INSTRUCTIONS:IMPORTANT: Write your answers on the PRINTED COPY of this document.Ensure you have completed the weekly tasks up to and including Monday of Week 10, including the Self Checks and the MapleLabs.Print and complete the following project INDEPENDENTLY. NO GROUP WORK IS PERMITTED. You should considerthis project as similar to an open-book mid-term exam.You must show enough work to justify your solution. Marks will be awarded for showing and justifying your work,not for the nal solution. You can use Maple to check your solutions, if applicable.WEIGHTThe weight of this assignment is 10% toward your nal grade. No late assignments are accepted (no exceptions).Projects received at the CEL O ce or UW Campus Drop Box after the due date/time will be marked for the student’sfeedback, but not included in the nal grade calculation.SUBMITTING YOUR PROJECTIt is the student’s responsiblity to ensure that projects are RECEIVED at the Centre for Extended Learning (CEL) O ceor in the Campus Drop Box BEFORE NOON on the due date. Any project received after the due date/time will not beincluded in the student’s nal grade calculation.All Projects must be dropped o in person or sent to the Centre for Extended Learning (CEL) O ce on Gage Avenue (notyour instructor) by mail, courier, or fax. (See instructions on the next page.)IMPORTANT: REFERENCE DECLARATIONYou must complete the Reference Declaration section below in order for your assigment to be graded.Part 0: Reference Declaration (weight: ZERO GRADE on the project if not completed )All of the solutions you submit must be your original work. NO GROUP WORK IS PERMITTED.You cannot discuss or exchange solutions with any other person. The objective of this project is to determineif you have mastered the course material.You may only use the course materials stated in the course outline (text, course notes, lectures, course website, orMaple). Indicate which of these resources you used to complete this project on on the lines below.IMPORTANT: Your submitted work must be your original work. By submitting this assignment you understandand agree to the University’s policies regarding academic honesty. Any violation of the University of Waterloo’s Policy71{ Student Academic Discipline Policy (see http://www.adm.uwaterloo.ca/infosec/Policies/policy71.htm) onacademic honesty will result in an academic penalty.Family Name: First Name:I.D. Number: Signature:Declared Aids (check all that apply):Course Lectures Course Notes Course Text: edition # Maple Software: version #Other Declared Aids:I did not use any aids or seek any help to complete this project.2SUBMISSION METHODSIMPORTANT: The CEL o ce will send your projects to me just after NOON on the due date. Any projectsreceived after this time will not be included in the nal grade calculation.If you can not use one of the methods listed below, please contact your instructor immediately (wellin advance of the deadline).To submit your project, use one of the following methods:1. CANADA POST or COURIER:Address your envelope as:Centre for Extended Learning (CEL)c/o Learner Support ServicesUniversity of Waterloo200 University Avenue WestWaterloo, Ontario, Canada N2L 3G1Attention: DE Math 128Re: Your Name, Your UW IDIMPORTANT: Projects will be sent to me after NOON on the due date. It is the STUDENT’S responsibilityto ensure their project arrives before the due date. Any projects received after this time will not be included inthe nal grade calculation.2. DROP OFF - Campus Drop Box:You can drop o your project in a designated drop box in the (old) Math Building at the University of Waterloocampus before NOON (Waterloo, Ontario time) on the due date. The location of the drop box will be postedunder the ANNOUNCEMENTS section of the LEARN course website during the term.IMPORTANT: Dropped o projects will be sent to me at NOON on the due date. Any projects received afterthis time will not be included in the nal grade calculation. The University is not responsible for lost submissions(e.g. projects placed in the wrong box). If you are uncomfortable using the campus drop-boxes, please send yourproject to the CEL O ce as noted above.3. FAX:FAXing is not recommended. However, if you use this method you must FAX your assignment to the Centre forExtended Learning O ce at (519-746-4607) before NOON (Waterloo, Ontario