10. Given the function f(x) = x4-1 > x²-x use Maple to evaluate the limit as x approaches 1 algebraically (use a spreadsheet), numerically (use the limit command), and graphically (use

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.

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.

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

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 ,∝

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 connected

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.

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.

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.

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.

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.