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Berkeley MATH 1B - Math 1B - First Exam

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page 1 of 10Math 1B: First ExamTuesday, 24 July 2009Instructor: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Name:Problem Number 1 2 3 4 5 TotalScoreMaximum 20 20 20 20 20 100Please 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 examby myself, without any help during the exam from other people, or from sources other thanmy allowed one-page hand-written cheat sheet. Moreover, I have not provided any aid toother students in the class during the exam. I understand that cheating prevents me fromlearning and hurts other students by creating an atmosphere of distrust. I consider myselfto be an honorable person, and I have not cheated on this exam in any way. I promise totake an active part in seeing to it that others also do not cheat.Signature:Name: Math 1B First Exam: 7 July 2009 page 2 of 101. (a) (15 pts) Find the centroid of the region bound by the curves y =√x and y = x2.(b) (5 pts) Recall the Theorem of Pappus, that the volume of the solid of revolution formedby rotating a region R around a line ` is the area of R times the distance that the centroidof R travels as it revolves around the line `. Use this theorem and some geometry tofind the volume of the solid of revolution formed by rotating the region bound by thecurves y =√x and y = x2around the line y = −x.Name: Math 1B First Exam: 7 July 2009 page 3 of 102. A trough is constructed with vertical ends and flat sides that meet at a 45◦point, so thateach cross section is a right triangle, as shown in the picture. One end of the trough can moveby sliding along the trough.sliderxh(a) (10 pts) If the trough is filled with water to a depth h, find the hydrostatic force on oneend 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 thatthe trough is one meter long, the water comes to a depth of ten centimeters. If you movethe slider so that the trough has length x, how deep will the water now be?(c) (5 pts) The work required to move the slider from a length a to a length b is defined tobe W =RabF (x)dx, where F (x) is the hydrostatic force on the slider when the slider isat length x. Find the work required to shrink the trough from a length of one meter toa length of fifty centimeters.Name: Math 1B First Exam: 7 July 2009 page 5 of 103. The class sizes at a certain large university are distributed exponentially: in a given semesterthere are N(x) = Ae−x/Bclasses of size x, where A and B are positive constants. Sinceuniversities are large, we approximate x and N(x) (which actually can only take integervalues) 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 hassize 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 endsup in any particular class is proportional to the number of students in that class, theprobability that his first-period class has size x is proportional to xAe−x/B. Find theexpected size of Jimmy’s first-period class.Name: Math 1B First Exam: 7 July 2009 page 7 of 104. A certain population with constant carrying capacity K has a relative growth rate k(t) thatvaries 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 sayabout 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 thepopulation at time t.Name: Math 1B First Exam: 7 July 2009 page 9 of 105. (20 pts) A spring has mass m = 1 kg, damping constant c = 2 kg/s, and spring constantk = 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 10References: All the problems on this midterm are due to the instructor, although they areloosely based on the material in Single Variable Calculus: Early Transcendentals for UC Berkeleyby 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 examsby Zvezda Stankova.Feel free to use this page for extra scrap


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