Math 1A Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Spring1A Find two or three classmates and a few feet of 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 in particular the last few exercises may be 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 or are independently marked Logarithms 1 Differentiate a ln 5 x b 1 ln t 1 ln t c 1 ln x r a2 x2 d ln a2 x2 2 e ln 1 ex f log2 e x cos x 2 Compute the derivative of ln xn in two different ways by using logarithm identities and by using the chain rule 3 Find y 0 if xy y x 4 Prove the product rule in the following manner a Let f and g be two functions and consider the function ln f x g x Use logarithm rules to rewrite it as the sum of two functions b Take derivatives of each side of your logarithm identity using the chain rule c Multiply both sides by the common denominator 5 Compute y 0 by first simplifying ln y and then using the fact that ln y 0 y 0 y where r 2 4 x 1 y x2 1 Word problems 6 If a stone is thrown vertically upward from the surface of the moon with a velocity of 10 m s its heigh in meters after t seconds is h 10t 83t2 a What is the velocity of the stone after 3 seconds b What is the velocity of the stone after it has risen 25 meters 7 a Sodium chlorate crystals are easy to grow in the shape of cubes by allowing the solution of water and NaClO3 to evaporate slowly If V is the volume of such a cube with side length x calculate dV dx when x 3 mm and explain its meaning b Show that the rate of change of the volume of such a cube with respect to its edge length i e dV dx is equal to half the surface area of the cube Explain geometrically why this is true 1 8 Boyle s Law states that when a sample of gas is compressed at constant temperature the product of the pressure and the volume remains the same P V C a Find the rate of change of volume with respect to pressure b A sample of gas is in a container at low pressure and is compressed steadily constant rate of increase in pressure at constant temperature for 10 minutes Is the volume decreasing more rapidly at the beginning or the end of the 10 minutes 9 Assume that in a particular reaction one molecule of the product C is formed from one molecule of the reactant A and one molecule of the reactant B and that the initial concentrations of A and B have the common value A B a moles per liter Let x be the concentration of molecule C Then a2 kt x akt 1 for some constant k a Explain how x relates to the concentrations of molecules A and B at time t b Find the rate of reaction at time t c Show that x satisfies the differential equation dx k a x 2 dt Show that the right hand side is proportional to the product of the concentrations at time t of molecules A and B d What happens to the concentrations of A B and C as t e What happens to the rate of reaction as t Questions on earlier material 10 What s the derivative of sin2 x What s the derivative of cos2 x What happens when you add them together and why 11 Find numbers A and B so that d Aex cos x Bex sin x ex cos x dx 12 Find numbers and so that y e x sin x is a solution to the differential equation y 00 4y 0 5y 0 Check that y e x cos x is also a solution 13 Show that every curve in the family y ax3 is orthogonal to every curve in the family x2 3y 2 b where a and b range over all real numbers 2
View Full Document
Unlocking...