管院微積分聯合教撃　含孝三 Jam 13, 2015

考試時間120分鐘･題目巻島南張紙･共四頁,満分120分｡所有題目的答

案都請依題競順序依序幕在答案巻上,而是非輿填充題必須罵在第一頁｡答案

巻務必罵撃競､姓名,題目巻不必緻回｡考試開始30分鐘後不得入場,開始

40分鐘内不得離場o考試期間禁止使用字典､計算機及任何通訊器材,監試

人員不得回答任何閥於試題的疑問｡ Questions are to be answered on the

answer sheet provided.

是非題Trueor False(20points) ･請答T(Ⅱue)或F (助Ise) ｡毎題2分o (不需

詳列過程･講依題競順序依序蔦在答案巻第-頁上e )

千 l･ Let A(i) represent a qua舶y growing exponentially, then A′(i)/A(i) is con~

stant.

ノ･',雄, 3倍〟笥叩entTamd.

I

∴ A(千)ニek･A･C#),伊純化k Ju a Po3,-5-Vクao柾a走

ああ,Att/l

--~~一一　一へ　二二

閉園堕墾田団長ALtI

牢2･藍-aha-1,-hereb>0,b≠1･

堕雄図暢

うま拒-曇詔b

本坊らニ　Anb･ a, -

掬･bXY

I 3･ The relative growth rate of P is the derivative of ln(P(i))I

計雄一C,)

-市･ (諾-/))

_巨,_I ･

PC封

W姐当確S河+Lta yetGTrv色Jr｡励一Y-te-

1

千 4" If ∫ is not integrable on囲then ∫ can冊e a continuous揖ion on [a,b]･

Th75　-iS'an CFAul'1,gaCt,,･d sで諦宛短*でo

lI

otnd,aT,詰諾∵嵩誓Ts擁

T　5･ Iff'isinFigurebelow,thenf(1)FSf(3)･

).詰み士｡十｡｡,- _①

畠-- ∫(3,守｡'｡ ㊨-㊨-①

'Thea-of⑤桂Th9 --㊨ ∴手｡)叩3)

F　6･ IfF,(i) isconsta-en j5F,(i)d-

ASSu枇L F't#ノ= C一宏砲巳ノ力み∽シ彫着席

.'告,擁aニI:ceG

四四

二　rc,易, rCヂe言f Cヂ0

2

紗

　

8

　

0

　

助

　

側

　

勘

　

助

　

棚

　

勘

　

畑

　

的

Ⅴノ

i 7･ Irj2g,i,dt-5then/2g(2-i,dt--5l

Je予〔′ご2-を,哀れ　cc0--番

Gと2　訪れてー～0, CLニO utub･て学2.

F!en`互.

j:-3,2-- ･j:3-du, JoeSでU,ek-- ∫

F　8･ IndeEnite integral is an antiderivative･

披石垣でniL,5詔TS履函(J at

ALL ｡府6fey1--87で'･:V色

9. ∫(¢) - 1- (1-8)2 isaLorenzcurve.

仁-af bd/ot･,宛)履-∫-X･･

･f … Iff(3, -∫(-3,･ then /_22f(B席-0･

Fcr exo-r/i'言出,-÷砿j詰諸声

(下頁還有試題)

3

填充題Short answer questions (40 points) ･毎題5分｡

(不需詳列過程･僅牌答案依題郷原序依序菊在答案巻第-頁上印可o )

1. Find the limit li血a3-÷O

Answer :

/3a?2+1- 1a;2

･ノ盈-壇∴上場適塾二⊥.虹Lfづo tr'　　差,e･　ズ老　　師1 4日

ー~ノ六､

Y､-多の

ごAr,Y,･

語法青十日

ィ｡一一　　一_　　　lノ

ヽ-Li　　　●一一一一一､

四1"fo'虜詰丁当

2. Find the absolute extrema of ∫(a) -省一4J云on the the interval [0, 9],

Answer:

!,I"ニI一言, 43c c履p n{J",,,bew右耳七両ノ

庇鼓字詰=o

ftaJ-午4'rjニー平

声i,ニケ4(aこ-3

4冒ia1-5ei･4-t蛋躯乱諦筋　=辛

4度C/L･tte　勅a′fG74m, ･b･i 8

4

3. A line goes through the orlgln and a point on the curve y - a:2e~3縞, ror縞≧ 0.

Find the maximum slope of such a line.

