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.Separable Differential EquationsIf a differential equationdydx= F (x, y) can be written as a productdydx= f(x)g(y), wheref(x) does not depend on y and g(y) does not depend on x, then the solutions can be found bycross-multiplying and integrating:R(g(y))−1dy =Rf(x)dx. Don’t forget to add a constant ofintegration, and then solve for y in terms of x if you can.1. § Solve the differential equation:(a)dydx=yx(b)dydx=√xey(c) (x2+ 1)y0= xy(d) y0= y2sin x (e) (1 + tan y)y0= x2+ 1 (f)dudr=1 +√r1 +√u2. § Solve the initial value problem:(a)dudt=2t + sec2t2u, u(0) = −5 (b) xy0+ y = y2, y(1) = −1(c) y0tan x = a + y, y(π/3) = a, 0 < x < π/2 (d)dLdt= kL2ln t, L(1) = −1In (c) and (d), a and k are constants.3. § Find an equation of the curve that passes through the point (0, 1) and whose slope at (x, y)is xy.4. § Recall that two lines are perpendicular if their slopes are negative reciprocals. Thus, twocurves y = f(x) and y = g(x) that intersect at the point (x, y) are perpendicular at thatpoint if and only if f0(x) = −(g0(x))−1. Thus, for each of the following families of curves,write a differential equation that a curve must satisfy in order to be perpendicular at everyintersection with every member of the family, and then solve the corresponding differentialequation:(a) x2+ 2y2= k2(b) y =kx(c) y =x1 + kx5. Let n be an arbitrary number. Find the family of curves perpendicular to the family y = kxn.For a few different values of n, graph several members of each family.16. §When chemists consider a reaction, say A+B → C, they write [A] (etc.) for the concentrationat time t of reactant A. For example, consider the reaction H2+ B2→ 2HBr. Experimentsshow that this reaction satisfies the rate lawd[HBr]dt= k[H2][Br2]1/2when [HBr] is not too big.(a) In mathematics notation, let x be the concentration of HBr, which we assume to startat x(0) = 0. If a and b are the initial concentrations of H2and Br2, explain why theabove equation is equivalent to:dxdt= k(a − x)√b − x(b) Assume that a = b. Find x(t) such that x(0) = 0.(c) Assume that a > b. The last step — solving the algebraic equation to get x as a functionof t — is very hard. Find an equation expressing t as a function of x. Hint: substituteu =√b − x.7. § A sphere with radius 1 m has temperature 15◦C. It lies inside a concentric sphere withradius 2 m and temperature 25◦C. The temperature T (r) at a distance r from the commoncenter of the spheres satisfies the differential equationd2Tdr2+2rdTdr= 0Solve this boundary value problem, i.e. find a function T (r) satisfying the above equationwith T (1) and T (2) the prescribed amounts. Hint: first find the general form, by solving thefirst-order differential equation in S = dT/dr.8. § The air in a room with volume 180 m3contains 0.15% carbon dioxide initially. Fresher airwith only 0.05% carbon dioxide flows into the room at a rate of 2 m3/min and the mixed airflows out at the same rate. Find the percentage of carbon dioxide in the room as a functionof time. What happens in the long run?9. § 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 if km(t) for some positive constant k. Newton’sLaws specify, moreover, that (mv0) = 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.10. § Find all functions f such that f0is continuous and[f(x)]2= 100 +Zx0[f(t)]2+ [f0(t)]2dtfor all real x.11. § 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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