Solving Differential Equations: Practice Questions & Answers

Incomect Question 1 0/1pts (Lesson 3.1: Solving a Differential Equation.) Suppose that f(:z) = . We know that if p is small, then 3 ~ flath)-f(z) flle) =—=—-. Using this expression with h = 0.01, find an approximate value for f'(1) a1 b.2.72 c.7.38 d. 14.93 We have f(z) ~ f(2+hz—f(2) = 2la+h) g2z h So using h =0.01, we have €202_g2 (1)~ St =14.93 Thus, the answer is (d).

(Lesson 3.1: Solving a Differential Equation.) Suppose that f(z) = e, What is the actual value of f(1)? % | b.e~ 2.72 c.e? ~ 17.39 d.2¢2 ~ 14.78 f'(z) = 2€*,s0 that f'(1) = 2¢?, and thus the answer is (d). e 14.93 f'(z) = 2e*,s0 that f'(1) = 2¢2, and thus the answer is (d).

Question 3 1/1pts | (Lesson 3.1: Solving a Differential Equation.) Consider the differential equation f'(z) = (z + 1) f(x) with f(0) = 1. What is the exact formula for f(z)? a. f(z) = €* b f(z) = e This takes a little work. The good news is that you can actually get the true answer using the technique of separation of variables. We have f@ _ f@) so that 1@ gy — ff(z)dz—fa:+1dz Which implies z+1, In(f(e)) = Z + 2 +C. 2 so that f(a:) — Ke> +z, where C and K are arbitrary constants. Setting f((]) = 1 implies that K = 1, so that the exact answer is , 22 . - =-+z, i.e., choice (c). the answer is f(z) =1 d. f(z) = cxp{uv2 + 21}

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Question 4 1/1pts (Lesson 3.1: Solving Differential Equations.) Consider the differential equation f'(z) = (z + 1) f(z) with £(0) = 1. Solve for £(0.20) using Euler's approximation method with increment & = 0.01 for z € [0,0.20]. a. £(0.20) ~ 0.0 b. £(0.20) &~ 1.0 c. £(0.20) ~ 1.24

By previous question, the true answer is the answer is L fl@)=er™"" But our job is to use Euler to come up with an iterative approximation, s0 here it goes. As usual, we start with f@+h) = f(z) + hf'(@) = f2) + bz + 1) f(z) = f@)[L+ h(z +1)], from which we obtain the following table. z | Euler approx true f(z) 0.00 1.0000 1.0000 0.01 1.0100 1.0101 0.02 1.0202 1.0204 0.03 1.0306 1.0309 0.04 1.0412 1.0416 0.05 1.0521 1.0526 0.06 1.0631 1.0637 0.07 1.0744 1.0751 0.08 1.0859 1.0868 0.09 1.0976 1.0986 0.10 1.1096 1.01107 0.19 1.2287 1.2313 0.20 1.2433 1.2461 ‘Wow, what a good match! In any case, the answer is (c). Question 5 1/1pts (Lesson 3.2: Monte Carlo Integration.) Suppose that we want to use Monte Carlo integration to approximate I = fls 1% dz. If Uy, U, ..., U, areiid. Unif(0,1)'s, what's a good approximation [, for r

1 n 1 a i 0, In the notation of the lesson, the general approximation we've been using is — + e~ + <) S s T 1 1+U; Ll n 2 n 1 n kS so that the answer has simplified very nicely to (a). 2 n 1 b. 5 i 1+U; —~n 1 n Z<i=1 1327, 2 o 1 4= Y o

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(Lesson 3.2: Monte Carlo Integration.) Again suppose that we want to use Monte Carlo integration to approximate I = fls 1+x dz. You may have recently discovered that the MC estimator is of the form _1ywn 1 Lo=1%n i Estimate the integral J by calculating fﬂ with the following 4 uniforms: Uy=03 Uy=09 U;=02 Uy=07 c.0.321 tructure.com/courses/165326/quizzes/213841 Week 4 Homework - Spring 2021: Simulation - ISYE-6644-0AN/001 I,=1 E?:l fi = 0.679, so the answer is (d).

Question 7 1/1pts (Lesson 3.2: Monte Carlo Integration.) Yet again suppose that we want to use Monte Carlo integration to approximate I = fla HLZ dz. What is the exact value of J? a.0.197 b. 0.693 I=In(1+z)} =1In(4) — In(2) = 0.693.Thus, the answer is (b). c. 1.386 d. 2773 I=1In(1+z)|} = In(4) — In(2) = 0.693.Thus, the answer is (b).