9

Ans-er: ge-i II

*2e-13ガ- o砧隼oI最/r･u G-上告中-毎竜一

叩/,-cs -4 x･(9㌢

(

擁･謹-i-告1坤隼脇･隼†S f/-or,-19-3･,_# pwl毒　口千十校†-,_Tt3-cm　芽e,-3力

A防･-e-Sx(3介() >0 ,号,W,(･《生-I),初eyj嵩-,)<o

膏　7'tJ(i-･･W)- 7~履　謡∵ZaA_偽　8LseI履頼,-榊m,

囲　遊園

一-i励まI予言9-14

4. Find the area of the region bounded by the graphs of the mnctions

1

g(a)--e年-5号=1,

I

Answer: e --4,,i/ - eTE

÷-,-Qx車手記ふ台,i

姐工片柳el･Xi;I

こ布JgY/.I

ウ

=tun/ -仁(塵+〆)

/

三　g-AlT/-　eそ

こ重,

5

l

一

鋼

二

〇い'~○細り

題のH○Hn

f

J

5. Find the e胱ctive rate corresponding to the given nominal rate.

(a) 20%/year compounded semiannually (twice a year)･

(b) lo瑞/year compounded continuously･

Write you answer in a pair: (the answer of (a), the answer of (b)).

If youぬil to do so, you WILL su餓∋r紐om NO POINTS.

Answ｡r:( , )

時好出塁打二上回-D誹雄)

胴囲

鞘ニ90-I-/

生硬rこ(0-ll, eO･日)

6. Compute the average value off(縞) - 6a'2-17-e鯵over i-1, 21･ If your answer

has a term eα where a is a number, please just leave that there.

Answer:

i, I:il'中e唖

- ÷(zX3-I-e引

i

函llL3年CL+-I十eつ]

卓骨品一弾･一冊亨一旦

＼　J

3

A両手_TI十%'｢ゼ3

6

7. Find the coe範cient of inequality (or Gini Index) of the Lorenz curve y - ∫(a:) -

鍬2. Answer:

三晶又膏摘

-i(÷史勘/'b

I

ごつ~T

l.-　　　-〟-iiiii -　~-

~　3

i

A佃Ner二言

8. Suppose that we know the following facts: (1)/4f(8,d8 - 2, (2) f6f(8,da -

-1, (3) f6f(8,d8 - 2･ Find /54f,are-ada･

Answer :

∫,a 4チ-~油

事由)加理祖-4伸子(フ()厨J勘串十J隼雄

河出輔車拒-庸庸雄告串本

-症埴)珂高宮a_-脅す詰-証蜜

(下頁選有識題)

鉦航サミー41 e十色J

計算問答讃明題Please show all your work (60points) ･毎題10分,請依題競順

序依序寮在答案巻上,可以用中文或英文作答o請詳列計算過程･否則不予計分｡需

標明題競但不必抄題o

1. (10 points) Neilsen Cookie Company sells its assorted butter cookies in containers

that have a net content of 1 Ib〟 The estimated demandめr the cookies is 1,000,000

1-1b containers per year･ The setup costめr each production run is $500, and

the manufacturing cost is $0･50 for each container of cookies. The cost of storing

each container of cookies over the year is $0.40. Assume uniめrmity of demand

throughout the year and instantaneous production.

a. Find the total cost hlnCtion.

b. How many containers of cookies should Neilsen produce per production run in

order to minimize the total cost?

諭告為奇箸坐雇政教素劣者勅tlr,毎†ニ　SToo･(

/_ eleJ. ec. Ll

芽 ) - Jo,,I I-"-.I,"/I-a･

璃･)

胴囲

-ctt･,宵闇二　時(阜)I--- t･･雷グ

偽梱読勧e. C珂ニ　G,㌻仕0-)｡し羽- 3-椋-i:.

鼓/おれ生鮮千　石-門

生加二--　聖書-,--Lil-i-｡十笹子-　(,ぐ･3台高･,Cごく,

C,(弟｡ ♂.皐一三二三呈｡'｡,○ノ｡"二〇/qご

()･<'亮一Jtlラし･,.i/) ,l･.∂

符　〆ごみ∫鉦㌧航.…｡諸点言

辞　ガこ　し∫~0の-　選一∫0-　tjrそり

詰高二　証人等掴,i,･虚上端

扱高T,:i彊rogt,"諦尊,摺最小窪&,寿,

8

2･ (10 points) Four months alter discontinuing advertising on national television, a

manuぬcturer notices that sales have dropped五〇m 1007000 MP3 players per month

to 807000 MP3 players･ If the salesめllow an exponential pattem of decline, what

will they be a請er another 4 months?

