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• Q1:Question 1 Consider the function f(x) = e^-x/2.cos(x). Compute the Taylor polynomials T2(x), T3(x) and T4(x) of degrees 2,3,4 about x = 0 in Maple and graph the differences f(x) -T2(x), f(x) - T3(x) and f(x) — T4(x) for 0≤x≤2 on the same plot. The differences tell us the error of the Taylor polynomial as an approximation to f(x). To get a Taylor polynomial see ?taylor Use Maple to calculate (1) the maximum error and (ii) the average error of T2(x), T3 (x) and 74(x) on the interval 0 ≤ x ≤ 2. What is the average value of a function g(x) on an interval a ≤x≤ b you ask? See Section 6.5 of the Stewart Calculus text or look it up on the web.See Answer
• Q2:Question 2 This question is related to Newton's law of cooling. Let T(t) be the temperature of a body of liquid at time t. Let Am be the room (ambient) temperature of the surrounding medium (air). The DE is T' (t) = k·(Am - T(t)) where k is the cooling rate constant. Solve the differential equation in Maple for Am = 20 degrees and an initial temperature of 50 degrees. Given also that T(20) = 35, determine k. Now compute T(60). Do all the calculations in Maple. Finally graph I(t) for 0 ≤ t ≤ 100 together with the room temperature on a suitable domain/range.See Answer
• Q3:Question 3 Carbon 14 decays into Nitrogen 14. Using Google, find the half life H of Carbon 14. The differential equation modeling radioactive decay is y' (t) = -ky(t) where k is the decay rate constant and y(0) is the initial concentration of Carbon 14. Given the half life is H, that is, given that y(H) = y(0)/2, determine k. You can do this one by hand at first but then do it in Maple. [Solve the DE in Maple and graph the solution for y(0) = 1 on a suitable domain.See Answer
• Q4:Question 4 Suppose we have a 400 liter tank. Suppose 8 litres per minute of salt water (brine) flows into the tank at the top and then flows out of the tank at the bottom. Assume for simplicity that the salt water in the tank is stirred so that its concentration is uniform in the tank. Let S(t) be the amount of salt, in grams, in the tank at time t minutes. Suppose the salt water flowing into the tank has concentration 100 grams per liter. Find the differential equation to model the change in S(t). Assuming there is no salt in the tank at time t=0 solve the differential equation using Maple. What is S( ∞ )? That is, how much salt is in the tank after a long time? Now graph S(t) for a suitable domain.See Answer
• Q5:Question 5 The logistic growth with harvesting model for a population y(t) at time t is given by y' (t) = a∙y(t) (max-y(t)) - H Here Ymax is the maximum sustainable population of the environment, a is a constant and H is a constant harvesting rate. For Ymax=8000, a=0.0001, and H= 1000, using the DEplot command. graph y(t) for 0 ≤ t ≤ 10 for the initial values y(0) in 1000, 4000, 8000 and 10000. Now determine populations y for which y' = 0, i.e., find the initial polulations for which there is no growth or decline. You should get two. Graph these on the same graph - you should get two straight lines.See Answer
• Q6:Question 6 Consider a random walk in the XY plane where at each time step you walk one step (one unit) either to the left, right, up or down, at random. Starting from the origin, generate plots for two random walks with at least n=1000 random steps (n=10,000 is much better). So first create a list of n values P = [[0, 0], [x1,y1]· [X2,y2 ]---; (xn,yn ]] - You can also use an array of points here here instead of a Maple list. Then you can simply graph them using the plot(P, style=line ); command. To get random numbers from 1,2,3,4 use the following > R = rand (1..4): Now when you call R() you will get one of 1,2,3,4 at random, e.g.. > R(), R(), R(); 3,3,2 See Answer
• Q7:Question 7 Input and evaluate the following four definite integrals using Maple's 2 dimensional input. Use the Expression pallette for a definite integal and the Common Symbols pallett for , \pi ,∝ See Answer
• Q8:Question 9 The NonIsomorphicGraphs command in the Graph Theory package generates graphs. Two graphs which are structurally different are said to be non-isomophic. In the example below I have created all the non-isomorphic graphs with 5 vertices and 6 edges which are connectedSee Answer
• Q9:Maple Lab #3: Project Remember to submit a detailed lab report including an introductory description on the lab's main goals and concepts. Include also a list of the main commands you learned and what they're used for, as well as the correct syntax. Make sure each exercise should be well explained by paragraphs. Remember to use restart before each problem to make sure your variables were not previously assigned. 2 1. Enter the functions f(x, y) = √ (x − 2)² + y² + √(x + 2)² + y² and g(x, y)=√(x-2) compute C = dg əx 2 2 2)² + y² -√(x + 2)² + y as expressions. Then a²g and L af ду 2 + 𐐀х og evaluated at (a,b). მ 2 dy 2. Plot the graph and the contour plot of the function 2 f(x, y) = y +√√√x² + (y-2)². Discuss the shape of the contours and any local minima and maxima of the function. Notice that f is the sum of the distances from (x, y) to the point (0, 2) and the line y = 0.See Answer
• Q10:OVERDAMPED MOTION The initial- value problem x" + 3x + 2x = 0 x(0) Xo, x'(0) = x₁ is a model for an overdamped spring/mass system. = (a) Let xo 3. Use maple solver to determine experimentally a range of values for the initial velocity x₁ such that the mass passes through the equilibrium position. (b) Repeat part (a) for xo=2, Xo = 1 Xo = -2 Xo = -0.5 (c) For any xo> 0, use part (a) and (b) to conjecture a range of values for x₁ such that the X1 mass passes through the equilibrium position. Then prove your assertion analyticallySee Answer
