# Derivative

MAT1330 X : Instructor Dr. Hai Yan Liu (Jack) July, 27, 2021 : Final Exam Duration: 3 hours Please read the following instructions carefully. • You have 3 hours to complete this exam. It has 13 pages, 9 questions and is out of 46 points (plus three bonus point). • This is a closed book exam, but you are permitted four (4) single-sided, hand-written formula sheets ( non-mechanically reproduced, letter size). • All work done during the exam will be done entirely by yourself, with no help from others. • You will not communicate with anybody except the professor during the exam (for exam-related questions). • You will not consult any people, sources, writings , and websites. • You will not share information about the exam’s contents before the exam period is over on August 17. • You certify that all solutions are entirely your own work, that you did not consult people or unauthorized resources during the examination, and did not share information with others during the exam. • Good luck! 1 2 In your uploaded exam files, you must include a copy (photograph or scan) of a page with the following student declaration, written in your own handwriting and signed and dated with your student id card: Student’s Declaration: I Your full name printed here of the MAT1348 exam. Signature: have read and I agree to honour the terms Student #: Date: Include image of your student id card on your student declaration page. Failure to comply with the terms of this exam may result in academic fraud allegations being filed and may result in you obtaining zero for this exam. 3 Question 1. (5 point) In the following question choose True or False. (1) Derivative of ln |x| is x1 . False True. (2) If any function f has a local maximum at x = a, then it must be the case that f 0 (a) = 0. False True. (3) It can happen thatf 00 (a) = 0, but that f does not have an inflection point at x = a. False True. (4) If f is continuous function on [−1, 1] and f (−1) = 3, f (1) = 5. Then there is a number 3 ≤ c ≤ 5, such that f (c) = 0. False True. (5) Every continuous function attains a global maximum and global minimum on each closed interval in its domain. False True. 4 Question 2. (3 points) Differentiate the following functions. Show all your steps. (a) f (x) = arcsin(x2 ). (b) g(x) = ln(3x sec3 (4x)). 5 Question 3. (total: 6 points) The drug Xlocyn is eliminated from a patient’s bloodstream at a constant rate of 30% per day. They receive a daily dose of 4 µg. Let xt denote the amount of Xlocyn in the patient’s bloodstream on day t, just after the daily dose is administered. We start counting t ≥ 0 from the first dose the patient receives. (a) (1 point) Give a linear discrete-time dynamical system (DTDS) modelling xt+1 in terms of xt . (b) (1 point) Write down the updating function of the dynamical system. (c) (1 point) Identify the equilibrium value on your graph or confirm this by computing the equilibrium of your DTDS. (d) (1.5 points) Suppose we switch to a new longer-lasting variant of the drug, XlocynPlus, which is eliminated at half the rate of Xlocyn. What daily dose of XlocynPlus should we administer in order to maintain the same equilibrium in the long term? 6 (e) (1.5 points) Draw a cobweb for this DTDS starting with initial condition x0 = 0. Label your axes and functions. 7 Question 4. (total: 4 points) (a) (3 points)The size of population of Bacillus subtilis ( in average number per petri dish ) introduced to a new nutrient can be described in the short term by the function N (t) = 50e4t + 300e−3t where 0 ≥ t ≤ 1 is measured in hours. At what time in the interval t ∈ [0, 1] is the population the largest? the smallest? (b) (1 points) Between time 0 and 1, the absolute minimum average population per dish is attained when t = the absolute maximum average population per dish is attained when t = e2x + 4ex − 21 . x→ln(3) ex − 3 (a) (2 points) Is this an indeterminate form? If so, what type (examples: Question 5. (total: 4 points) Consider lim ∞ 0 , , ∞ − ∞, · · · )? ∞ 0 (b) (2 points) Evaluate the limit or show it does not exist, using methods from algebra and Calculus. Show and justify all your steps. 8 Question 6. (total: 7 points (0.5 points each and 1.5 for part (l)) Evaluate the following about the function f (x) = 2 + 3e−3/x , and then use that information to sketch its graph. (a) Domain of f : (b) lim f (x) = x→∞ (c) lim f (x) = x→−∞ (d) lim f (x) = x→0− (e) lim f (x) = x→0+ (f) f 0 (x) = (g) Critical points of f : (h) (Remember to think about the domain of f !) On what interval(s) is f (x) increasing? (i) f 00 (x) = (j) (Remember to think about the domain of f !) On what interval(s) is f (x) concave up? (k) Are there any inflection points? If so, where? (l) Sketch the graph of f , being sure to include all the features identified in (a)-(k) above. 9 You can draw graph here. 10 Question 7. (10 points) Consider a population of albino laboratory mice (species Mus musculus) raised for research. Each month, a certain fraction of the mice are taken away to lead productive lives as researchers. The resulting discrete-time dynamical system (DTDS) describing this population of albino mice is xt+1 = 3xt (1.2 − 4xt ) − hxt , where h > 0 is a certain parameter, xt is the size of the population (in units of hundreds) and where the time t = 0, 1, 2, . . . is measured in months. (a) (3 points) Write the updating function for this DTDS and then find all of its fixed points. Your answers may depend on h. (b) (1 point) Determine all value(s) of h > 0 that give rise to a positive fixed point. (c) (3 points) State the stability theorem for DTDS. Then use it to determine all value(s) of h > 0 that give rise to a stable positive fixed point. Make sure your solution shows your mathematical reasoning, especially as it concerns inequalities and absolute values. 11 (d) (3 points) In the long term, the number (in hundreds) of mice that are sold to laboratories each month is given by N = hx , where x is the positive stable fixed point (corresponding to that value of h). How should we choose the harvesting rate h to maximize N ? Justify that your answer is a maximum. 12 Question 8. (total: 7 points) Compute the following indefinite and/or definite integrals. Your work must be legible, logical and correct to earn full marks. Identify the technique of integration that you use, and show all your work. In particular, you will not receive credit for using memorized integration formulas — you must derive the answer from the techniques of integration and the standard formulas for derivatives of basic functions. Z 3/2 x + 2×6 + 1.2 (a) (2 points) dx x Z (b) (2 points) Z (c) (3 points) 1 ln(6×3 ) dx 2 √ 1.2x 2 + 3×2 dx (notice the limits of integration) 13 Question 9. (3 bonus points) The acceleration of a molecule constrained to a linear path is described by the function a(t) = 4 cos(3t) + 1.2e−6t . Given that acceleration is the rate of change of velocity v(t) with respect to time, and that velocity is the rate of change of position x(t) with respect to time, solve for the position of the molecule, given that it begins at time t = 0 in a resting state at position 5, and v(0) = 0.

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