all questions need to show the work and solution. not only the final resuts.
MTH105 1 Assignment 3 Series, Trigonometry, Vectors, Differentiation and Max and Min problems. Subject: MTH105 Due date: 31 May 2021 Value: 15% This is a written assignment that covers; TOPIC 7 - Arithmetic and Geometric Series TOPIC 8 - Trigonometry and Vectors TOPIC 9 - Differentiation TOPIC 10 - Applications of differentiation. Show all working as marks are awarded for working as well as the answer. The assignment is to be uploaded to EASTS as a single Word or PDF file. Assignments submitted in non-printable formats such as a ZIP file or as a collection of images will not be marked. If your scanner produces separate graphics files please paste them into a Word document before submitting to EASTS. Hints: Read the questions AND the marking criteria carefully. They indicate what is expected of you. Note that since the dispatch date for Assignment 3 is after the examination period commences you may not receive your marked assignment until after the final exam. Rationale Assignment 3 is designed to assess your learning of the following Subject learning outcomes which are; Be able to use exponential, logarithm, and trigonometric functions in applications; Be able to calculate the sums of arithmetic and geometric series and use them in simple financial calculations; Be able to use vectors in scientific applications. Be able to use basic rules of differentiation and calculate derivatives of simple functions; Be able to interpret and write about applications of mathematics in their field of study; Marking Criteria All questions in this assignment involve problems with a sequence of several steps. These are marked using the following criteria 2 Criteria Description Correctness Arithmetic, algebra and calculations are correct (except possibly some minor rounding errors) Process/Method The indicated/correct method is selected and carried out completely. Communication/Working You have made it clear what you have done using an appropriate mix of text, mathematical notation, neat diagrams and code excerpts. The mark for each question is determined by the proportion of your solution that satisfies these criteria. Full marks Partial marks Zero marks Correct answer written as a clear response to the original question. Full worked solutions provided that are clear, adequate and legible and use the correct mathematical notation and reasoning, with neat diagrams and code excerpts where appropriate. Also make sure you give final answers with appropriate units and (where specified) rounding. Correct answer with incomplete working which lacks the appropriate notation or reasoning. OR Incorrect or partially correct answer with working that uses the correct method and appropriate notation and reasoning. Method and solution are incorrect. No working to arrive at the solution is shown. Grade allocations: Marks for all questions will be totalled and grades awarded based on percentage of the total available marks. (FL <50%; ps 50-64%; cr 65-74%; di 75-84%; hd 85-100%) presentation the assignment can be typed and/or neatly handwritten. the handwritten method of presentation is allowed because it can be difficult and time consuming to type mathematical formulas in word. however, if the handwriting is illegible then you do run the risk of it not being marked. also if you write the solutions to the assignment around the printed assignment questions, the assignment will not be marked. use fresh paper to handwrite the solutions. it is not necessary to copy the actual questions in an assignment, but every part of a solution for a question must be numbered exactly following the question in the assignment. include a separate cover page with your name, student number, subject code, assessment number and assessment question. number your pages (except the cover page). use a footer (or write on the bottom of the page) with your name and student number on each page. requirements for this subject, because the only sources are the topic modules, tutorials and online lectures, you are not required to provide references in your assessments. 