MATH2010 MATH2100 Assignment 1 Use the Gradescope assignment submission link on Blackboard to submit this assignment. The submission must consist of a single PDF file containing a printout of your Mathematica notebook and your handwritten/typed solutions. It is important that your name and student number appear on the first page of your submission. Your pdf file should be legible and not too large. Files that are poorly scanned and/or illegible may not be marked. Once you have uploaded your file, you must assign page number(s) to each question! Assignments submitted after the due date will attract a penalty, as outlined in the course profile. Total: 40 marks, allocated as indicated on each problem. Problem 1. To be completed in Mathematica 1. Obtain a 20- decimal approximation of e/T, where e is the base of the natural loga- rithm. (1 mark) 2. Express 4 - 2 + + + - as a single fraction, and obtain and approximation accurate to 4 decimal places. (1 marks) 3. Compute - sin(T/3)+v47 exactly and approximately to 8 significant digits. (1 marks) Problem 2. To be completed in Mathematica 1. Define f(x) = sin 6x + 2tan x and g(x) = 3sinx - cos 3x. Construct (f.g)(x) and to evaluate this expression at ". The result should be simplified. (2 marks) 2. Factorise the polynomial -14x3-9x2y+6xy2+y3+23x2z+4xyz-y2z-10xz2-yz2+23. (1 mark) Problem 3. To be completed in Mathematica Define a function f(x) = x3 - 2x2 - 3x + 1 in Mathematica. 1) Find the critical points using Solve and its numerical values using NSolve. Identify whether they are local maxima, local minima, or neither. (2 marks) 2) Using different colours and two separate coordinate systems, plot the graphs of f(x), and f'(x) for x E [-5, 5]. Then, combine the graphs into one single coordinate system using the commands Show and the option PlotRange-> {-50, 50}. (Hint: The option PlotRange may be useful.) (2 marks) 3) Calculate the definite integral of f(x) over the interval [0, 3], and find a numerical approximation of the definite integral to 3 significant digits. (2 marks)
MATH2010 MATH2100 Assignment 1 Problem 4. To be completed in Mathematica. Solve the system of equations: x + 2y - z = 5 2x-y+3z = - 4 3x + y + z = 2 1. Find the solution for x, y, and z. (2 marks) 2. Verify the solution by substituting it back into the original equations. (1 mark) Problem 5. To be completed in Mathematica 1. Define a function that represents the distance between the points (x, y) to (-1, 2) and use this function to find the distance between the points (1, -1) and (-1, 2), approximately to 3 significant digits. (2 marks) 2. Use this function and the command Solve to identify all points on the y-axis which are at distance 8 from the point (-1, 2). (2 marks) Problem 6. Mathematica should be used for parts 2 & 3 only 1. Compute the solution of the differential equation dy() = -