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 Introduce yourself to your new friends and write all of your names at the top of the chalkboard As a group try your hand at the following exercises 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 Improper Integrals infinite domains Occasionally and especially in physics problems one wants to compute integrals over infinite domains In these cases the integral might be infinity but it might be finite Usually you can evaluate improper integrals e g an integral over the domain 1 just like any other integral find an antiderivative and then plug in the endpoints Of course is not a number so plugging it in takes some skills You learned these skills in 1A really means a limit When R you have to be careful is for domains that are infinite in both directions In a situation like it s important to remember that they are different s 1 To move an object against a force requires energy also known as work If the object moves in one dimension from point a to point b against a force field F x then the amount of work Rb required is W a b a F x dx An object of mass m e g a spaceship is at distance R e g the radius of the Earth from a gravitating body e g the Earth of mass M The force of gravity on the object when it is at distance x is F x GmM x2 where G is a physical constant that is there only because humans don t work in units natural for doing gravitational physics we work in units natural for everyday life instead a Find the work required to move the object from its current positing R to a position infinitely far away from the planet b Remember that the kinetic energy of an object of mass m and velocity v is mv 2 2 For what v is the kinetic energy enough for the object to escape the gravitational pull Recall that the acceleration due to gravity is g M G R2 write your answer v as a function of g and R The velocity v is called the escape velocity at radius R Incidentally the orbital velocity at radius R is v gR How do the orbital and escape velocities compare 2 The average speed of molecules in an ideal gas is 4 v M 2RT 3 2 Z 0 1 v 3 e M v 2 2RT dv where M is the molecular weight of the gas T the temperature and R is the ideal gas constant Find v Hint first perform a u substitution R R 3 Find all values of p for which 1 xp dx converges If xp dx converges to what does it converge Your answer should of course be a function of p R 4 Let n be a nonnegative integer Using integration by parts find 0 xn e x dx 5 The Laplace Transform of a function f t is the function of s given by Z f t e st dt L f s 0 if this integral converges Find the Laplace transforms L 1 s L t s L et s and L tn s The last one requires your answer to the previous question What are the domains of these functions for what s values do the integrals converge Improper Integrals infinite discontinuities If a function has a finite discontinuity integrating it is no problem you break up the integral into pieces But an infinite discontinuity can be deadly Again the answer to defining such integrals requires a limit In general if a function f x on a b is continuous except for an infinite discontinuity at c then we define Z b Z f x dx lim a s c a s Z f x dx lim t c t b f x dx The left hand side is only defined if each limit on the right converges independently Otherwise we say that the integral diverges R1 R1 1 Find all values of p for which 0 xp dx converges If 0 xp dx converges to what does it converge answer should of course be a function of p Are there any values of p for R Your p which 0 x dx converges R 2 Is 0 tan x dx well defined i e does the integral converge If so to what R1 3 Show that 0 ln x dx converges and find the limit More generally use integration by parts R1 to find 0 ln x n dx for any nonnegative integer n 2
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