3 question 1 (16 marks) a. show that the sum of the first n -terms of the series 1 3 5 7 .... is given by 2ns n . hence find the sum of the first 100 odd numbers. hint: let the first term = 1, and the last term, which is odd be 2 1n . (3 marks) b. find sum of the geometric series 112 1 1 8 2 n n . if instead we summed to infinity, what would the sum be? (3 marks) c. if in a geometric sequence the sum of the second and third terms is 20 and the sum of the fourth and the fifth term is 320, find the common ratio and the first term. assume that the common ratio is positive. (3 marks) d. consider an annuity with a regular quarterly deposit of $2000 and an interest rate of 5% p.a. compounded quarterly. if the annuity is invested for 5 years, find i. the future value of the annuity, (3 marks) ii. the amount of interest earned, and (1 mark) iii. the present value of the annuity. (3 marks) question 2 (17 marks) a a triangle has sides of 35, 21, and 28 centimetres. is this a right-angled triangle? with this question you must show your working by use of diagrams, and/or explanation of your mathematical thinking. (2 marks) 4 b for the right angled triangle below, find x and y . (3 marks) c a person on top of a 15 metre high cliff looking out at a ship, determines that the angle of depression from the top of the cliff to the boat is 5 . find the distance that the ship is from the base of the cliff. a well labelled diagram must be included with the solution. (3 marks) d. for 0 2x , sketch the graph of 2cosy x . state the period and amplitude of this function. (4 marks) e. an aeroplane has a speed of 550 km/hour in calm air, and is has a bearing of 075 . that is, the velocity of the plane relative to the air, v, is 550km/hour at a bearing of 075 . a wind is blowing from the north east at 85 km/hour. that is, the velocity of the wind, w, is 85 km/hour at a bearing of 225 . x 10cm 060 y 5 the velocity of the plane relative to the ground is the vector sum v + w. i) draw a diagram showing the vectors v, w and v + w. (3 marks) ii) find the magnitude and direction of v + w. (2 marks) question 3 (22 marks) a. find the derivatives of: i. 3 1 1 f t t t t (2 marks) ii 3 2 2( 6 )( 1)y x x x using the product rule. (2 marks) iii. 4 3 1 x y x (2 marks) iv. 2 3( 5)y x (2 marks) v. 3sin 3y x (2 marks) vi. lny x x (2 marks) b. sketch the graph of 3 9y x x you must also include points where the graph crosses the x and y axes as well as the max/min points. (5 marks) 6 c. a rectangular piece of aluminium, 3m wide and 8m long. four equal squares are to be cut from each corner. the resulting piece of metal is to be folded and welded to form an open-topped box. (i) show that the volume of the box is given by the equation 3 24 22 24v x x x . (2 marks) (ii) what should be the size of the squares cut from the corners if we wish to maximize the volume of the box ? note: the max/min point must be tested. (3 marks) question 4 (5 marks) a. 2( 4) 2 x x dx (2 marks) b. 2 1 1 ( )dx xx (3 marks) ps="" 50-64%;="" cr="" 65-74%;="" di="" 75-84%;="" hd="" 85-100%)="" presentation="" the="" assignment="" can="" be="" typed="" and/or="" neatly="" handwritten.="" the="" handwritten="" method="" of="" presentation="" is="" allowed="" because="" it="" can="" be="" difficult="" and="" time="" consuming="" to="" type="" mathematical="" formulas="" in="" word.="" however,="" if="" the="" handwriting="" is="" illegible="" then="" you="" do="" run="" the="" risk="" of="" it="" not="" being="" marked.="" also="" if="" you="" write="" the="" solutions="" to="" the="" assignment="" around="" the="" printed="" assignment="" questions,="" the="" assignment="" will="" not="" be="" marked.="" use="" fresh="" paper="" to="" handwrite="" the="" solutions.="" it="" is="" not="" necessary="" to="" copy="" the="" actual="" questions="" in="" an="" assignment,="" but="" every="" part="" of="" a="" solution="" for="" a="" question="" must="" be="" numbered="" exactly="" following="" the="" question="" in="" the="" assignment.="" ="" include="" a="" separate="" cover="" page="" with="" your="" name,="" student="" number,="" subject="" code,="" assessment="" number="" and="" assessment="" question.="" ="" number="" your="" pages="" (except="" the="" cover="" page).="" ="" use="" a="" footer="" (or="" write="" on="" the="" bottom="" of="" the="" page)="" with="" your="" name="" and="" student="" number="" on="" each="" page.="" requirements="" for="" this="" subject,="" because="" the="" only="" sources="" are="" the="" topic="" modules,="" tutorials="" and="" online="" lectures,="" you="" are="" not="" required="" to="" provide="" references="" in="" your="" assessments.="" 3="" question="" 1="" (16="" marks)="" a.="" show="" that="" the="" sum="" of="" the="" first="" n="" -terms="" of="" the="" series="" 1="" 3="" 5="" 7="" ....="" ="" ="" ="" is="" given="" by="" 2ns="" n="" .="" hence="" find="" the="" sum="" of="" the="" first="" 100="" odd="" numbers.="" hint:="" let="" the="" first="" term="1," and="" the="" last="" term,="" which="" is="" odd="" be="" 2="" 1n="" .="" (3="" marks)="" b.