2021 Practice Final PDF

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APPLMATH1201B – Practice Final Exam (2021)Test Date: April 19th, 2021 from 9:00AM until 12:00PM on MobiusThe actual final exam will be between 18 - 24 questions in length, with a variety of multiple choice, fill-in-the-blank (or list), or equation-input style questions. Final answers will be posted ...

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APPLMATH1201B – Practice Final Exam (2021) Test Date: April 19th, 2021 from 9:00AM until 12:00PM on Mobius The actual final exam will be between 18 - 24 questions in length, with a variety of multiple choice, fill-in-the-blank (or list), or equation-input style questions. Final answers will be posted by Wednesday, April 14th at the latest. Attempt this as practice before looking at the answers! -tp.

1. Suppose that you have discovered a new element that you have been researching for some time. You identify that this new element decays following the recursion model, ut+1 = a ut , where ut denotes the number of grams of the element on day t. If the element has a half-life of 15 days, and you initially start with 100 grams of the element, how much element will be present on day t? (A) 100 (2)15t (D) 100

2. If u′ (t) +

 t/15 1 2

(B) 100

 15t 1 2

(C) 100 (2)t/15

(E) There is not enough information.

1 u(t) = 3 with u(1) = 3, what is u(3)? t+2

3. Suppose a population’s growth is modelled by the differential equation, dN = N (N − 2) (5 − N ) dt where N (t) is the density of individuals in the population at time, t. If N (0) = 4, then what is the long-term population density? 4. Consider the initial-value problem, u′ (t) = 6 t − 2u(t) with u(0) = 3 . (a) If we wished to approximate a solution on the interval (0, tf ] using Euler’s method with 4 subintervals and h = 1, then what should we set tf to be? (b) What is the value Euler’s method approximates for u(tf )? (c) In Python, we can quickly print onto the screen the value of u(tf ) for any amount subintervals (remember that increasing the subintervals will increase the size of the array that we produce in our code). What print statement would you use in Python to extract u(tf ) for any amount of subintervals used? 1

5. We learned about another numerical method to approximate a solution to an initial-value problem, the Midpoint method. What is the main advantage to using Midpoint method over Euler’s method? (A) It is a more accurate method for similar S values. (B) It is a quicker method when computing the approximate solution by hand. (C) Increasing S has less of an effect on the accuracy than it does for Euler’s Method. (D) Python understands Midpoint method easier than Euler’s method. (E) None of the above are true. 6. Consider two events E and F in the sample space Ω. Which of the following statements is not true? (A) P(E) ≤ 1

(B) 0 ≤ P(F )

(C) P(E) = 1 − P(E C )

(D) P(E ∩ F ) = P(E) + P(F ) − P(E ∪ F )

(E) All are true.

7. Heart diseases and ulcers can occur together. Suppose you study a large group of patients and determine that 20% have heart disease, 30% have ulcers and 5% have both. If we choose an individual in the study at random, what is the probability that they have heart disease or an ulcer but not both? (Give your answer as a fraction to be marked correct) 8. E. coli bacteria are commonly found in the intestines of humans and Lambda is a virus that attacks E. coli. Suppose that 84% of the subjects tests had E. coli in their gut and 65% had both E. coli and Lambda. What is the probability that a subject will have Lambda virus in his gut given that he has E. coli? (Give your answer as a fraction to be marked correct) 9. In a population composed of an equal number of males and females, a certain disease occurs in 15% of males. What is the probability that a randomly chosen individual is a male with this disease? (Give your answer as a fraction to be marked correct) 10. A discrete random variable X takes the value 1 with probability 0.5, the value 2 with probability 0.1 and the value 3 with probability 0.4. What is the expectation of X ? (A) 1.9

(B) 1

(C) 1.5

2

(D) 0.5

(E) 2

11. Consider the cumulative distribution function for a continuous random variable X,   0 if x < 0   1 2 F (x) = 2 (x + x) if 0 ≤ x ≤ 1    1 if x > 1 (a) What is P(−1/2 < X < 1/2)?

