numbers a and b. Show that f0(x) = f(x) for all x and deduce that f(x) =


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