Assignments are very important part of your studies. They serve two purposes: Feedback and Diagnostic. Assignments allow students to find out how they are going in developing the skills and understanding the new ideas as the unit goes along. They are also intended to determine whether you get specific knowledge and skills which are needed constantly, so that you can remedy this if necessary. Feedback and diagnostic assessment intentionally do not carry a great deal of weight in calculating your final mark. This means that you are not overly penalised when you are still working towards developing certain skills and knowledge. But don't think a low weighting means that submitting these forms of assessment is not important – they carry value well above their weight, because they give you so much information about what you may need to work on, and what standards are expected. Similarly, it is not intended that the time it may take you to complete an assignment is reflected in its weighting; the work required to get to the stage where you can answer the questions is also a major contribution to your preparation for the final assessment. It is clearly important that feedback is provided while the task is fresh in your mind and in time for you to act on it before your next assessment. I undertake to return assignments to you in a timely manner, generally within two weeks of the submission deadline. I will put master solutions to CloudDeakin one week after the submission due date. The number and spacing of assignments ensures that you get feedback throughout trimester. On each assignment, your tutor will mark in detail some of the arguments you put forward in your solutions. You will be provided with complete model answers, and you will get most benefit from the assignment if you self-assess the other parts of your work against these. You need to develop confidence in your ability to check your own work both for the final assessment, and in your future career. You will also be assessed on how well you have planned your work to complete the whole assignment. Your use of writing and the grammar of mathematics in presenting your reasoning makes up the third component of your mark. Finally I suggest you make a copy of your submission, especially of Assignment 3, as the feedback may come after the exam date, which is unknown at this stage. SIT291 Mathematical Methods 2015 Assignment 1 Due 5 p.m. March 31, 2015 1. Show that the equation represents a sphere and find its centre and radius 4x 2 + 4y 2 + 4z 2 + 16y − 8z = 1 2. Given ⃗a = ⟨1, 4, 4⟩ and ⃗b = ⟨0, −1, −2⟩, find: (a) ⃗a −⃗b, (b) 2⃗a + 3⃗b, (c) | ⃗b|, (d) |⃗a −⃗b|, (e) ⃗a · ⃗b, (f) ⃗c = ⃗a ×⃗b, (g) proj⃗a⃗b, (h) determine whether ⃗a and ⃗b vectors are orthogonal, parallel or neither. (i) verify that ⃗c is orthogonal to both ⃗a and ⃗b. 3. Find equations of the two planes parallel to the plane x + 2y + 3z = 1 and located at the distance 1 unit from the origin. 4. Write down the following equation in spherical and cylindrical coordinates. Identify the surface given by that equation. x 2 − y 2 + z = 0 5. If ⃗r(t) = ⟨t 2 ,sin t−t cost, cost+tsin t⟩, find T⃗(0), ⃗r ′′(0) and ⃗r ′ (t)·⃗r ′′(t) 6. Find the length of the curve ⃗r(t) = ⟨t 3 , t, 1 2 √ 6t 2 ,⟩, 1 ≤ t ≤ 3. 7. Find the partial derivatives of the functions (a) f(x, y) = y ln x + xy (b) f(x, y) = x y + x 2 e y SIT291 Mathematical Methods 2015 Assignment 2 Due 5 p.m. May 5, 2015 1. Use the chain rule to find ∂z/∂s and ∂z/∂t. z = x 2 /y2 , x = set , y = se−t 2. Find the directional derivative of the function at the given point in the direction of vector ⃗v. f(x, y, z) = x/(y + z), (2, 2, 1), ⃗v = ⟨3, 2, 1⟩ Find the maximum rate of change of f at that point and the direction in which it occurs. 3. Find the local maxima and minima and saddle points of the function f(x, y) = x 3 − 6xy + 8y 3 4. Find the extreme values of f(x, y) = 4x 2+(y−1)2 on the circle x 2+y 2 = 25 5. Calculate the integral ∫ 1 0 ∫ 2 1 xex y dydx 6. Calculate the integral ∫ ∫ R xydA, R is bounded by y = √ x, y = 6 − x, and y = 0