This examination has 13 pages of questions excluding this cover

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1 This eamination has pages of questions ecluding this cover The University of British Columbia Final Eam - April 2, 207 Mathematics 0: Integral Calculus with Applications to Life Sciences 20 (Hauert, 202 (Mazumdar, 20 (Hauert, 206 (Kreso, 207 (Lohmann, 208 (Dahlberg, 209 (Dahlberg, 22 (Bade Closed book eamination Time: 2.5 hours (50 minutes Last Student Number: First Section: circle above Rules governing eaminations:. No books, notes, electronic devices or any papers are allowed. To do your scratch work, use the back of the eamination booklet. Additional paper is available upon request. 2. You must be prepared to produce your library/ams card upon request.. No student shall be permitted to enter the eamination room after 5 minutes or to leave less than 5 minutes before the completion of the eamination. Students must ask invigilators for permission to use the washrooms. 4. You are not allowed to communicate with other students during the eamination. Students may not purposely view other s written work nor purposefully epose his/her own work to the view of others or any imaging device. 5. At the end of the eam, you will put away all writing implements upon instruction. Students will continue to follow all of the above rules while the papers are being collected. 6. Students must follow all instructions provided by the invigilators. 7. Students are not permitted to ask questions of the invigilators, ecept in cases of supposed errors or ambiguities in eamination questions 8. Any deviation from these rules will be treated as an academic misconduct. The plea of accident or forgetfulness shall not be received. I agree to follow the rules outlined above (signature Question: Total Points: Score: Important. Simplify all your answers as much as possible and epress answers in terms of fractions or constants such as e or ln(4 rather than decimals. 2. Show all your work and eplain your reasonings clearly!. Questions in a section are weighted evenly unless otherwise stated. 4. Formula sheet at the back (you may tear it off and use it for scratch work. 5. Additional sheets of paper for calculations are available upon request.

2 Math 0. Multiple-choice problems (Full marks for correct answer. No partial marks. (a (6 points Given the following general terms a n, determine whether the corresponding sequences {a n } n are converging, diverging, monotone (increasing or decreasing and/or bounded. Check all boes that apply. (do not calculate the limit of converging sequences. i. n 2 : ii. ( n + : convergent divergent monotone bounded iii. + 2 n : iv. ln(n: (b (4 points Determine whether the following series converge or diverge. Check appropriate bo. (do not calculate the value of converging series. i. ii. iii. iv. n= n= n= n=0 n 2 : n 2 + n 2 : n 2 n! : n + 4 : converging diverging (c (4 points Determine whether the following integrals converge or diverge. Check appropriate bo. (do not calculate the integrals. converging diverging i. ii. iii. iv. 0 e d: ( + d: ln( 2 d: + d: 2 Final Eam Page of

3 Math 0 (d (4 points For each graph (i and (ii identify the graph (A-D, which depicts its antiderivative A( = a f(t dt and determine a consistent starting point a of the integral. y 5 (i (ii y 5 5 Answer: 5 Answer: A a = a = y 5 B y C y 5 D y Final Eam Page 2 of

4 Math 0 (e (4 points Plots (A-D depict probability density functions, pdf s, for < < at the same scale. List all pdf s that satisfy the following criteria: ( denotes the mean and med the median A y B y C y D y i. = med? Answer: ii. med >? Answer: iii. 0 p( d > /2? Answer: (f (6 points Consider the differential equation dy dt = 4t y. Check all solutions in the following list. (Note: different solutions refer to different initial conditions/values. true false i. y(t = t 4 is a solution ii. iii. y(t = t 4 + 4t is a solution y(t = t 2 is a solution Final Eam Page of

5 Math 0 2. Short-answer-problems (Full marks for correct answer. Work must be shown for partial marks. Simplify your answers. (a (4 points Write the following in -notation: i = ii = (b (4 points Consider the functions A i ( and calculate their derivatives A i ( = da i( d. i. A ( = 0 e y2 dy, A ( = ii. A 2 ( = e e (c (4 points Evaluate I = ln y dy, A 2 ( = π 0 (Work must be shown for full marks. f(f ( d using f(0 = 2, f(π = 2π. ANSWER: I = Final Eam Page 4 of

