Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Find two or three classmates and a few feet of chalkboard. Introduce yourself to your newfriends, and write all of your names at the top of the chalkboard. As a group, try your hand atthe following exercises. Be sure to discuss how to solve the exercises — how you get the solutionis much more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe 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.Improper Integrals: infinite domainsOccasionally, and especially in physics problems, one wants to compute integrals over infinitedomains. In these cases, the integral might be “infinity”, but it might be finite. Usually, you canevaluate “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 pluggingit in takes some skills. You learned these skills in 1A — “∞” really means a limit.When you have to be careful is for domains that are infinite in both directions. In a situationlikeR∞−∞, 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 movesin one dimension from point a to point b against a force field F (x), then the amount of workrequired is W (a, b) =RbaF (x) dx.An object of mass m (e.g. a spaceship) is at distance R (e.g. the radius of the Earth) from agravitating body (e.g. the Earth) of mass M. The force of gravity on the object when it isat distance x is F (x) = GmM/x2, where G is a physical constant that is there only becausehumans don’t work in units natural for doing gravitational physics (we work in units naturalfor everyday life instead).(a) Find the work required to move the object from its current positing R to a positioninfinitely far away from the planet.(b) Remember that the kinetic energy of an object of mass m and velocity v is mv2/2.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 afunction 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 escapevelocities compare?2. § The average speed of molecules in an ideal gas is¯v =4√πM2RT3/2Z∞0v3e−Mv2/(2RT )dv1where M is the molecular weight of the gas, T the temperature, and R is the ideal gasconstant. Find ¯v. (Hint: first perform a u-substitution.)3. Find all values of p for whichR∞1xpdx converges. IfRxpdx converges, to what does itconverge? (Your answer should, of course, be a function of p.)4. Let n be a nonnegative integer. Using integration by parts, findR∞0xne−xdx.5. § The “Laplace Transform” of a function f (t), is the function of s given byL[f](s) =Z∞0f(t) e−stdtif 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 thesefunctions (for what s values do the integrals converge)?Improper Integrals: infinite discontinuitiesIf a function has a finite discontinuity, integrating it is no problem: you break up the integralinto pieces. But an infinite discontinuity can be deadly. Again, the answer to defining suchintegrals requires a limit. In general, if a function f(x) on [a, b] is continuous except for an infinitediscontinuity at c, then we defineZbaf(x) dx = lims%cZsaf(x) dx + limt&cZbtf(x) dx.The left-hand side is only defined if each limit on the right converges independently. Otherwise wesay that the integral “diverges”.1. Find all values of p for whichR10xpdx converges. IfR10xpdx converges, to what does itconverge? (Your answer should, of course, be a function of p.) Are there any values of p forwhichR∞0xpdx converges?2. IsRπ0tan x dx well-defined (i.e. does the integral converge)? If so, to what?3. Show thatR10ln x dx converges, and find the limit. More generally, use integration by partsto findR10(ln x)ndx for any nonnegative integer
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