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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
Q11:37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?/n Assignment Instructions: Use Maple to help you solve and analyse the problem. Include at least! one relevant graph and a verbal analysis. It may take some experimentation and adjusting of the horizontal and vertical axis to get graphs that show the pertinent features. Treat this as a standalone report. Please include the problem statement, clearly labelled mathematical work, clearly labelled 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). NOTE: There is only 1 question, to be done using Maple. Need the graph and analysis too. All the instructions must be followed. Need a paragraph at the end too. An example is attached in reference sections (do have a look to know how it should be done)See Answer
Q12: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
Q13:Convergence and Divergence Which of the sequences {an) in Exercises 31-100 converge, and which diverge? Find the limit of each convergent sequence. 35. 39. an = 1+ (-1)"See Answer
Q14:Determining Convergence or Divergence your Which of the series in Exercises 13-46 converge, and which diverge? Give reasons for answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) 15. ∞ n=1 n n+1See Answer
Q15: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
Q16: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
Q17:3. Given the function f(x) a) Graph the function. b) As you zoom in on the graph nearx = 1, what does the limit appear to be? Describe the nature of the graph. Use Maple, insert a spreadsheet, and show the x-values and output values for f(x). c) Is f(x) continuous at x=1? What does the graph appear to tell you versus what you know about f(x) =See Answer
Q18:5. 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
Q19:8. Let f(x)= (x+1, x < 4 lx-4, x ≥ 4 a) Explain why f is (or is not) continuous at x = 4. Explain why fis (or is not) differentiable at x = 4. In general, what is the relationship between continuity and differentiability?See Answer
Q20:Let f(x) = |x² + 3x + 21. Graph f(x). Using the graphical evidence together with your conceptual knowledge of how f(x) should behave, answer the following questions, and explain your answers. a) For what values of x, if any, is f(x) not defined? b) For what values of x, if any, is f(x) not continuous? c) For what values of x, if any, is f(x) not differentiable?See Answer

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