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Unformatted text preview: Math 102 Old exam questions dealing with (roughly) Chapters 12 and 13 of Keshet’s Course Notes For Math 102 Section 102 Fall 2009, this is good practice related to Homework #5 due December 2nd. metres. Average velocity =
Decemb er 2006 Mathematics 102 Name Page 7 of 11 pages (b) Use a linear approximation to estimate (65) 1/3 , given that (64)1/3 = 4. The answer 7. Writeshouldanswers in the not decimals. your use fractions, b oxes. /3 (65)1(a) ≈ An ob ject moves along a line. If the ob ject’s p osition at time t is x(t) = 3t 2 − 2t + 1, dec2006q7 what is its average velocity from 0 to time 3? Time is measured in seconds, p osition in Average velocity = metres. (c) In trying to solve the equation x 3 + 5x − 3 = 0 using Newton’s method, our initial guess is x0 = 1. What is x1 ? The answer should use fractions, not decimals. (b) Use a linear approximation to estimate (65) 1/3 , given that (64)1/3 = 4. The answer should use fractions, not decimals. 1/3 (65) ≈ 3 (c) In trying Average velocity = to solve the equation x + 5x − 3 = 0 using Newton’s method, our initial guess is x0 = 1. What is x1 ? The answer should use fractions, not decimals. x1 = (b) Use a linear approximation to estimate (65) 1/3 , given that (64)1/3 = 4. The answer (65)1/3 ≈ should use fractions, not decimals. (d) Using Euler’s method to solve the diﬀerential equation y = y 2 with y (0) = 2 and step size 0.1, what is the approximate value of y (0.2)? The answer should b e given to at least (c) 3 decimal to solve the equation x 3 + 5x − 3 = 0 using Newton’s method, our initial guess In trying places. is x0 = 1. What is x1 ? The answer should use fractions, not decimals. x1 = 2 (d) (65)1/3 ≈ Using Euler’s method to solve the diﬀerential equation y = y with y (0) = 2 and step size 0.1, what is the approximate value of y (0.2)? The answer should b e given to at least 3 decimal places. ecemb er 2006 dec2006q10 Mathematics 102 Name dT 1 =2− T dt 5 Page 10 of 11 pages 0. Consider Newton’s law of cooling with initial condition T (0) = 37. (a) Find values of the constants a, b and k such that T (t) = a + be−kt is a solution to the initial value problem given ab ove. (b) Using the solution obtained in (a) to ﬁnd the time τ at which T (τ ) = 13. Express the answer in terms of m, where m = ln 3. (c) What is the steady state for this diﬀerential equation? Is it stable or unstable? dec2000q2e dec2000q6 net gain in energy zero? dec1999q5 c) Determine the length of time t for which the net gain in energy is greatest. dy = 2 y (1 " y ) describes the growth of a certain population dt [10] 5. The differential equation y(t). a) What is the maximal growth rate of the population? b) Use Euler’s method with !t = 0.1 and y(0) = 0.1 to find an approximation of the solution at t = 0.3. [10] 6. Cultures of two types of cells, A and B, are studied in an experiment. Initially, at time t = 0, it is found that the population levels are yA(0) = yB(0) = 1000. Population A is observed to double every day. Population B increases linearly by adding 100 new individuals every hour. a) If this growth continues indefinitely, what will the population levels of A and B be at some time t (in days) later? (i.e., find yA(t) and yB(t)). b) How long does it take for population A to reach 100 times its initial level. c) How long does it take for population B to reach 100 times its initial level. [10] 7. IN THIS QUESTION YOU MUST SHOW ALL YOUR WORK TO GET CREDIT. x " " b) Find all intervals where f(x) is increasing. c) dec1999q8 ind all intervals where the function f(x) is concave up. d) F 10] 8. A certain function f(x) satisfies the following conditions: f(1) = 2, f!(1) = 1. a) What is the equation of the tangent line to the graph of y = f(x) at x = 1? b) Find an approximation for f(1.01). Now assume in addition that f"(x) > 0 for all x. c) Sketch the graph of y = f(x) in the neighborhood of x =1. d) d) Will your approximation in (b) be larger or smaller than the actual value of f(1.01)? Why? 10] 9. a) Use Newton’s method to find the critical point(s) of the function g ( x ) = x 2 + e ! x to 3 decimal places. (Steps must be displayed to get credit). b) Derive a formula for x1 in terms of x0 in the diagram below. Show all steps in the derivation. dec2004q4 dec2004q9 dec2008q2b