Question 8 171 pts (Lesson 3.3: Making Some 7r.) Inscribe a circle in a unit square and toss n = 1000 random darts at the square. Suppose that 760 of those darts land in the circle. Using the technology developed in class, what is the resulting estimate for 7? b. 4.0 (UGA answer) c.32 d.3.04 The estimate #n =4 x (proportion in circle) = 4(760/1000) = 3.04 Thus, the answer is (d). e.3.12

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Question v 1P (Lesson 3.3: Making Some 7r.) Now suppose that we can somehow toss n random darts into a unit cube. Further, suppose that we've inscribed a sphere with radius 1/2 inside the cube. Letp,, be the proportion of the n darts that actually fall within the sphere. Give a Monte Carlo scheme to estimate 7. ture.com/courses/165326/quizzes/213841 Week 4 Homework - Spring 2021: Simulation - ISYE-6644-0AN/O01 The probability that a dart falls inside the sphere is the volume of the sphere divided by the volume of the unit cube, i.e.,%ﬂ”f’a = 1r/6. Thus, for large n, we have p,, ~ 7/, so that 7 = 6p,, should do the trick. Therefore, the answer is (d).

Question 10 1/1 pts (Lesson 3.4: Single-Server Queue.) Consider a single-server Q with LIFO (last-in-first-out) services. Suppose that three customers show up at times 5, 6, and 8, and that they all have service times of 4. When does customer 2 leave the system? ture.com/courses/165326/auizzes/213811 e. 19 Let's make a version of our usual table. Thus, the answer is (d).

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Question 12 1/1 pts (Lesson 3.6: Simulating Random Variables.) If 7 is a Unif(0,1) random number, what is the distribution of —0.5¢n(U)? a. Who knows? b. Exp(2) By the Inverse Transform Theorem, we know that —3én(1 - U) ~ Exp()). Butsince U and 1 — U are botn Unif(0,1) (why?), we also have —36n(U) ~ Exp(N). In particular, —0.5¢n(U) ~ Exp(2), so that the answer is (b). c. Exp(1/2) d. Exp(—2) o x(-1/2)

Question 13 0/1pts | (Lesson 3.6: Simulating Random Variables.) If ; and U, are i.i.d. Unif(0,1) random variables, what is the distribution of 7, + U, ? Hints: (i) | may have mentioned this in class at some point; (i) You may be able to reason this out by looking at the distribution of the sum of two dice tosses; or (iii) You can use something like Excel to simulate U; + U, many times and make a histogram of the results. a. Unif(0,2) icture com/courses/165326/quizzes/213841 Week 4 Homework - Spring 2021 Simulation - ISYE-6644-OAN/001 b. Normal c. Exponential d. Triangular By any of the hints, you get a Triangular(0,1,2) distribution, i.e., answer (d).

Question 14 1/1 pts (Lesson 3.7: Spreadsheet Simulation.) | stole this problem from the Banks, Carson, Nelson and Nicol text (5th edition). Expenses for Joey's college attendance next year are as follows (in $): Tuition = 8400 Dormitory = 5400 Meals ~ Unif(900,1350) Entertainment ~ Unif(600,1200) Transportation ~ Unif(200,600) Books ~ Unif(400,800) Here are the income streams the student has for next year: Scholarship = 3000 Parents = 4000 Waiting Tables ~ Unif(3000,5000) Library Job ~ Unif(2000,3000) Use Monte Carlo simulation to estimate the expected value of the loan that will be needed to enable Joey to go to college next year.

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a. $2500 instructure.com/courses/165326/quizzes/213841 Week 4 Homework - Spring 2021: Simulation - ISYE-6644-0AN/001 b. $3250 c.$3325 An easy spreadsheet simulation (or an almost-as-easy exact analytical calculation) reveals that the expected loan amount is $3325, or answer (c). If you don't believe me, here's some Matlab code (if you happen to have Matlab)... m = 1000000; % reps Income = 7000 + unifrnd(3000,5000,[1 m]) + unifrnd(2000,3000,[1 m]); Expenses = 13800 + unifrnd(900,1350,[1 m]) + unifrnd(600,1200,[1 m]) + unifrnd(200,600,[1 m]) + unifrnd(400,800,[1 m]); Totals = Income - Expenses; hist(Totals, 100) ‘mean(Totals) var(Totals) d. $3450 e. $4000

HW4

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