1 Department of Chemical Engineering PRC3701: Advanced Process Control 2022 Timed Online Assessment 1: Minor test Duration: 24 Hours Total: 50 Marks First Examiner: Dr. K Mphahlele Second examiner: Prof. AV Kolesnikov Third examiner: Mr. K Ledwaba Instruction to candidates: • Answer all the questions • If the information available is not complete please assume, and specify the required value and continue with the exam paper • Submit in PDF format only2 Question 1 Linearize 𝑑𝑓𝑑𝑥= 𝑥2− 1 when 𝑥(0)= ±1 [8 marks] Question 2 Find the Laplace transform for the following equation below: 2.1. 𝑓(𝑡)= 4𝑡 (5) 2.2. 𝑓(𝑡)= {1 𝑓𝑜𝑟 𝑡 ≥ 𝑎0 𝑓𝑜𝑟 𝑡 < 𝑎 (7) [12 marks] Question 3 Find the inverse Laplace transform for 𝐹(𝑠)=𝑠2+ 9𝑠+ 2(𝑠+ 3)(𝑠− 1)2 [10 marks] Question 4 Convert this differential equation into an algebraic equation using Laplace Transforms to solve the equation and then invert the equation back into the time domain. 𝑑2𝑥𝑑𝑡2+ 4𝑑𝑥𝑑𝑡+ 3𝑥 = 1 Given that: 𝑥0= 0 𝑎𝑛𝑑 𝑥0′= 0 [20 marks] [Total 50 Marks] @ UNISA 20223 Table 1: Table of Laplace transforms (adapted from the study guide for MAT3700) ( )ft= L( ) 1Fs− ( )Fs=L( ) ft Unit impulse (t) 1 a as, a = 1, 2, 3, …. nt 1!, 1,2,3,nnsn+= bte 1sb− sinat 22asa+ cosat 22ssa+ sinhat 22asa− atcosh 22ssa− btnet 1!(), 1,2,3,nnsbn+−= sint at 2 2 22()assa+ cost at 222 2 2()sasa−+ sinht at 2 2 22()assa− cosht at 222 2 2()sasa+− sinbte at ( )22as b a−+ atebtcos ( )22()sbs b a−−+ atebtsinh ( )22as b a−− atebtcosh ( )( )22sbs b a−−− ( )H t c− cses− ( ) ( )ctFctH −− . ( ).cse f s− ( )at − ase− Study guide 2 page 55:4 L  ( ) ( )f t F s= L  '( ) ( ) (0)f t sF s f=− L  2"( ) ( ) (0) '(0)f t s F s sf f= −