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 The Indefinite Integral R 0 0 R b f x dx 1 Let f x be a continuous function What is the derivative a f x dx What is How does this illustrate the difference between definite and indefinite integrals 2 What is wrong with the following Z 3 x2 x dx 2 1 3 1 9 1 8 4 C 2 2 2 3 Find the general indefinite integral a Z Z x3 3 3 x2 Z dx 2 d y 1 8y 2 4y dy b Z c v v2 2 2 Z 1 x 1 2 x 1 2 Z dx csc2 t 2et dt e Z dv f sec t sec t tan t dt sin 2x dx sin x Z 1 h cos x x dx 2 Z i ex 2x2 dx g 4 Evaluate the definite integral Z 3 1 2x 4x a 3 Z dx d 1 Z 2e 4 cos x dx 0 u5 u3 u2 du Z Z 9 sec tan d h 4 2t dt 0 Z f 0 3 10 g 3 e 2 c x 0 b Z 5 10 Z 1 3 2 t t4 0 sin sin tan2 d i sec2 2ex dx sinh x cosh x Z 1 dt 1 2 x 2 x dx 1 5 R Draw the curve x 2y y 2 and explain why the area between the curve and the y axis is 2 2 0 2y y dy Evaluate this integral The Net Change Theorem 6 If w0 t is the rate of growth of a certain child in pounds per year what does represent R 10 5 w0 t dt 7 Recall that the marginal revenue function R0 x is defined as the derivative of the revenue function R x which returns R 5000 the total revenue received if a company sells x units of a given commodity What does 1000 R0 x dx represent 8 If the units for x areR feet and the units for a x are pounds per foot what are the units for 8 da dx What about 2 a x dx 9 A particle starts with velocity v 0 4 and has acceleration a t 2t 3 where t is measured in seconds and a t is in meters per second squared For far does the particle travel in the first three seconds 10 The marginal cost of manufacturing x yards of a certain fabric is C 0 x 3 0 01 x 0 000 006 x2 in dollars per yard Find the increase in cost if the production level is raised from 2000 yards to 4000 yards Rb 11 Find the interval a b for which the value of the interval a 2 x x2 dx is a maximum 2
View Full Document
Unlocking...