• Q11:Problem 1 f(x)=x²-1 g(x) = -x + 2, x=0, x=2 a) Sketch the graphs of the equations given above in the same Cartesian Plane. b) Find the area of the region bounded by the graphs of the given equationsSee Answer
• Q12:Problem 2 For each of the questions a), b), c) and d) find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines and display the graph of the solid as seen in class. g(x)=√x y=0, x=3 a) x-axis b) y-axis c) x = 3 d) y = -2See Answer
• Q13:Question 1 sConsider a chemical reaction involving chemicals A, B, and C in which A is converted to B at a rate k and chemical B is converted to C at a rate of k₂ as illustrated in the compartment model below. I created the figure in Maple using the "Drawing" option under the Insert menu. Note, in older versions of Maple it's called the "Canvas" option. Letting A(t), B(t), C(t) be the amount of chemical A, B, C at time t we can model the chemical reactions with the differential equationsSee Answer
• Q14:Question 2 Shown in the figure below is a house with three rooms, A, B and C. Rooms B and C are the same size and shape. There is a furnace F in room C which heats room C. Let A(t), B(t), C(t) be the temperature at time t in rooms A, B, C respectively and let Am be the outside temperature. Shown in the figure are the cooling rate constants k₁, k₁, ką - k₁, k₁ for how heat moves through the walls of the house.See Answer
• Q15:Question 3 Shown in the figure below is lake Erie and lake Ontario and the main rivers flowing through them (the arrows). Google says that the volume of lake Erie is about 500 km³ and lake Ontario is about 1500 km³ and the amount of water flowing through the lakes is about 60 km³ per year. Yes, that's kilometers cubed. The goal is to model the amount of pollution in the two lakes at time t (years). We will assume that initially, there is no pollution in either lake and that the river flowing into lake Erie is polluted and is bringing in 30 tons of pollutant per year. Let Er (t) be the amount of pollutant (in tons) in lake Erie at time t and let On (t) be the amount of _pollutant (in tons) in lake Ontario at time t (years). Part (a) Set up two differential equations, one for the amount of pollution in lake Erie at time t and the other for the amount of pollution in lake Ontario at time t. This problem is very much like the tank problem in the last assignment, except that here we have two tanks (two lakes). Parb (b) Solve the differential equations and plot the solutions for a suitable time domain. You should see that the amount of pollutant in each lake increases from 0 to a maximum. What are the _maximums? Part (c) Using the DEplot command in the DEtools package, generate a field plot with solution curves for initial values Er (0) = 0, On (0) = 0 and Er(0) = 500, On (0) = 0 and Er (0) = 500, On(0) = 1500 on the same plot.See Answer
• Q16:Question 4 The Kermack-McKendrick virus spread model (where we partition the individuals in a population into those which are susceptible, infected, and recovered), is given bySee Answer
• Q17:Question 5 Let M(t) be the amount owed on a 30 year mortgage of $200,000 at time t years. Suppose the annual interest rate on the mortgage is 7=4%. Suppose the term of the mortgage is 30 years, i.e., M(30) should be 0. Suppose we pay$P per year. We can model the change in what we owe the bank with the differential equation M (t) = r.M(t) - P dollars per year and initial values M(0) =$200000. This model assumes the interest is charged continuously (banks usually charge interest daily which is approximately continuous) and if we assume we make the payments continuously (banks usually require us to pay monthly or weekly which is approximately continuous over 30 years). So the values we obtain with it will be approximations.See Answer • Q18:Question 6 In Question 5 the 30 year mortgage was$200,000 and the interest rate was 4% per year compounded daily. We modelled the change in what we owe the bank with the differential equation M (t) = r-M(t) - P dollars per year We solved the differential equation with initial value M(0) = 200000. We worked out if we set P= \$11448.10 per year then M(30) = 0. But the differential equation assumes a continuous process in which we are continually being charged interest and we are continually paying down the mortgage. In reality the bank charges us interest once a day and we make payments once every two weeks or once a month at the end of the month. Let's assume monthly payments. Assuming no leap years for simplicity, over a 30 year period there will be 30-365 = 10,950 interest charges which is almost continuous. But there are only 30-12 = 360 payments, less continous.See Answer
• Q19:Question 7 The following is a solution to the random walk exercise in Assignment 4 by a student. It is correct but very slow for large values of n.See Answer
• Q20:Question 8 The study of Bohemian matrices is a current research topic. A family of Bohemian matrices is a set of matrices where the entries of the matrices are restricted to a finite set of values, for example, {0,1} or {-1,0,1}. One is interested in the set of eigenvalues of such a family of matrices. In the Maple code below I've constructed all 2 by 2 matrices whose entries are 0 or 1. Since a 2 by 2 matrix has 4 entries and each entry can be a 0 or a 1, there are 24 = 16 such matrices.See Answer

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