(b) What is the expectation of X ? 12. Suppose you run a clinical trial for a new prescription drug on 20 patients, and determine that the sample mean of resting heart rate is +4 beats per minute, with a variance of 9 beats per minute squared. Your drug, however, was expected to have little to no effect on resting heart rate. You are now intrigued to find out if this happened by random chance, or if there is some major problem with your test. (a) You wish to run 10,000 simulated experiments using Python through the use of random number generation to test for the chance of type-I errors. Specifically, we will use the normal distribution to generate random trials for each experiment. Assuming you have already run import numpy as np beforehand, what single line of code would you type into Python to generate all your experiments? (Assume you want to store your experiments in an array called experiments) (b) According to William Sealy Gosset, we can construct a benchmark from our original data to identify type-I errors. Gosset called this benchmark t (i.e., Student’s t-test). What value of t should we set as our benchmark? 13. Suppose we sample a population of meerkats, and determine that their body length has a sample mean of 10 inches, and a sample variance of 25 inches squared. You go to the zoo and spot a meerkat that is 18 inches tall. How many standard deviations away from the mean is this meerkat?

3





2 3 −1 2    ~  1  ~ (~x) = A ~x where A = , what is the value of F  ? 14. If F −1 4 0 −1 (Note: on Mobius, we won’t ask you to input a vector, but we may ask you for the elements individually!) #

"

15. Which of the following fields matches with the figure below?

(A) F~ (

h i u v

)=

h i

(B) F~ (

u v

(D) F~(

h i u v

)=

h

h i u v

)=

i

h

i

(E) F~ (

−u −v

16. True or False: The vector field, "

#

2 −3 F~ (~x) = 4 −6 is invertible.

4

(C) F~ (

−u v

h i u v

)=

h i v u

h i u v

)=

h

u −v

i

~ (~x) = A ~x, where 17. Consider the vector field F 

2 0 2  k 1 A = 0  . 1 k+2 0 

What value(s) of k ensure that F~(~x) does not have an inverse? 18. Suppose that female hedgehogs in a wild population live only 3 years. The first year, they are immature and do not reproduce. The second year, they are adolescent and reproduce at a rate of 0.8 female offspring per female hedgehog. The last year, they are adults and produce 3.5 female offspring per female hedgehog. Further, suppose that 90% of the immature hedgehogs survive to adolescents, and that 90% of adolescents survive to adulthood. All adult hedgehogs pass away. For simplicity, we are only interested in the female portion of the population. (a) Suppose that u1 (t) represents the number of immature hedgehogs in year t, u2 (t) represents the number of adolescent hedgehogs in year t, and u3 (t) represents the number of adults in year t. What must be A in the recursion model,     u1 (t + 1) u 1 ( t)      u2 (t + 1) = A  u2 (t) u3 (t + 1) u 3 ( t) that defines how the structure changes year-to-year?

(b) Suppose we project an initial population structure many generations into the future using Python. We then print onto our screen the ratio between the most recent generation ~u(t + 1) and the second most recent generation ~u(t). The array that prints to the screen has three very similar numbers. What are these three numbers approximating? 19. (Fill-in-the-blanks) Consider the matrix, "

#

2 4 . A= 3 3 The largest eigenvalue of A is λ1 = , and the smallest eigenvalue is λ2 = . If we set the first element of ~v1 to be 1, then the second element will be . If we set the first element of ~v1 to be 1, then the second element will be .

5

20. (Fill-in-the-blanks) Consider the matrix, #

"

3 −1 . A= 1 3 This matrix has complex eigenvalues, λ = a ± bi, where a = and b= h i. The eigenvector associated with the positive imaginary part is given . by ~v = 1x , where x =

21. Consider the system of recursion equations, "

#

"

#"

4 1 u1 (t + 1) = u2 (t + 1) 2 3

u1 (t) u 2 ( t)

#

If u1 (0) = 5 and u2 (0) = 1, then the solution for u1 (t) is given by which of the following? (5)t − 34(−2)t (A) u1 (t) = − 11 3 (B) u1 (t) = (5)t − (2)t

(C) u1 (t) = − 34 (−5)t +

(D) u1 (t) = (E) u1 (t) =

11 (5)t 3 11 (5)t 3

11 (2)t 3 t

+ 34(2)

