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Q1:Math-Help-Services.com Introduction to Polynomials 3 Shannon MacLean Question 1 Question 3 Which of the following polynomials are written with their Indicate the degree of the polynomial given below. terms in the proper order? 1. 4x-1 2. x4 - 4x2 3. 2x+5x2 + 9×4 16x+4-x2 Question 4 Indicate the degree of the polynomial given below. Question 2 Which of the following polynomials are written with their terms in the proper order? 7-5x2 -12x -- 1. 3x2 -> + 5x4 + 10 2. 4x3 - x2 +9x-7 3. 5x3 -x+12 Question 5 indicate the degree of the polynomial given below. 9+5x4 http://math-help-services.org 20/04/2018 09:42:46 Page 1 of 3 Math-Help-Services.com 8 Question 6 Which of the following polynomials are written with their terms in the proper order? Question 8 Which of the following polynomials are written with their terms in the proper order? 1. x - x4 -x3 2. 7x-4x2 -2x4 3. 5x+ +9x2 - 8 1. x2 - 4 2. 5x2 - x-15x4 3. 81-x4 Question 7 Question 9 Which of the following polynomials are written with their terms in the proper order? Indicate the degree of the polynomial given below. 6x-5x3 http://math-help-services.org 20/04/2018 09:42:46 Page 2 of 3 1. 6x5 -3x4 -x3 2. 2x4 +7x- 422 - x3 3. 3x2 + 7x4 -12x6 9 Math-Help-Services.com Question 10 Which of the following polynomials are written with their terms in the proper order? 1. x3 -7x2 + 12x 2. x2 - 19+ 11x3 3. x+1 http://math-help-services.org 20/04/2018 09:42:46 Page 3 of 3 10 Math-Help-Services.com Introduction to Polynomials 3 Shannon MacLean - MTH-S42 Module 1 (16719) Answer Sheet Question 1 1 and 2 only Question 2 2 and 3 only Question 3 2 Question 4 2 Question 5 4 Question 6 3 only Question 7 1 only Question 8 1 only Question 9 3 Question 10 1 and 3 only - http://math-help-services.org 20/04/2018 09:42:46 Page 1 of 1See Answer
Q2:Math-Help-Services.com Introduction to Polynomials 1 Shannon MacLean Question 1 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 4 Identify the variable and the constant in the polynomial given below. 2-x2 +3x2 - 4x+2x +6+ 4x2 = ? 9x3 +5x2 - 5 Question 2 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 5 Combine the like terms of the polynomial given below and. identify the leading coefficient of the resulting polynomial. 2x2 -2 -3+2x+6+4x2 -10-2 = ? Question 3 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 6 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. 3x2 +5x+6-2x2 +2x-1 =? 2×3 -3x-2x3 - 1+3x+52 - 1 =? 9x3 -x-2x3 - 1+3x+5x2 - 5 =? http:/math-help-services.org 20/04/2018 09:39:56 Page 1 of 2 Math-Help-Services.com Question 7 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 9 Identify the variable and the constant in the polynomial given below. 4-3x2 +x+5-4x+ 7x2 = ? ₹ -4x+5 Question 8 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. Question 10 Combine the like terms of the polynomial given below and identify the leading coefficient of the resulting polynomial. ≥ +3x+2x3 +4+23 - 2x2 -3x+6 =? 2x3 +5% +2x-62 -1 =? http://math-help-services.org 20/04/2018 09:39:57 Page 2 of 2 Math-Help-Services.com Introduction to Polynomials 1 Shannon MacLean - MTH-S42 Module 1 (16719) Answer Sheet Question 1 6x2 - 2x+8 leading coefficient is 6 Question 2 7x3 + 5x2 + 2x-6 leading coefficient is 7 Question 3 5x2 -2 leading coefficient is 5 Question 4 variable is x constant is - 5 Question 5 -2x3 +6x2 +2x-7 leading coefficient is - 2 Question 6 x +7x+5 leading coefficient is 1 Question 7 4x2 - 3x+ 9 leading coefficient is 4 Question 8 4x3 - 2 + 10 leading coefficient is 4 Question 9 variable is x constant is 5 Question 10 2x3 - x2 +2x-1 leading coefficient is 2 http:/math-help-services.org 20/04/2018 09:39:57 Page 1 of 1 3See Answer