Jt+I -- /0000c cTk-'-

go00 - Sc4/a /｡cooo壱

ラ　長　く-41-4,1･讐

So,車,っoo("o証言/4)g

蛋

1-,.P,.孝

二/eaeoO a

a /00-クi(阜了

二_　6争OCC

9

3･ (10 points) The value of good wine increases with age. Thus, if you are a wine

dealer) you have the problem of deciding whether to sell your wine now, at a prlCe

of SP a bottle, Or to sell it later at a higher price. Suppose you know that the

amount a wine-drinker is willing to pay for a bottle of this wine i years from now is

Sp(1 + 20vG)･ Assuming continuous compounding and a prevailing interest rate of

5% per year, when is the best time to sell your wine? Note: Round the answer

to the 2md decimal place･ (Vを両面兎FS 6･ 32475296)

fed yttj- p自-2｡Jテ)

7-/当蘭訪壷'of +lee- -宮部常"r u･,irG/

音　符塞拓殖-J rS

_♂.L･壬千

昨'こV',年を0両-- p(lt20Tt)珍

Tv声,g庇-,,I- -o･んe of pet,,- C-咋ね

証,叩[計･両手] a-0'aS~ビ-四国 こ: ○

7諒3,.,es　虚-ニ竺竺ユ血盟笠二一".,･?緒i･4-8之

7-hut

七鳥P-雅3/3((背

症′ a p･eT"5-''上宝0 --∴ノ中oJ車名'0-7･843旬,77,

50笹J a的1-nS i,･iS h･L1--/64Ju一柳1-.-､8,L2if隼/G･843/3/ ′7.7

lJle一組　-/kd ll,ua_　bLt,千　で,-pp退　w se,〟

7･g43/3/i啓Jears fro- -㌶

10

r`イr~　軌-rfW ec,i"I-

4. (10points) Find

禄t

il章〇度　　　　一

側　曾

/ Ede･u-e雫牛Ien　右-卑語萄⊃(

十十●関開田i一-四囲

I･ I ･ i

葦⊥晶囲

Iれ国ヰ⊂一

帖請出｡/

ll

5. (10points) De丘ne ∫(3) - a?3 and

g(鯵) -

〈J云　ifa≧0-√石ifa<0

(a) Sketch the graphs of the mnctions ∫ and g,

(b) Find the area completely enclosed by the graphs of the請nctions ∫ and g,

ra)航　温熱　十･工　poi壷S of壷捗擁n

上々.)万,リ-et*),

(,)掠　ccA5,e　捉仁王シ〇 ･ノ

(3=膚=>真二よ　う珪Iラ享/

(､I)七五　roL;疋　功匁,丸く｡

_症,｢/庫⇒

･言､-{

3(.(ニーズう真二-I =>

5'.Lm,a I(i)=3亮0, f'/I)=dd,

チ‥高み牛車-′

雄幸ぐみ履　叩脇目-

†

o"d : -- I-･-I+-

oh (-∞ノ0コ

I

貢~モ　畠ラe仁

(一㌦, if ,しくo

う

-÷草江jL,｡

÷(-i)-3=両-｡ ,

9 ＼iS lhC-ふっ　読　∂iS…　めれ…せ　び昨yd　-

(-eoJol aれd　-cavu d0-0vJd oh Co,GQ)

ぐのA二鳥厄一項十宮西掠

･専一中性需品L:12

･三十三ニ丁5日躍

LJ年末

毘iiiliE

｢'8　両

⊥ユ_〓

{二

回賀O

T<

/

I

-

-

aIu

〓

ヽ

ノ九

l

′

I

I

-

ヽ

(　(｢V

~~~"○

点"囲

6. (10 points) Consider the demand細nction

p-D(a')ニーa'2+20,0≦a'≦ 2J言

and the supply mnction

p-S(a')-鯵2+2%+8,0≦a'≦2J5.

(a) Find the equilibrium price and quantity. Please make sure you do those

correctly since you need them fbr part (b) and (C).

(b) Find consumers'surplus if the market price is set at the equilibrium price,

(C) Find producers'surplus if the market price is set at the equilibrium price･

｢盤繕霊草露盤料-(Y　シフ晶*-+-

(九,)は十ajニ0./ 51唖

L SL｡JA be　工､の完

30/

0-'オー<-瓦,

捉n pニ/(

閥Cs - J∴高射Ⅹ一再

-上土*3+-)l詰-

･一斗-0-

的■oo←　　一~　　-~~~~-

｡C'p5ニ吉子詰誹*=31-仕損+埴

(試題結束)

･ 31-(?+41136)

_哩る