ASSIGNMENT 3 Huiyu Chen 89679716 James Liang 58287723 Sharon Jiao 64517188 Tony Zhang 37406642 Daniel Mahmud 86411485 1. One important component of problem solving is to extract general insights from specific mistakes. Find one problem that you solved incorrectly on a previous assignment. In one paragraph, describe the specific mistake you made. In a second paragraph, describe a general modification you can make to your problem solving approach to avoid making similar mistakes in the future. (If you achieved a perfect score on all of your previous assignments, describe a general part of your problem solving approach that helps you do so well!) One of the problems we solved incorrectly was Problem 4. Although the general approach we applied was mostly correct, logical arguments and explanations were lacked, which led to loss of points. Specifically, knowing that the function p(t) represents the size of the population of parasites t days before and after the treatment is introduced, t is meant to be a variable rather than a constant. However, by assigning t a certain value, a mistake was made due to our lack of consideration upon the property of t. Then, a similar mistake was made in the next step, when March 1st was set as the x day instead of the xth day, leading to a misunderstanding that x might be a fixed value. Both mistakes were caused due to our insufficient understanding to the fact that a variable should have changing values. In the future, before solving problems, we need to first establish a clear vision regarding the proper- ties of the variables we will be applying or solving for. For instance, we ought to understand that a function describe a general relationship between the dependent and independent variables, rather than single values. In addition, clear explanations should be provided to support the purposes and reasons of the key steps in our solutions. After solving the problems, proofread on both grammar and logic are necessary to eliminate any potential fallacies. The rectifier function r(x) = { 0 if x < 0 0 is used in artificial neural networks to model the firing of neurons. However, r(x) is not differentiable at 0. Differentiability can improve the stability and performance of neural networks. Two common differentiable approximations to r(x) are the softplus function p(x) = log (1 + e2) and the swish function s(x) = x 1 + e-x In this assignment, you may use without proof the facts that p(x) > r(x) and s(x) < r(x) for all x, and p(x) and r(x) are both continuous.
2. (a) Explain why p(x) approximates r(x) well for large (positive and negative) values of x. Take the derivative of p(x): x p'(x) =- e eª + 1 For large (negative) values of x: As the x approaches -oo, the graph of eª approaches 0. Therefore, the denominator 1 + eª approaches 1. According to the properties of logarithmic function, as 1 + eª approaches 1, log(1 + e2) approaches 0. Thus: lima-+-0