− 38(2)t

22. Suppose that a population of tree frogs following an age-structured recursion model, #" #" # " u1 (t + 1) 0.7 1.5 u1 (t) . u2 (t + 1) 0.2 0.8 u2 (t) (a) What is the long-term growth rate for this population? (b) For every 2 individuals in the u2 class, how many individuals do you expect to be in the u1 in the long-term? 23. (Select all that apply) Which of the following are primitive matrices? wish to sketch out the network for each.      0 # # " " 1 0 1 1 1 1  1 1 1 0   1 (C)  (B) (D)  0 1 0   1 1 0 (A) (E)  1 0 1 0 1 1 0 1 1 0 0 0

6

You may 1 0 0 1

0 1 0 0

0 0   1 0 

24. Suppose (insert your name here) is a regular at the cafeteria in Ontario Hall; so much so, that the servers have figured out a clear pattern to your ordering pattern. On a regular basis, you order only three of the options: burgers, pizza or hotdogs (all have vegan options, just in case). The servers determine the following pattern to your order: • If you ordered a burger the day before, you are equally likely to order a burger or hotdog the next day, but three times as likely to order pizza in comparison to the other two values (i.e., probability of burger and hotdog are equal, but pizza is three times as likely); • If you ordered pizza the day before, you will not order it again the next day, and will order a hotdog twice as likely as you would a burger; • If you ordered a hotdog the day before, you won’t order pizza but you order a burger or hotdog with equal chance. If you order every day for a very long time, what is the chance of you ordering each of the food items? 25. Consider the system of differential equations, u1′ (t) = u1 (t) + 3 u2 (t)

.

u2′ (t) = −5 u2 (t) − 3 u1 (t)

If u1 (0) = 2 and u2 (0) = −1, then what is u1 (t) and u2 (t)? 26. Consider the system of differential equations, u′1 (t) = u1 (t) − 3 u2 (t) u′2 (t) = 2 u1 (t) − 3 u2 (t)

.

(a) If u1 (0) = 3 and u2 (0) = 2, then what is u1 (t)? (b) Describe the shape the solution curve makes in the u1 -u2 plane.

7

27. Consider the system of differential equations, "

#

"

#"

4 −2 u1′ (t) = −3 5 u2′ (t)

#

"

#

u1 (t) α + 1 . u 2 ( t) α2

(a) Suppose that the equilibrium of this system occurs at u1∗ = 2 and u2∗ = 1. What are the values of α1 and α2 ? (b) After a change of variables to the variable ~x(t), we are left with a homogeneous system of differential equations. What is that homogeneous system in terms of ~x(t)? (c) What is the solution to this system of differential equations assuming u1 (0) = 3 and u2 (0) = 4?

8

Final Answers for Practice Test: Note that full solutions will not be made available for these problems, only final answers. If need assistance, please email or come to Student Hours! 1. (D) 2.

33 5 (Note that Nˆ = 5 is a stable equilibrium point for this model)

3. 5

4. (a) tf = 4

(b) u(4) = 15

(c) print(u[-1])

5. (A) 6. (E) 7.

40 100

8.

65 84

9.

15 200

10. (A) 11. (a)

3 8

(b)

7 12

12. (a) experiments=np.random.normal(0,np.sqrt(9),[10000,20]) √ 4 20 (b) 3 13.

8 5

14.

" #

3 2

15. (C) 16. False. 17. k = −1 



0 0.8 3.5  0  18. (a) 0.9 0  0 0.9 0

(b) the dominant eigenvalue of A

9

19. In order, the blanks are →

6 , −1 , 1 , and −3/4.

20. In order, the blanks are →

3 , 1 , and i.

21. (D) 22. (a) ρ = 1.3

(b) 5

23. (D), and (E) are primitive. 24. P(burger) =

5 , 14

P(hotdog) =

3 14

and P(pizza) =

6 14

25. u1 (t) = e−2t (2 + 3t) and u2 (t) = −e−2t (1 + 3t) √ 26. (a) u1 (t) = 3e−t cos( 2 t) (b) counter-clockwise spiral inward 27. (a) α1 = −6 and α2 = 1 "

#

"

#

4 −2 ~x(t) (b) ~x (t) = −3 5 ′

" #

" #

2 −2 1 9 2 (c) ~u(t) = e7t + e2t + 3 1 1 5 5

10...