time) on the due date. Projectsthat are faxed are often UNREADABLE; portions of the project that are unreadable will not be graded. It isrecommended that you use a DARK leaded pencil to complete your project.It is the STUDENT’S responsibility to ensure the project is received on time. Late submissions due to technicalproblems with FAX machines will not be accepted. If you intend to FAX your assignment it is recommendedthat you submit at least one business day before the due date to avoid any di culties.IMPORTANT: The CEL O ce DOES NOT proofread faxed assignments.The time the fax is received is the time that will be used for your project submission to determine if it wassubmitted before the deadline.4. EMAIL:Emailing your submission is not an option for this project.IF YOU CAN NOT USE ONE OF THESE METHODS, CONTACT YOUR INSTRUCTOR IMMEDIATELY(WELL IN ADVANCE OF THE DUE DATE).3Part 1: Reference Declaration (weight: ZERO on the project if not completed)Please ensure that you have completed the Reference Declaration section on the front page of this assignment andinclude your signature to indicate that you have read and understand the project instructions.Part 2: Short Answer (weight: 2 marks each)Provide an answer to the following questions. For any true or false questions, correct the given statement if the answer isfalse or explain why it is not true. For any questions that require a calculation, you must show enough work to justify youranswer.1. Describe the curve represented by the following parametric equations: x = sin(2t);y = cos(2t);0 t 2 . Include alabelled diagram to illustrate your answer.2. Draw a picture to show that1Pn=21n1:5 K:P 1.b) What happens to the convergence of the series1Pn=1xnn2 when x = 1 and when x = 1. (Hint: Substitute the valuesx = 1 and x = 1 value for &"x&" and consider the two speci c series.)149. (3 marks each)a) Use the Integral Test to show that the series1Pn=11n12diverges.b) Use the formula for estimating sums from the Integral Test (see page 266 in the course notes) to show that2p101 2 100Xn=11n12 191510. Find the radius and interval of convergence for each of the following power series. (3 marks each)a)1Pn=1xnnb)1Pn=0xnn!1611. (3 marks each) Let f(x) = ln(x+ 1).a) Find f0(x);f00(x) and f000(x).b) Show that T1;0(x) = x and that T2;0(x) = x x22 .c) Use Taylor’s Theorem to show thatx x22 0.(Hint: R代做数据结构作业、SQL语言调试、帮做R实验、 PRINTED COPY 程序代写留学生ecall that f(x) Tn;0(x) = Rn;0(x) = f(n+1)(c)(n+1)! xn+1 for some c between 0 and x. Calculate R1;0(x) and R2;0(x)and then use these formulas to show that no matter what x> 0 you choose the remainder R1;0(x) = ln(x+ 1) x 0.)1712. Recall that 11 x has a power series representation11 x =1Xn=0xna) (3 marks) Show that f(x) = x1+x6 =1Pn=0( 1)nx6n+1.b) (3 marks) Let H(x) = Rx0 t1+t6 dt. Use integration of power series to show thatH(x) =1Xn=0( 1)n x6n+26n+ 2c) (1 mark) Express H(0:1) = R0:10 t1+t6 dt as a (numerical) series by substituting 0:1 for x in the series from part (b).18d) (3 marks) The series you derived in part (c) satis es the conditions of the Alternating Series Test. Use the error in theAlternating Series Test to show that Z 0:10t1 +t6 dt 1200108.e) (2 marks) Is 1200 greater than or less than the true value of R0:10 t1+t6 dt? Explain. (Hint: What is the sign in the nextterm in the alternating series for H(0:1)?)f) (3 marks) We know that H(k)(0) = k!ak where ak is the coe cient of xk in the series representation of H(x) frompart (b). Find H(8)(0) and H(11)(0).1913. The position of an ant in centimeters at time t is given by F(t) =hcos(2t);sin(2t)i.a) (5 marks) Evaluate F(0);F( 4 );F( 2 ) and F( ) and use this to sketch the path of the ant on the axes below. Labelthese points on the path. Use arrows to indicate the direction in which the ant is travelling.b) (4 marks) Find F0(t) and use this to evaluate F0( 2 ). Include a sketch of F0( 2 ) on your diagram above. What physicalquantity does F0( 2 ) represent?c) (2 marks) Express the distance that the ant would travel over the interval [0;1] as an integral and then nd the distancetravelled.20Maple LabsUse Maple CLASSIC to complete the following labs which are found in your weekly/daily tasks on the course website.Please attach the requested items to the BACK of this project.1. (10 marks) Di erential Equations in Maplea) Attach the printed