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.
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.
(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.
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.