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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 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
Q4: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
Q5: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
Q6: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
Q7: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
Q8: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
Q9:Use Maple to help you solve and analyze the problems. For the second problem, include at least one relevant graph and while you may utilize Maple to check the integral, please clearly identify and describe the method and steps to evaluate the integral as if one were using pencil and paper. Treat this as a standalone report. Please include the problem statement, clearly labeled mathematical work, clearly labeled and relevant plots, and explanations for all (mathematical work, plots, and conclusions). For formatting within the Maple file, please include at the top your name and the problem number you are presenting. At the end of your Maple file, please include a paragraph sharing what you have learned and any difficulties you found (mathematically or with Maple). Attach this Maple file to your main discussion post (which will be as a response to your empty main post where you claimed your problem)./nEvaluating Improper Integrals The integrals in Exercises 1-34 conserge. Evaluate the integrals without using tables. 3./nNonrepeated Linear In Exercises 9-16, express the integrand as a sum of partial fractions and evaluate the integ € -./. 12. 2x+1 x² - 7x+12 dxSee Answer
Q10:Using Maple, evaluate the following limit. Verify your results graphically, numerically, and algebraically, if possible. x³-1 x5-x a) limx-13 X-9 √√x-3) d) limx-91 읊 b) limx-o 1 X limx-23 2. Evaluate the following discontinuities: 1-x √x+x-2 x-3 c) limx-31 f) limx-07 x 1See Answer
Q11:CONT 2. Evaluate the following discontinuities: a) Find the discontinuities, if any, for the following functions. b) For each discontinuity, indicate whether it is removable or non-removable. c) Using Maple, verify your results graphically, then confirm your results algebraically by evaluating the appropriate limit. a) f(x): = x²-x x²+x-2 d) f(x) = 2-x x²-4 b) f(x) = x245246 x+2 -1 < x < 2, x 1 x² + 3x-2,2 ≤ x < 5 -- e) f(x) = x¹-4 c) f(x) = -1 x³+x=2 (x²-2, x < -2 f) f(x)=5,-2 ≤x≤0 8x-5, x > 0 JSee Answer
Q12:5. Given the function: g(x) = (8-2x if x < 4 √x-4 if x>4 a) Find the limit of g(x) as x approaches the value "4" indicated below. To do this, compute a table of values of "x" and the corresponding f(x) where "x" takes on the values getting closer and closer to "4" from both sides of "4". b) For the function g(x) given above, determine if g(x) is continuous at c = 4. c) If discontinuous, state whether the discontinuity is removable or non-removable, and explain why. d) At the point x = 4, identify if there is a cusp or a corner..See Answer
Q13:Distances In Exercises 33-38, find the distance from the point to the line. 35. (2, 1,3); x=2+21, y=1+6t, z=3See Answer
Q14: Finding Tangent Vectors and Lengths In Exercises 1-8, find the curve's unit tangent vector. Also, find the length of the indicated portion of the curve. 7. r(t)=(t cost)i +(t sin t)j + (2√2/3) +³/2k, 0 ≤1≤ π Instructions Include at least one relevant graph and while you may utilize Maple to work the problem, clearly identify and describe the method and steps to evaluate the problem as if one were using pencil and paper. include the problem statement, clearly labeled mathematical work, clearly labeled and relevant plots, and explanations for all (mathematical work, plots, and conclusions). include a paragraph sharing what you have learned and any difficulties you found at bottom of fileSee Answer
Q15: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
Q16: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
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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