6 Math 0. Calculate the following integrals: (Work must be shown for full marks. Simplify your answers. sin ( (a ( points I = d ANSWER: I = (b ( points I 2 = y ln(y dy ANSWER: I 2 = Final Eam Page 5 of

7 Math 0 (c (4 points I = 2 2 cos d ANSWER: I = (d (4 points I 2 = π 4 0 sin y cos 2 y dy ANSWER: I 4 = Final Eam Page 6 of

8 Math 0 (n 2 ( 2 n 4. (8 points Find all such that the series n n (Work must be shown for full marks. n= converges? ANSWER: Final Eam Page 7 of

9 Math 0 5. (5 points In a petri dish with a radius of r = cm scientists placed a bacteria killing agent along a diameter. They found the density of bacteria, ρ(, increased with distance cm from the diameter according to ρ( = k bacteria/cm 2 for some constant k. Determine the total population size of bacteria, N, in the petri dish. r diameter ANSWER: N = Final Eam Page 8 of

10 Math 0 6. Consider the function y = f( = 2. (Work must be shown for full marks. (a (4 points Calculate the area, A, bounded by the -ais, y-ais and y = f( (shaded area in figure or show that it does not eist. (b (5 points Calculate the volume of the solid of revolution, V, obtained by rotating the above area (shaded area in figure around the y-ais or show that it does not eist. y ANSWER: A = V = Final Eam Page 9 of

11 Math 0 7. The figure below shows the graph y = f( and the diagonal y = (dashed line. y 0 y = f( 5 0 A 5 0 B 0 Use this plot to answer the following questions: (a (2 points Clearly mark and label all fied points of the iterated map a n+ = f(a n. (b ( points Determine the stability of each fied point. (Eplain your reasoning. (c (2 points Use the plot above to draw cobwebs with at least three steps (or until the trajectory eits the graph for each of the two initial values a 0 = A, and B. Final Eam Page 0 of

12 Math 0 8. Suppose the Taylor series y = initial condition y(0 = 4. (Work must be shown for full marks. n=0 a n n solves the differential equation dy d + 2y = 2 with the (a (6 points Calculate the first four coefficients a 0, a, a 2 and a. ANSWER: a 0 = a = a 2 = a = (b (2 points Find the recursive relation a n = f(a n for n 4. ANSWER: a n = (c (optional 2 bonus points save for last Find the closed formula a n = f(n for n 4. ANSWER: a n = Final Eam Page of

13 Math 0 9. The probability density function (pdf for the mortality of a jellyfish (Turritopsis dohrnii, p(, at age is given by p( = 2 π + 2, for 0 <. (a (2 points Find the probability, C(, that a jellyfish dies before reaching the age. ANSWER: C( = (b (2 points Find the median mortality, med. ANSWER: med = (c ( points Find the mean mortality,. (Setup and evaluate the improper integral for the mean mortality. ANSWER: = (d ( point Consider a large colony of Turritopsis dohrnii jellyfish. After how many years do we epect that half of the animals of the original colony have died? (Circle correct answer. (i Year (ii 2 Years (iii π 2 Years (iv 2π Years (v Never (e ( point The average lifespan of any single jellyfish is arbitrarily long. (Circle correct answer. (i True (ii False Final Eam Page 2 of

14 Math 0 Useful Formulæ Summation k= N k = k= N(N + 2 N k 2 = k= N ( N(N + 2 N k = r k = rn+ 2 r k=0 N(N + (2N + 6 Trigonometric identities sin(α + β = sin α cos β + cos α sin β; for α = β: sin(2α = 2 sin α cos α cos(α + β = cos α cos β sin α sin β; for α = β: cos(2α = 2 cos 2 α = cos 2 α sin 2 α sin 2 α + cos 2 α = tan 2 α + = sec 2 α = cos 2 α Some useful trigonometric values sin(0 = 0, sin = 2 6 2, sin = 4 2, sin = 2, sin =, sin(π = cos(0 =, cos = 6 2, cos = 4 2, cos = 2, cos = 0, cos(π = 2 tan(0 = 0, tan =, tan =, tan =, tan = DNE, tan(π = Derivatives d d arcsin = 2 d d arccos = 2 d d arctan = d + 2 d tan = sec2 Final Eam Page of