b) A diﬀerential equation describing radioactive decay of a substance is dy = −ky, dt where y (t) = Ce−kt is the amount of radioactivity remaining. A plot of y (t) versus t is found to be a curve that goes through the point (0, 10), and has tangent line of slope −5 at that point. What are the values of the constants? C= , k= c) A linear approximation for y = sin(0.05) that uses the fact that sin(0) = 0 is sin(0.05) ≈ S and December 2008 that could haveName: f are constants Math 102 diﬀerent values for diﬀerent people. [Your answers toout of 13 Page 9 Knowledge can be acquired by Name: studying, but it is forgotten over time. A simple 9 out offor model 13 December 2008 Math will constants or f .] dthe following questions 102 containknowledge, such as S a person has at timePage years) by ec2008q7 learning represents the amount of y (t), that t (in a diﬀerential equation Problem 7: “Live and Learn” 3pts](a) Mary Problem 7, Continued dy does this imply about the constants S and/or f ? never forgets anything. What Problem 7: “Live and acquired by studying, but it is forgotten over time. A simple model for Knowledge can be Learn” = S − fy Mary can studies at rate of at How Knowledge Jane bestudying a byschool10 units peris 0 with no over person atesimple model per year. by studying,time it forgotten knowledge hasof time tmuch [5pts](c) starts acquired in amount atdt but t =year (t), that a time. r A all.0.2 units for learning representshave after 4 years (i.e. at t = and forgets at a the of knowledge, y 4)? at (in years) knowledge will “direction of knowledge,ﬁeld”) that aof forgetting. We will assume that she learning ≥ 0 is theathe amountﬁeld” (“slope 0 y (tthefor the diﬀerential equation(in years) by represents rate of studying and f ≥ is ), rate person has at time t describing Jane’s Sketch equation whereaSdiﬀerential a and f are equation Add a few curves y (t) to show for Jane’s people. [Your answers time. knowledge. S diﬀerentialconstants that could have diﬀerent values howdiﬀerentknowledge changes withto dy dy −f the following questions will contain constantsSdt =yS S oryf .] = such as −f Mary starts studying in school atdt time t = 0 with no knowledge at all. How much where0S ≥will sheofrate after 4 years≥ 0 isat t0rate of rate of forgetting. We will assume that 0 is thehave of studying and f the = 4)? forgetting. We will assume that ≥ is the whereMary 100 S ≥ never forgets studying and f (i.e. is 3pts](a) knowledgethe rate anything. What does this imply about the constants S and/or f ? f S andSf and 90 are constants that have diﬀerent ﬀerent for diﬀerent people. people. [Your answers to are constants that could could have di values values for diﬀerent [Your answers to the following questions will contain constants S or f S the following questions will contain constants such assuch as .] oryf .] = (4)
80 [3pts](a) never forgets anything. What What does this imply he constants S and/or and/or f ? 3pts](a) Mary Mary never forgets anything. does this imply about tabout the constants S f ? 70
3pts](b) Tom learned studying in school at time t = knowledge when entering all. How much preschool that his 0 with no knowledge at school at time Mary starts so much 60 y (4) = t = 0 is y = 100. However, once Tom(i.e.school, he stops studying completely. What knowledge will she have after 4 years in at t = 4)? 50 does this imply about the constants S and/or f ?
Mary Mary studying in school at time t time with 0 when e knowledge How time starts so much 3pts](b) Tom learnedstarts studying in school athis knowledgewith no ntering all. at at much much preschool that = 0 t = no knowledge at school all. How 30 knowledge However, December 2008 she have after 4after 4(i.e.school, he stops studying completely. What Page 10 out of 13 tknowledge= 100.Mathshe have Name: in at(i.e. 4)?t = 4)? = 0 is y will will 102 onceyears years t = at Tom does this imply about the constants S and/or f ? 20 How long will it take him to forget 75% of what he knew? y (4) = 10 Problem 7, Continued
0 1 2 3 4 5 6 7 8 5pts](c) Jane studies at a rate of 10 units per year and forgets 9at a 10ate of 0.2 units per year. r 3pts](b) Sketch a “direction ﬁeld” (“slope ﬁeld”) for the diﬀerential equation describingat time Tom learned so much in preschool that his knowledge when entering school Jane’s How0long will it take him to forgetto showschool, he knew? studying complet t = is y Add few curves y (t 75% what he stops y (4) changes with What knowledge.= 100.a However, once)Tom inof how Jane’s knowledge = y (4) = ely. time. does this imply about the constants S and/or f ? How much knowledge would Jane have if she keeps studyingTime = (and forgetting) for a very long 100 3pts](b) time? learned so much much in preschool that his knowledgeentering ntering at time at time Tom Tom learned so in preschool that his knowledge when when e school school [3pts](b) t90 0 t = 0 is y =However, once Tom in school, school, hestudying completcompletely. What = is y = 100. 100. However, once Tom in he stops stops studying ely. What does this imply about the constants S and/or f ? 0 40 ...
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 Winter '09
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