Assignment 1
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Assignment 1
IMPORTANT INSTRUCTION: Present your solution so that the sequence of logical steps is clear, with succinct justfication for each step where it is not obvious. Note that marks will be deducted for solutions that do not show all the important logical steps and reasoning in reaching a conclusion. Marks can also be deducted if solutions are much more convoluted than required.
COVER SHEETS: Note that online submissions through the standard Moodle assignment submission system do NOT require a cover sheet.
1. (a) Show carefully how to use the rules of logarithms given in lectures to deduce that ln(x3z+1 ) = (3x + 1) ln(x) whenever x > 0. State which rule(s) you are using in general form.
(b) Giving full details of all steps, show how to use the product rule for derivatives to find the derivative of (3x + 1) ln(x).
(c) Show how to find the derivative of x3z+1 by first writing it in a different exponential form and then using the results of parts (a) and (b) above. State which rules you use along the way.
2. Sketch a neat the graph of a function with domain [0, 4] that clearly has the required properties in each case, without the graph containing a straight line on any significant interval:
(a) f (x) is increasing and concave on [0, 1], increasing and convex on [1, 2], and has a global minimum at x = 3.
(b) g(x) is convex on [0, 1], convex on [1, 3], and has global maxima at x = 1, x = 3 and x = 4.
(c) h(x) is convex on [0, 1], concave on [1, 2], convex on [2, 4], and has no stationary points.
3. A function h : A _ B is observed to have an inverse function h− 1 such that h− 1 (x) = x10 + x4 + 1 for all x ∈ B . Given this information:
(a) Compute h− 1 (x) for x = 2, 1, 0, -1, -2.
(b) Is it possible that B contains both -1 and 1? If so, give an example of one possible such set B, and if it is not possible, explain why not.
(c) You are given additional information: h is concave on the interval [1, 3]. Using this fact, explain why h(2) > eh(1) + h(3)_/2. (We advise you not to try to compute a formula for h(x).)
4. A rumour is being spread by students in Melbourne. At the start of day 1, a set of 10 students know the rumour. Let S0 be this set of students, and for n > 1, let Sn denote the set of students who are told the rumour during day n. Also, write sn to stand for the number of students in Sn . Assume that students in Sn pass on the rumour on day n + 1 to the following numbers of new students: 0.2sn in cafes, 0.3sn in public transport, 0.4sn at work, and 0.3sn in university classes. Assume that no student is told the rumour twice, and students in Sn do not pass on the rumour any more after day n +1. In the following, assume the above specifications hold precisely, ignoring the fact that the numbers of students of each type should be integers.
(a) For every integer n > 1, derive a formula for sn using exponential notation.
(b) Show how to use logarithms to find the first value of n for which sn exceeds 10,000.
(c) The government is embarassed by the rumour, so it introduces social distancing laws at the end of day k, to reduce the spread of the rumour. This restriction is only imposed in public transport and in university classes. As a result, the number of students told the rumour from these two sources changes to 0 .1sn each (on day n + 1, when n > k, as described above). The other sources follow the same rule as before. Find the largest value of k that will permit s50 to be less than 10.
5. For each of the following functions, find all local and global extrema. Give full reasons for your answers. Also find the second derivative in each case and explain how it can be useful for this purpose.
(a) f : [0, 2.99] _ R where f (x) = x2 + 4 ln(3 - x).
(b) g : [-2, 2] _ R where g(x) = xe−北 .
2023-03-21