Q3:Question 2 Consider the gambler's ruin with draws: Alice starts with fa and Bob with £(m - a), and at each time step Alice wins £1 off Bob with probability p, loses £1 to Bob with probability q, and no money is exchanged with probability s, where p +q + 8 =1. We consider the case where Bob and Alice are equally matched, so p= q and s = 1- 2p. (We assume 0<p <1/2.) Let ry be Alice's ruin probability from the point she has Li. (a) By conditioning on the first step, explain why pri+1-(1-s)r.+pr ;- 1 = 0, and give appropriate boundary conditions. [2] (b) Solve this linear difference equation to find an expression for ry- [2] Let d, be the expected duration of the game from the point Alice has &i. (c) Explain why pd141 - (1-8)d, + pd1-1 =- 1, and give appropriate boundary conditions. [2] (d) Solve this linear difference equation to find an expression for d .. [2] (e) Compare your answer to parts (b) and (d) with those for the standard gambler's ruin problem with p = 1/2, and give reasons for the similarities or differences. [2]See Answer
Q4:1. Let A be the area between the x-axis, the curve y=x+, and x = 2 and x 5. Find a such that the line xa divides this area into two regions of equal area. 2. Find the volume of the solid of revolution when the area between y x and y =x³ between x=0 and x = 1 is rotated around the x-axis.See Answer
Q5:Problem 2 Consider the problem minimize f(x) = -x² s.t. x₂ ≤ x² X1 20 where #1, #2 € R. a. Does the point (1, ₂) = (0,0) satisfy the FD-FONC? b. Is the point (₁,₂)= (0,0) a local minimizer, a strict local minimizer, a local maximizer, a strict local maxi- mizer, or none of the above? Problem 3 =See Answer
Q6:Problem 7 Let A € Rmxn, m≤n, rank(A) = m, and ceR", beR". Show that the problem minimize ||z-c||² s.t. Arb has a unique solution and find the expression for r*.See Answer
Q7:Problem 5 Write the following function as a quadratic one (a) f = (2x - y)² + (y-2)²+(2-1)² (b) f = x² + 16xy + 4yz + y² (x¹Qx+2b¹x+c).See Answer
Q8:Problem 4 Consider the LS problem with A = 1 1 -1 0 1 and b = 1 (a) Solve the LS problem. (c) Write down the matrix that represents the orthogonal projection onto col(A).See Answer
Q9:Problem 3 Rewrite each problem minimize, f(x) as a LS problem. Specify matrix A and vector b, and then solve the problem. (a) f(x) = (2x₁ - x₂ + 1)² + (x₂ − x3)² + (x3− 1)² (b) f(x) = (1 - ₁)² + Σ₁=₁(x₂ − xj+¹)²See Answer
Q10:Problem 2 Given the experimental data y = 13.26 14.15 13.86 14.81 15.68 15.64 15.58 15.67 16.08 16.36 16.14 .. 16.68 16.00 15.66 16.05 16.22 16.17 16.03 16.69 15.83 16.21 16.13.. 16.36 16.42 16.92 16.65 16.94 17.47 18.07 17.52]; x= 1.00 1.13 1.27 1.41 1.55 1.68 1.82 1.96 2.10 2.24 2.37 2.51 2.65 .. 2.79 2.93 3.06 3.20 3.34 3.48 3.62 3.75 3.89 4.03 .. 4.17 4.31 4.44 4.58 4.72 4.86 5.00]; 1. Find the least-squares line that fit this data. 2. By inspecting the residuals, can you determine if the previous model is suitable? 3. Can you find a better model than 1.? Use residual plot to measure "betterness".See Answer
Q11:MA576 - Midterm All answers must be motivated and clearly formulated. Explain each step in your solutior Your solutions should make very clear to the instructor that you understand all of the steps and the logic behind the steps. Correctly interpreting the statement of each proble is part of the test. Each problem is 25 points. Very importantHonor code applies fulgu must submit your own work only. This means you cannot consult any other person (except me) directly or indirectly abou anything related with the topics of this midterm during the time you alre solving it. particular, it is prohibited to post the following problems or ask anything on any website/forum/tutoring system or any other virtual means. Question 1 Let f(x) = log Xk Question 3 Consider the problem: i=1 for given, € R¹, B₁ € Randx € R¹. a) Give a formula for the gradient and the Hessian. b) Show it is convex. ex+B₂ Question 2 Let X CR¹ convex. Prove thatis an extreme pointXoff and only if the sets convex, where = CU {}. minimize x+y s.t. x+ 2y ≤9 (x-2) + (y- 4 ≤2 a) Describe the normal cone to the feasible set at the point (1 b) Use the necessary optimality conditions to solve the problem. Are these condition sufficient? Question 4 Compute the subdifferentifilxof= max(1x,0) forx € R.See Answer