copy of the pages entitled \Assignment: DEs in Maple&" containing your hand-writtenanswers for grading.b) Attach the printed copy of your Maple worksheet for this assignment for grading.Important Note: Both the completed Part (a) with your hand-written answers and Part (b) theprinted copy of the Maple worksheet must be submitted for credit on this lab. If either part ismissing, no marks will be awarded.2. (5 marks) Direction Fields in Maplea) Attach the printed copy of the pages entitled \Assignment: Direction Fields in Maple&" containing yourhand-written answers for grading.b) Attach the printed copy of your Maple worksheet for this assignment for grading.Important Note: Both the completed Part (a) with your hand-written answers and Part (b) theprinted copy of the Maple worksheet must be submitted for credit on this lab. If either part ismissing, no marks will be awarded.3. (10 marks) Taylor Polynomials in Maplea) Attach the printed copy of Assignment: Taylor Polynomials in Maple containing your hand-written answersfor grading.b) Attach the printed copy of your Maple worksheet for this assignment for grading.Important Note: Both the completed Part (a) with your hand-written answers and Part (b) theprinted copy of the Maple worksheet must be submitted for credit on this lab. If either part ismissing, no marks will be awarded.4. (5 marks) Sketching Vector Valued Functions (Space Curves) in Maplea) Attach the printed copy of Assignment: Space Curves in Maple containing your hand-written answers forgrading.b) Attach the printed copy of your Maple worksheet for this assignment for grading.Important Note: Both the completed Part (a) with your hand-written answers and Part (b) theprinted copy of the Maple worksheet must be submitted for credit on this lab. If either part ismissing, no marks will be awarded.21MATH 128 : Sample Solutions for Problems on SeriesInstructor: B. ForrestAddendum for ProjectUse the following questions and their solutions to help you complete this Project.Sample Questions: Either the the Comparison Test or the Limit Comparison Test can be used to determinewhether the following series converge or diverge.For each series clearly state which test (CT or LCT) you would use to show convergence or divergence and the speci cseries you could use in the comparison. You should brie y justify your choice. In the case of the Limit ComparisonTest, you do not explicitly have to evaluate the limit limn!1anbn ). Finally state whether or not the series converges ordiverges.a)1Pn=13n+1n3+nSolution:Since this is a positive series we want to nd the order of magnitude of the terms in this series. To do this we lookonly at the highest power terms in the numerator (3n) and the denominator (n3).Observe that for large n we have3n+ 1n3 +n= 3nn3 =3n2:This means that we could use the Limit Comparison Test with bn = 1n2 .(Note that we can use 1n2 rather than 3n2 because these di er only by a constant. In fact, we can show that ifan = 3n+1n3+n and bn = 1n2 , then limn!1anbn = 3.You are not required to do this for this question but here is how this would look:limn!1anbn= limn!13n+1n3+11n2= limn!1n2(3n+ 1)n3 +n= limn!1n3(3 + 1n2 )n3(1 + 1n2 )= limn!1 3 +1n21 + 1n2= 3We know from the p-seies test that the series1Pn=11n2 converges. Thus, the Limit Comparison Test shows that1Pn=13n+1n3+nalso converges.Alternatively, we could use the Limit Comparison Test directly with bn = 3n2 since we also know that1Pn=13n2 converges.22b)1Pn=1pn3+1n2+4Solution:We want to nd the order of magnitude of the terms in this series. To do this we again look only at the highest powerterms in the numerator and the denominator.In this case, the numerator is pn3 + 1. For large n we can replace pn3 + 1 with pn3 = n32 when calculating theorder of magnitude.Again for large n, we see that pn3 + 1n2 + 4= n32n2 =1n12 :We can use the Limit Comparison Test with bn = 1n12.In this case, the p-series test shows that1Pn=11n12diverges. It follows from the Limit Comparison Test that1Pn=1pn3+1n2+4diverges as well.c)1Pn=1jcos(n2)jn5Solution:Since jcos(x)j 1 for any x, we see that jcos(n2)j 1 andjcos(n2)jn5 1n5for every n.Since we know from the p-series Test that the series1Pn=11n5 converges, we can apply the Comparison Test with bn =1n5to get that1Pn=1jcos(n2)jn5 converges.& 转自：http://ass.3daixie.com/2018052158549156.html