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MATH 101 ASSIGNMENT
创建日期：2024-09-06 15:23:52
所属分类：数学mathematics
标签： MATH 101
ASSIGNMENT

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MATH 101  ASSIGNMENT 

Learning objectives

• Use differential equations to understand exponential decay

• Use numerical integration to estimate error intervals from data

• Compute volumes of real-life objects

• Calculate distance travelled along a non-linear curve.

• Apply the concept of expectation

• Compute partial fractions for roots of degree > 1.

• Learn about a UBC alumni

Contributors

On the first page of your submission, list the student numbers and full names (with the last name in bold) of all team members. Indicate members who have not contributed using the comment “(non-contributing)” .

Reflection questions

Reflection questions encourage you to think about how mathematics is done. This is an important ingredient of success. Reflection  questions contribute  to  your engagement grade.

1. In written assignments you are asked to solve difficult problems as a team. Understanding how to approach or begin a problem is often the most challenging aspect of problem solving. Discuss  the following prompts together and write a one or two paragraph response addressing all three.

(a) What strategies does your team use when you are stuck and do not know how to approach or begin to solve a problem?

(b) Which strategies have you found to be the most effective?

(c) Is the way you approach difficult problems in math similar to how you approach difficult problems in other subject areas? Explain why or why not.

2.  Each member of your team should submit an individual response for part (a). Part (b) should be done as a group.  You will not be evaluated on what you write here; use this question as a prompt for your own preparation.

(a)  Make a realistic study plan for yourself for the final exam. You may want to consider: timelines (for your schedule), topics (in order of importance), resources, different types of preparation (review versus exam practice), and mental and physical health (sleep, movement breaks).

(b)  Compose a brief message to the rest of the class to motivate them for the final exam. You may use words, drawings, memes, or any other medium of your choice (The best messages may be posted anonymously on Piazza!).

Assignment questions

The questions in this section contribute to your assignment grade. Stars indicate the difficulty of the questions, as described on  Canvas.

First, some notes:

• In all your answers, when giving a result that is a numerical value, specify the unit when possible.

• Also, take a moment to review the difference between decimal places and significant digits.

• Some of the non-mathematical facts in this assignment are true, some are completely made up.

3.  (10 marks) Radioactivity

The radioactive isotope wanowanium-18 is present in small quantities in many UBC students and alumni, and is constantly replenished until the student graduates.  After this, the wanowanium-18 decays into stable wisium-22 at a rate proportional to the amount of wanowanium-18 present.  Suppose that C(t) is the amount of wanowanium-18 present in a student at time t, and that the half-life of wanowanium-18 is 4.6 years.  (The half life is the time taken for the amount of wanowanium-18 to decay to exactly one half of what it was originally.)

(a)  (★☆☆☆) Find the general solution C(t) of the relevant differential equation below, as well as the particular solution for an initial amount C(0) = C0 :

dt/dC = -kC

(b)  (★☆☆☆) Based on the half-life of wanowanium-18, find the value of the constant k and specify its unit.

(c)  (★☆☆☆) From the differential equation, and the value and unit of k, give an interpretation for the meaning of this constant. Justify your answer.

(d)  (★☆☆☆) In 2021, a team of scientists measured that Bjarni Tryggvason had 0.062% of the amount of wanowanium-18 found in fresh UBC graduates. When did Bjarni Tryggvason graduate according to this data?

4.  (16 marks) Waste in space

The waste compartment of Bjarni Tryggvason’s space shuttle  Discovery  had  a  leak.   The  following measurements, made at the start of each day of the mission, show the rate at which waste was escaping (in kg/day) into space.

Time (days)      0 1  2   3   4  5   6

Rate r(t)           4 8 12 19 29 41 55

(a)  (★☆☆☆ ) In the table, find an overestimate and an underestimate of the quantity of waste that escaped during the first day.

(b)  (★☆☆☆ ) Make an overestimate and an underestimate of P,  the total quantity of waste that escaped during the whole period for which we have data.

(c)  (★☆☆☆ ) How often would measurements have to be made to find an overestimate and an under- estimate for P, which differ by exactly 1 kg from each other?

(d)  (★☆☆☆ )  Enter the table values in a spreadsheet, plot the data points on a graph, and  fit a quadratic function to these data points (also called quadratic regression). Show the plot you obtain, and write the equation of the fitted function in terms of t.  Keep 4 significant digits for decimal numbers.

(e)  (★★☆☆ ) We now assume that in reality, the rate r(t) followed a smooth quadratic function (such as the fitted one in the previous question). Using the trapezoidal rule and its error bound, find a tighter interval for the true value of P, than the one found in question (b).  Give the results with 6 significant digits.

(f)  (★☆☆☆ ) Does the Simpson approximation fall in the interval found in the previous question? Given the current assumption that r(t) follows a quadratic equation, can we find an even tighter interval for the true value of P?

(g)  (★☆☆☆ ) We now change our assumption, and assume that in reality, r(t) follows the equation of a polynomial of degree 4.  In your spreadsheet, change the fit function to a polynomial of degree 4.  Show the plot you obtain, and write the equation of the fitted function in terms of t.  Keep 4 significant digits for decimal numbers.

(h)  (★★☆☆ ) Given our new assumption, can we find an even tighter interval for the true value of P?

5.  (4 marks) Fixing the leak

On day 6 of the STS-85 mission, Bjarni Tryggvason identified the leak.  It was due to a broken part in the waste compartment pipe system.  Unfortunately, they didn’t have a spare for this part (how foolish!), so they had to 3D print it, with one of the first portable prototypes of selective laser melting.  In order to gauge how much metal was needed to print the part, Bjarni Tryggvason needed its volume.

(★★☆☆ ) Calculate the volume of the solid obtained by rotating the region bounded by the curves y = 9 − 5x + x2  and y = −9 + 5x − x2  and the lines x = 1cm and x = 4cm, around the line y = 6cm. Don’t hesitate to use Desmos to help you visualize this.

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MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 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