page 1 of 10 Math 1B First Exam Tuesday 24 July 2009 Instructor Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Name Problem Number Score Maximum 1 2 20 20 3 4 5 20 20 20 Total 100 Please do not begin this test until 2 10 p m You may work on the exam until 4 p m Please do not leave during the last 15 minutes of the exam time You must always justify your answers show your work show it neatly and when in doubt use words and pictures to explain your reasoning Please box your final answers Calculators are not allowed Please sign the following honor code I the student whose name and signature appear on this midterm have completed the exam by myself without any help during the exam from other people or from sources other than my allowed one page hand written cheat sheet Moreover I have not provided any aid to other students in the class during the exam I understand that cheating prevents me from learning and hurts other students by creating an atmosphere of distrust I consider myself to be an honorable person and I have not cheated on this exam in any way I promise to take an active part in seeing to it that others also do not cheat Signature Name Math 1B First Exam 7 July 2009 1 a 15 pts Find the centroid of the region bound by the curves y page 2 of 10 x and y x2 b 5 pts Recall the Theorem of Pappus that the volume of the solid of revolution formed by rotating a region R around a line is the area of R times the distance that the centroid of R travels as it revolves around the line Use this theorem and some geometry to find the volume of the solid of revolution formed by rotating the region bound by the curves y x and y x2 around the line y x Name Math 1B First Exam 7 July 2009 page 3 of 10 2 A trough is constructed with vertical ends and flat sides that meet at a 45 point so that each cross section is a right triangle as shown in the picture One end of the trough can move by sliding along the trough slider h x a 10 pts If the trough is filled with water to a depth h find the hydrostatic force on one end of the trough Name Math 1B First Exam 7 July 2009 page 4 of 10 b 5 pts Enough water is poured into the trough so that when the sliding end is such that the trough is one meter long the water comes to a depth of ten centimeters If you move the slider so that the trough has length x how deep will the water now be c 5 pts The R a work required to move the slider from a length a to a length b is defined to be W b F x dx where F x is the hydrostatic force on the slider when the slider is at length x Find the work required to shrink the trough from a length of one meter to a length of fifty centimeters Name Math 1B First Exam 7 July 2009 page 5 of 10 3 The class sizes at a certain large university are distributed exponentially in a given semester there are N x Ae x B classes of size x where A and B are positive constants Since universities are large we approximate x and N x which actually can only take integer values by continuous variables a 5 pts How many classes does the university offer in a given semester b 5 pts Write a probability distribution expressing the probability that a given class has size x Name Math 1B First Exam 7 July 2009 page 6 of 10 c 5 pts What is the average class size at this university d 5 pts Jimmy Stewart is a student at this university Since the probability that he ends up in any particular class is proportional to the number of students in that class the probability that his first period class has size x is proportional to xAe x B Find the expected size of Jimmy s first period class Name Math 1B First Exam 7 July 2009 page 7 of 10 4 A certain population with constant carrying capacity K has a relative growth rate k t that varies sinosoidally with time k t a sin bt c d for constants a b c d a 3 pts Assuming that the relative growth rate is always positive what can you say about the coefficients a b c d b 2 pts Write a differential equation modeling the population growth Name Math 1B First Exam 7 July 2009 page 8 of 10 c 15 pts Assuming that the initial t 0 population is P0 find a formula for the population at time t Name Math 1B First Exam 7 July 2009 page 9 of 10 5 20 pts A spring has mass m 1 kg damping constant c 2 kg s and spring constant k 2 kg s2 and the spring is driven by a force F t sin t s kg m s2 where t is the time Find the general solution describing the displacement of spring as a function of time Name Math 1B First Exam 7 July 2009 page 10 of 10 References All the problems on this midterm are due to the instructor although they are loosely based on the material in Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart The honor code language is adapted from the Stanford Honor Code http www stanford edu dept vpsa judicialaffairs guiding honorcode htm and from the exams by Zvezda Stankova Feel free to use this page for extra scrap work
View Full Document
Unlocking...