Math 2A In-Class Activity Sections 7.8: Laplace Transform 1. We have this spring-mass system with forcing function (): + = (), (0) = (0) = 0 Some possible forcing functions a. b. c. d. () = 1, t 0 () = {1, 0 < < 2 0 () = {1, 0 < < 6 0 () describes the impact of hitting the spring once with a hammer, in the positive direction, at some time 0 > 0. 13 Sketch the forcing functions. Classify each forcing function as continuous or piecewise continuous. Responses are shown below. Which response () matches which forcing function ()? Could the response () ever be piecewise continuous rather than continuous? Page 1 of 1 Information Festival III (Chapter 7, 8, 6) MATH 2A FALL 2014 11-25-2014 Registered Name (print): ___________________________________________ I pledge to support the mission of Foothill College and to demonstrate its core values by upholding academic integrity in this assignment. Signature: __________________________ Materials permitted for the Festival: one 3\" x 5\" note card (front and back), Ti-83 or Ti-84 calculator Time: 85 minutes Total: 150 points Directions: Please read the following directions carefully. 1. Turn off all cell phones and pagers now. 2. Please write your full name clearly which is registered for the class. 3. Read all questions carefully before responding. 4. Please show all your work clearly to get partial credit, unless instructed otherwise. If your work is hard to read, it will be considered wrong. You can earn significant points for showing clear, well-organized work even if your answer is not correct. Similarly, you will lose significant points if you don't show your work or display an answer with little or no work. 5. Make sure to go back and check your work including signs! 6. English to native-language paper dictionaries are permitted on the exam. 7. If you have a question, need extra paper, etc., please raise your hand. Page 1 of 10 1. (30 points) Consider a spring-mass system + = (), (0) = (0) = 0, with the following discontinuous forcing functions. (i) () = {1, 0 < < 0, > (ii) () = {1, 0, 0 < < 2 > 2 (iii) () = {1, 0, 0 < < and () has period 2. < < 2 0, 0<< (iv) () = { 1, < < 2 0, > 2 (A) Sketch each of the forcing functions over [0, 6]. (solution) (i) (ii) (iii) (iv) (B) Rewrite each of the forcing functions using Heaviside step functions (i.e., unit step functions). (solution) (i) (ii) (iii) (iv) Page 2 of 10 (C) Given the following choices, find the solution curve of each system and put your choice in the parenthesis provided. [You are not supposed to solve the IVP.] Use the qualitative nature of the solution to explain why clearly and briefly. ( ( ) ) 6 2.0 4 1.5 2 1.0 5 10 15 2 0.5 4 5 10 6 15 why why ( ) ( ) 2 1.0 1 0.5 5 0.5 10 15 5 10 15 1 1.0 2 why why Page 3 of 10 2. (20 points) Consider a spring-mass system + = (), (0) = (0) = 0, with the discontinuous forcing function () that describes the impact of hitting the spring with a hammer, in the positive direction, at = and = 2. (A) Rewrite the forcing functions using Dirac delta function. (solution) (B) Given the following choices, find the solution curve of the system. [You are not supposed to solve the IVP.] Use the qualitative nature of the solution to explain why briefly. (i) (ii) 1.0 1.0 0.8 0.5 0.6 5 0.4 10 20 25 20 25 0.5 0.2 5 10 15 20 25 1.0 (iii) (iv) 1.0 2 0.5 1 5 0.5 15 10 15 20 25 5 10 15 1 1.0 2 (solution) Page 4 of 10 3. (20 points) Consider 2 = 0 with (0) = 1. Find a power series solution about = 0 for the initial value problem. [You need to include the interval of convergence as a part of your answer.] (solution) Page 5 of 10 4. (40 points) Consider the following initial value problem: + 2 = , (0) = 0, (0) = 0 (A) Find {()}(). (solution) (B) Find the solution () using the method of partial fractions. (solution) Page 6 of 10 (C) Find the solution () using the convolution theorem. [Hint: Convolution Theorem: 1 {()()}() = ( )() ( )() 0 ( )() (solution) Page 7 of 10 5. (10 points) Use the annihilator method to determine the form of a particular solution for 5 + 6 = 3 2 . [You neither determine the coefficients nor solve the equation.] (solution) Page 8 of 10 6. (30 points) The mixing tank initially holds 1000L of a brine solution with a salt concentration of 0.1 kg/L. For the first 5 min of operation, valve A is open, adding 10 L/min of brine containing a 0.2 kg/L salt concentration. After 5 min, valve B is switched in, adding a 0.5 kg/L concentration at 10 L/min. The exit valve C removes 10 L/min, thereby keeping the volume constant. Find the amount of salt in the tank as a function of time. Your answer must be written in one equation. (solution) Page 9 of 10 Make sure to go back and check your work! Page 10 of 10