Q12: Problem 1 Compute the limit Problem 2 Find global minimizers of lim (x,y) →(0,0) using a Taylor expansion of order 2. on the set X = MA576 - Extra problems Problem 3 Find global minimizers of e²+y(y+1)(1+y+y²/2) x² + y² f(x₁, x₂) = 2x1 +X2, - {(x1, x2) € R² : x² + x² − 1 = 0}. Lagrange multipliers cannot be used. Problem 4 Find global minimizers of f(x₁, x₂) = x₁ - x², on the set X = {(x₁, x₂) € R² : x1 ≤ 2x2 − x², x1 ≥ −x², x1 ≤ 2 − 2x2 − x²}. Lagrange multipliers cannot be used. f(x₁, x2, 3) = exp(x₁ - x₂) + exp(x₂ − x₁) + exp(x²) + x3. ƒ(x1, State if the optima you found is strict or non-strict. Problem 5 The main hypothesis of the FONC condition we studied is that f € C¹ on its domain. We can still use the same optimality condition in the case the function is not C¹ on the whole domain as long as we study separately “problematic” points. For example, the points in which the gradient is undefined. The objective function f(x₁, x2) = √√√x² + x² is continuous in R², that is Cº, but it is not C¹. Find the global minimum of that function. Hint: no need to use FD optimality conditions. Problem 6 Consider minimize xERn = ₂T Ax xBx' X where A and B are symmetric and pd. This problem has infinite solutions, so we choose the solution that satisfies x¹ Bx X 1. This problem can be approached as a regularized least squares problem. The FONC leads to an eigenvalue problem. 1. Why this problem has infinite solutions? 2. Find the stationary points of this problem. 3. Check a second order condition. 4. Solve a particular example in R². 5. How the solution changes if A is psd instead of pd?See Answer
Q13: Homework 5 Important submission instructions: • All explanations and answers must be clearly and neatly written. Explain each step in your solution. Your solutions should make very clear to the instructor that you understand all of the steps and the logic behind the steps. You are allowed to discuss the homework problems with other students (in particular via Canvas discussion board). However, what you hand in should reflect your own understanding of the material. You are NOT allowed to copy solutions from other students or other sources. Also, please list at the end of the problem set the sources you consulted and people you worked with on this assignment. • The final document should be saved and submitted as a single .pdf file, and please be sure all problem solutions are presented starting from the first to the last (that is, the first solution must correspond to problem 1 and the second to problem 2 and so on). • Typed submissions (for example in LaTeX) will be positively considered in the grade. Overleaf is an easy avenue to start learning LaTeX. See the tutorials at https://www.overleaf.com/learn • Very important: Honor code applies fully. You must submit your own work only. In particular, it is prohibited to post the following problems on any website/forum or any other virtual means (for example Chegg). Problem 0 Which of the following sets are convex? Justify either by definition or some other formal way. 1. The set {x E R³ : ai ≤ xi ≤ bi, i 2. The set {x R²0 : x1×2 ≥ 1}. == 1, , n}, for ai, bi Є R. 3. The set of points closer to a given point than a given set: where SCR”. {x € R” : ||x − x0||2 ≤ ||x − y||2 for all y € S}, 4. The set {x ЄR" : x + S₂ ≤ S₁}, where S1, S2 CR" with S₁ convex. Problem 1 Consider the following sets: (a) A = {(x1, x2) ЄR²: |x1|+|x2| ≤ 1}; (b) B = {x € R³ : 0 ≤ x1 ≤ x2 ≤ x3}; Hint: Recall {x: Ax ≤ 0} is a convex cone; (c) C = {x R2 x2 ≤ x}}; : (d) D = {(x1, x2, x3) Є R³ : √√√x² + x ≤ x3}; For each set: 1. Prove or disprove it is convex. 