THE UNIVERSITY OF SYDNEY SCHOOL OF MATHEMATICS AND STATISTICS Assignment 2 MATH1061: Mathematics 1A Lecturers: Nathan Brownlowe, Chris Lustri, Brad Roberts and Haotian Wu Semester 1, 2024 This individual assignment is due by 11:59pm Sunday 12 May 2024, via Canvas. Late assignments will receive a penalty of 5% per day until the closing date. A single PDF copy of your answers must be uploaded in Canvas at https://canvas.sydney. edu.au/courses/54497. It should include your SID. Please make sure you review your submission carefully. What you see is exactly how the marker will see your assignment. Submissions can be overwritten until the due date. To ensure compliance with our anonymous marking obligations, please do not under any circumstances include your name in any area of your assignment; only your SID should be present. The School of Mathematics and Statistics encourages some collaboration between students when working on problems, but students must write up and submit their own version of the solutions. If you have technical difficulties with your submission, see the University of Sydney Canvas Guide, available from the Help section of Canvas. This assignment is worth 10% of your final assessment for this course. Your answers should be well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any resources used and show all working. Present your arguments clearly using words of explanation and diagrams where relevant. After all, mathematics is about communicating your ideas. This is a worthwhile skill which takes time and effort to master. The marker will give you feedback and allocate an overall mark to your assignment using the criteria available through Canvas. Copyright 2024 The University of Sydney 1 Please justify your answers. Correct answers without adequate justification will not receive full marks. A plot on its own is not considered adequate justification. 1. Consider f(x) = x³ + 3x² − 9x – 6. (a) Find and classify the critical point(s) for f(x) on [—4, 5]. (b) Find the global maximum and minimum values of f(x) on [-4,5]. (c) Does f(x) have any point of inflection on [-4,5]? If it does, locate this point of inflection; if not, explain why. 2. (a) For integer n ≥ 1, use upper and lower Riemann sums with n equal subdivisions to find an upper and lower bound for the value of 1 L'a (1 − x²) dx. You may use the following fact without proof: n k=1 k² n(n + 1)(2n+1) 6 (1 - x²) dx using the definition of Riemann integral. (b) Evaluate L ( 3. Let f(x) = (9+5x+x²)e¯x. (a) Find the 2nd order Taylor polynomial P2(x) of f(x) centred at x = 0. (b) Use the Lagrange form of the remainder to obtain an upper bound for the remain- der R2(x) when x = 1. 2 4. Let 1 and l2 be two lines in space defined by the parametric equations: l₁ : x = 3s, y = 4+ 5s, z = 3 + s (s Є R) l2: x = 2+t, y= −3+t, z = −2+2t (tЄ R) Let P be the plane that contains l₁ and l2. (a) Find the point of intersection of l₁ and l2. (b) Find a general equation for P. 5. Let R, and consider the system of linear equations in the variables x, y, z given by X - 5z = 4 Xy 2z = 2 X + 2y + Az = 2 х (a) Row reduce the corresponding augmented matrix to row echelon form. (b) Find the values of the constant \ for which the system has (i) no solutions (ii) exactly one solution (iii) infinitely many solutions 6. There are 2800 MATH1061 students, and they all do one of three things on a given night: they study linear algebra, they study calculus, or they watch netflix. We say a MATH1061 student ● is in State 1 if they study linear algebra; ● is in State 2 if they study calculus; and ⚫ is in State 3 if they watch netflix. MATH1061 students change their habits from night to night according to the following rules: • If a student studies linear algebra one night, they have an 80% chance of studying linear algebra the next night; a 10% chance of studying calculus the next night; and a 10% chance of watching netflix the next night. • If a student studies calculus one night, they have a 20% chance of studying linear algebra the next night; a 60% chance of studying calculus the next night; and a 20% chance of watching netflix the next night. ● If a student watches netflix one night, they have a 40% chance of studying linear algebra the next night; a 40% chance of studying calculus the next night; and a 20% chance of watching netflix the next night. We encode the collection of probabilities of moving from one state to another in the matrix P = (Pij)3×3, where Pij is the probability of moving from State j one night to State i the next night. This means P is the matrix P11 0.2 P13 P = P21 0.6 P23 [P31 0.2 P33 where the middle column has been filled in for you. 3 (a) Finish writing down the matrix P. (b) For night n we define the vector X₂ = • • x У where Z x is the number of students in State 1; y is the number of students in State 2; and • z is the number of students in State 3. This means that for night n + 1 we have = Xn+1 Pxn. Suppose initially we have 1000 students in State 1, 1000 students in State 2, and 800 students in State 3; or in other words, x0 = [1000] 1000 800 Find the number of students in each state on night two, i.e. find x2. (c) By considering the system (P = 13)x = 0, find all the vectors x that satisfy Px = x. (d) Suppose that instead of the initial conditions in part (b), we initially have 1600 students studying linear algebra, 800 studying calculus, and 400 watching net- flix. Find the number of students studying linear algebra, the number of students studying calculus, and the number of students watching netflix on night n = 100. End of assignment 4