="" find="" sum="" of="" the="" geometric="" series="" 112="" 1="" 1="" 8="" 2="" n="" n="" ="" ="" ="" ="" ="" ="" ="" ="" ="" .="" if="" instead="" we="" summed="" to="" infinity,="" what="" would="" the="" sum="" be?="" (3="" marks)="" c.="" if="" in="" a="" geometric="" sequence="" the="" sum="" of="" the="" second="" and="" third="" terms="" is="" 20="" and="" the="" sum="" of="" the="" fourth="" and="" the="" fifth="" term="" is="" 320,="" find="" the="" common="" ratio="" and="" the="" first="" term.="" assume="" that="" the="" common="" ratio="" is="" positive.="" (3="" marks)="" d.="" consider="" an="" annuity="" with="" a="" regular="" quarterly="" deposit="" of="" $2000="" and="" an="" interest="" rate="" of="" 5%="" p.a.="" compounded="" quarterly.="" if="" the="" annuity="" is="" invested="" for="" 5="" years,="" find="" i.="" the="" future="" value="" of="" the="" annuity,="" (3="" marks)="" ii.="" the="" amount="" of="" interest="" earned,="" and="" (1="" mark)="" iii.="" the="" present="" value="" of="" the="" annuity.="" (3="" marks)="" question="" 2="" (17="" marks)="" a="" a="" triangle="" has="" sides="" of="" 35,="" 21,="" and="" 28="" centimetres.="" is="" this="" a="" right-angled="" triangle?="" with="" this="" question="" you="" must="" show="" your="" working="" by="" use="" of="" diagrams,="" and/or="" explanation="" of="" your="" mathematical="" thinking.="" (2="" marks)="" 4="" b="" for="" the="" right="" angled="" triangle="" below,="" find="" x="" and="" y="" .="" (3="" marks)="" c="" a="" person="" on="" top="" of="" a="" 15="" metre="" high="" cliff="" looking="" out="" at="" a="" ship,="" determines="" that="" the="" angle="" of="" depression="" from="" the="" top="" of="" the="" cliff="" to="" the="" boat="" is="" 5="" .="" find="" the="" distance="" that="" the="" ship="" is="" from="" the="" base="" of="" the="" cliff.="" a="" well="" labelled="" diagram="" must="" be="" included="" with="" the="" solution.="" (3="" marks)="" d.="" for="" 0="" 2x="" ="" ="" ,="" sketch="" the="" graph="" of="" 2cosy="" x="" .="" state="" the="" period="" and="" amplitude="" of="" this="" function.="" (4="" marks)="" e.="" an="" aeroplane="" has="" a="" speed="" of="" 550="" km/hour="" in="" calm="" air,="" and="" is="" has="" a="" bearing="" of="" 075="" .="" that="" is,="" the="" velocity="" of="" the="" plane="" relative="" to="" the="" air,="" v,="" is="" 550km/hour="" at="" a="" bearing="" of="" 075="" .="" a="" wind="" is="" blowing="" from="" the="" north="" east="" at="" 85="" km/hour.="" that="" is,="" the="" velocity="" of="" the="" wind,="" w,="" is="" 85="" km/hour="" at="" a="" bearing="" of="" 225="" .="" x="" 10cm="" 060="" y="" 5="" the="" velocity="" of="" the="" plane="" relative="" to="" the="" ground="" is="" the="" vector="" sum="" v="" +="" w.="" i)="" draw="" a="" diagram="" showing="" the="" vectors="" v,="" w="" and="" v="" +="" w.="" (3="" marks)="" ii)="" find="" the="" magnitude="" and="" direction="" of="" v="" +="" w.="" (2="" marks)="" question="" 3="" (22="" marks)="" a.="" find="" the="" derivatives="" of:="" i.="" ="" ="" 3="" 1="" 1="" f="" t="" t="" t="" t="" ="" ="" ="" (2="" marks)="" ii="" 3="" 2="" 2(="" 6="" )(="" 1)y="" x="" x="" x="" ="" ="" using="" the="" product="" rule.="" (2="" marks)="" iii.="" 4="" 3="" 1="" x="" y="" x="" ="" ="" (2="" marks)="" iv.="" 2="" 3(="" 5)y="" x="" ="" (2="" marks)="" v.="" 3sin="" 3y="" x="" (2="" marks)="" vi.="" lny="" x="" x="" (2="" marks)="" b.="" sketch="" the="" graph="" of="" 3="" 9y="" x="" x="" ="" you="" must="" also="" include="" points="" where="" the="" graph="" crosses="" the="" x="" and="" y="" axes="" as="" well="" as="" the="" max/min="" points.="" (5="" marks)="" 6="" c.="" a="" rectangular="" piece="" of="" aluminium,="" 3m="" wide="" and="" 8m="" long.="" four="" equal="" squares="" are="" to="" be="" cut="" from="" each="" corner.="" the="" resulting="" piece="" of="" metal="" is="" to="" be="" folded="" and="" welded="" to="" form="" an="" open-topped="" box.="" (i)="" show="" that="" the="" volume="" of="" the="" box="" is="" given="" by="" the="" equation="" 3="" 24="" 22="" 24v="" x="" x="" x="" ="" ="" .="" (2="" marks)="" (ii)="" what="" should="" be="" the="" size="" of="" the="" squares="" cut="" from="" the="" corners="" if="" we="" wish="" to="" maximize="" the="" volume="" of="" the="" box="" note:="" the="" max/min="" point="" must="" be="" tested.="" (3="" marks)="" question="" 4="" (5="" marks)="" a.="" 2(="" 4)="" 2="" x="" x="" dx="" ="" (2="" marks)="" b.="" 2="" 1="" 1="" (="" )dx="" xx="" ="" (3="">