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 Separable Differential Equations dy dy If a differential equation dx F x y can be written as a product dx f x g y where f x does not depend on y and g y does not depend on x then the solutions can be found by R R cross multiplying and integrating g y 1 dy f x dx Don t forget to add a constant of integration and then solve for y in terms of x if you can 1 Solve the differential equation dy y a dx x dy x b y dx e d y 0 y 2 sin x e 1 tan y y 0 x2 1 c x2 1 y 0 xy du 1 r f dr 1 u 2 Solve the initial value problem a 2t sec2 t du u 0 5 dt 2u b xy 0 y y 2 y 1 1 c y 0 tan x a y y 3 a 0 x 2 d dL kL2 ln t L 1 1 dt In 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 two curves y f x and y g x that intersect at the point x y are perpendicular at that point if and only if f 0 x g 0 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 every intersection with every member of the family and then solve the corresponding differential equation a x2 2y 2 k 2 b y k x c y x 1 kx 5 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 1 6 When chemists consider a reaction say A B C they write A etc for the concentration at time t of reactant A For example consider the reaction H2 B2 2HBr Experiments show that this reaction satisfies the rate law d HBr k H2 Br2 1 2 dt when HBr is not too big a In mathematics notation let x be the concentration of HBr which we assume to start at x 0 0 If a and b are the initial concentrations of H2 and Br2 explain why the above equation is equivalent to dx k a x b x dt 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 function of t is very hard Find an equation expressing t as a function of x Hint substitute u b x 7 A sphere with radius 1 m has temperature 15 C It lies inside a concentric sphere with radius 2 m and temperature 25 C The temperature T r at a distance r from the common center of the spheres satisfies the differential equation d2 T 2 dT 0 2 dr r dr Solve this boundary value problem i e find a function T r satisfying the above equation with T 1 and T 2 the prescribed amounts Hint first find the general form by solving the first order differential equation in S dT dr 8 The air in a room with volume 180 m3 contains 0 15 carbon dioxide initially Fresher air with only 0 05 carbon dioxide flows into the room at a rate of 2 m3 min and the mixed air flows out at the same rate Find the percentage of carbon dioxide in the room as a function of 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 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 10 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 11 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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