12. Fechne'r's law describes the relationship between \"stimulus intensity\" 11:, such as the brightness of a light in lumens, and \"perceived intensity\" 3901:), such as the perceived brightness of a light by a viewer. It is given by ptm)=klog($), m where k and m are positive constants. The domain is restricted to 1: 2 m. (a) [3 marks] Find p'(m). and state all intervals in the domain where x) is increasing. (Answers should be written in the box. Justication should be outside the box.) Answer: (b) [3 marks] Find p\" (:13), and state all intervals in the domain where p(:17) is concave up. (Answers should be written in the box. Justication should be outside the box.) Answer: (c) [3 marks] Draw a large sketch of the graph of 12(1) on the axes below, clearly (e) [2 marks] Steveus' power law is sometimes taken by psychophysicists (who study labelling any intercepts, asymptotes, extrema and inection points. stimuli and perceptions) to supersede Fechner's law. It proposes that perceived intensity 1/;(1) is related to stimulus intensity 1 according to y W) = m where a is a parameter that depends on the type of stimulus, and k is the same positive constant as in Fechner's law. If 12(2) and Man) are meant to describe the same relationship, in which of the following intervals is the parameter 11? Remember to justify your answeri (d) [3 marks] At a given stimulus intensity :1}, the dzscn'mmatwn threshold is the mini- mum increase in stimulus intensity such that the corresponding increase in perceived intensity is equal to or greater than a constant Ap. According to this model, are discrimination thresholds at high stimulus intensities greater than discrimination thresholds at low stimulus intensities? Give your answer and justify it in a few sentences. 14' The differential equation (d) [4 marks] Draw the slope eld for the differential equation on the axes below. On Q 2 r P 1 _ E _ h R the slope eld, draw at least four solutions, dt K where r > h and K are positive constants, describes the growth rate of a population P P over time t (a) [4 marks] Find the steady states and indicate them clearly on the phase line below, along with arrows indicating the sign of % between steady states, (e) [2 marks] The term hP in the differential equation represents variable harvest, in which the population is \"harvested\" at a rate proportional to the present popula- P tion. In a few sentences, describe carefully what the model predicts will happen to populations if the parameter h is larger than 1". (b) [1 mark] Given an initial population P(0) = 1,5K, does the model predict that the population will increase or decrease? Answer: (c) [1 mark] Given a very small positive initial population 19(0), what will the popu lation be at the moment it is changing most rapidly? Your answer may include the terms 7*, h and K. (f) [2 marks] Suppose h : 0. In a few sentences, speculate what X represents in terms of population