2. Prove or disprove it is a cone. 3. Determine the extreme points (if convex). 4. Determine the recession cone (if convex). Problem 2 Consider the set of all non-decreasing functions ƒ : R → R. Prove or disprove it is a convex cone. Problem 3 Let be a set of n elements. 1. Describe the set P of all probability distributions (probability mass functions) sup- ported on . 2. Show that P is convex. 3. Determine the extreme points of P. Problem 4 K = {x R 0 ≤ x ≤ x2 < ... < xn} Compute the polar cone K°. Problem 5 Consider the following optimization problem: minimize -10x1 - 2x2 + x3 + 5x4 s.t. 2x1x3x4 = −5 4x1 x22x4 = 3 x > 0 1. List all basic solutions indicating which ones are feasible. 2. What is the solution of the problem? Problem 6 Consider the optimization problem: maximize (2xy)² + ( y − z)² + ( − 1)2 s.t. x+2y+ 3z = 1 x, y, z≥ 0 1. What is the solution of the problem? 2. What is the solution of the probem if the objective function is f(x) = z +1? 3. What is the solution of the probem if the objective function is f(x) = x+z+1? Problem 7 Consider the following problem minimize log(y2 — x²) - s.t. x² + y ≤1 2x - y ≤0 y ≥ 1/2 x ≥ 0 What is the solution of the problem? Problem 8 Consider P = {x = R^ : Ax = b,x > 0} where A = Rmxn. Assume the rows are l.2. For each one of the following assertions, state whether it is true or false. If true, provide a proof, else, provide a counterexample. 1. If n = m + 1, then P has at most two basic feasible solutions. 2. The set of all optimal solutions is bounded. 3. At every optimal solution, no more than m variables can be positive. 4. If there is more than one optimal solution, then there are uncountably many optimal solutions. 5. If there are several optimal solutions, then there exist at least two basic feasible solutions that are optimal. 6. Consider the problem of minimizing max{c¹x, d³ x} over the set P. If this problem has an optimal solution, it must have an optimal solution which is an extreme point of P. Problem 9 Implementing and Applying Support Vector Machines (SVM) Part 1: Implementing SVM with Synthetic Data • Implement a linear SVM classifier using CVXpy. Create a synthetic dataset with two features and a binary target variable. En- sure that the data is linearly separable. Visualize the dataset using a scatter plot, distinguishing the two classes with different colors. Your implementation should include the formulation of the hard and soft SVM optimization problem. Apply the SVM model to the synthetic data and find the optimal hyperplane. Visualize the optimal hyperplane along with the margin and support vectors on the scatter plot of the synthetic dataset. Discuss the role of support vectors in SVM. Calibrate the parameter (soft formulation). Part 2 (bonus credit): Applying SVM to a Real-World Dataset • Choose a well-known dataset that is suitable for binary classification. Examples include the Iris dataset (selecting two of the three species) or the Breast Cancer Wisconsin (Diagnostic) dataset. • Preprocess the dataset as necessary. • Apply the SVM model implemented in Part 1 to the selected dataset. • Evaluate the model's performance using appropriate metrics such as accuracy, pre- cision, and recall. Compare these results with a baseline model, such as logistic regression. • Discuss how well the SVM model generalized from the synthetic data to the real- world data. • Reflect on the strengths and limitations of SVMs based on your observations. • Reflect on the difference in results between synthetic and real-world data and what this implies for the application of SVM in practical scenarios.See Answer
Q14:Problem L5.6. Let B(t) be a sBM. Consider the process Y(t) =tB(1/t), for t> 0. It is clear that Y(t) has continuous path for t> 0, and using law of large numbers, we get Y(0) = 0. It is also easy to show that Y is a Gaussian process. Prove that Cov(Y(t), Y(s)) = min(t,s). Hence, conclude that Y is a sBM.See Answer
Q15:Problem L5.4. Prove Theorem L5.8(3)/nTheorem L5.8. 1) B(t) is a martingale 2) B2(t)-t is a martingale 3) euB(t) is a martingaleSee Answer
Q16:Problem 9 Consider the problem of minimizing the function f(x, y) = a(x² + y²) + bxy — (x + y) + c over R2, where a, b, c € R. Find all values of a, b and c such that the problem has a unique optimal solution.See Answer
Q17:Problem 3 Find all stationary points of the following R2 functions and classify them (only local): (a) f(x, y) = 1- y² — x¹. (b) f(x, y) = x² + y² − 2x² + 4xy – 2y² (c) f(x, y) = (ax² + by²)e-²+y², where a, b € R.See Answer
Q18:Problem 2 Let QE Rxn and R = Rmxm symmetric. Prove that Q, R are psd (2) is psel, R where 0 denote the zero matrix of dimension n x m.See Answer
Q19:4.9 Assignment: Culminating task Part of your evaluation for this course will be in the form of a Culminating Task. In this task, you will apply your advanced understanding of mathematical functions to addressing real-world problems, including mathematical models and interpretation of mathematical information. This assignment is worth 15% of your course grade. Instructions Overview You are going to make a pitch to the hosts of a reality TV show where entrepreneurs request funding for projects or products that are designed to improve communities. Your pitch must combine a compelling reason why your initiative should be supported, with evidence that justifies the need for action. You will research and collect secondary data, use the data to create graphical and algebraic models, make predictions based on your algebraic model, and create a visual presentation using the media/platform of your choice. Some example options are PowerPoint, WordPress, Prezi, Weebly, iMovie, Show Me, Blogger, Tumblr, YouTube, etc. The function used to create your algebraic model must be one of the types of functions covered in this course (polynomial of degree 3 or higher, exponential, logarithmic, or trigonometric)./nStep 1: Identify and describe a community issue and a proposed way to reduce impact and/or improve outcomes. As an example, the proposal could be a campaign to convince people to change daily habits that are harmful to the community. Or you may propose the development of a new product or process that can help to address a community issue. The proposal could also be an initiative that creates awareness of a local, national, or global issue. Step 2: Research secondary data from a reliable source and create a mathematical model to predict outcomes if no action to reduce harmful effects is taken. Step 3: Create algebraic and graphic representations of your model and provide a brief, but convincing, description of the benefits of your proposal Sample submission To get you started, you may choose to review [this sample report]. This is just an example, and should not be interpreted as limiting your choice of topic, format, model, or approach to the issue. Your teacher will grade and provide feedback on your work using the following success criteria and rubric. Before submitting your assignment, review the success criteria and rubric. If you are unsure about how to format your assignments, citation or what constitutes plagiarism, please review the following support pages: • Referencing and Formatting Submission Guidelines Avoiding Plagiarism Guide Crinnan and Mathematine Suhmissinn DanusiramantaSee Answer
Q20:How many pairs (x, y) satisfy the equalities x-1 3-x-y = xy-37-x²-y² y-2 xy-4 -?See Answer

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