Suggested+Solution+NPMaC Vt96 Eng

8/3/2019 Suggested+Solution+NPMaC Vt96 Eng

1/35

National Test in Mathematics course C

Spring 1996

Directions

Test period 3 May - 15 May 1996.

Test time 180 minutes without a break.

Resources Non-programmable calculator and formula tables. A formulasheet is attached to the examination paper.

Test material The test material should be handed in with your solutions..

Write your name, gymnasium programme/adult education and date of birth on the papers you hand in..

The test The test is made up of 13 questions.

The problems which are short answer problems (problemswhich can earn 1 point) require, for the most part, only an answer.Most of the problems are long answer problems (problemswhich can earn 2 or more points). where a shortanswer is not sufficient, but it is required that you write down what you do that you explain your train of thought that you draw figures when necessary that you show how you have used your resources when youhave solved problems numerically/graphically.

Some of the problems (where it is statedOnly an answer is required) need only an answer.

Try all of the problems. It can be relatively easy, even towardsthe end of the test, to earn some points for a partial solution or presentation

The score levels The teacher responsible will explain the grade levels which arerequired for Passed and Passed with Distinction. The tes canearn a maximum of 42 points.


2/35

Np MaC VT (Spring) 1996 NV-College

Skolverket [email protected] Not for sale. Free to use for educational purposes 1/35

1. a) Differentiate f x x x( ) = +3 4 34 (1p)

b) Evaluate f ( )2 (1p)

Suggested Solutions:

a. 443)(343)( 144 =+= x x f x x x f Answer: 412)( 3 = x x f [1/0]

b. 4812)2(4)2(12)2( 3 == f f Answer: 92)2( = f [1/0]

2. a) Differentiate4

)(2 x

xg = (1p)

b) Differentiate

c) Solve the equation (1p)

d) Calculate the geometric sum

1.072000.......1.0720001.0720002000 492 ++++ (1p)


a. 24

2)(4

)(122 x x

xg x

xg ===

Answer: ( )2 x

xg = [1/0]

b. x x x e xhe xhe xh 333 6)(32)(2)( === Answer: xe xh 36)( = [1/0]

c. 389.72ln 2 == e x x Answer: 389.72 = e x [1/0]

d. 49250 1.072000.......1.0720001.0720002000S ++++= ( ) 86.057813S

1-1.0711.072000S 50

50

50 =

= Answer: 000813S50 [1/0]


3/35



3. Some employees of a company which sells floor ball sticks, have examined thecompanys costs in relation to purchase and storage

If the company buys a large number of sticks every order, then there will be feworders. This leads to, amongst other things, lower administrative costs but higher storage costs.If instead the company buys fewer sticks with everyorder, then the orders will occur more frequently.Then the company will have higher administrativecosts, but lower storage costs.They concluded that the total annual costK x( ) can be expressed by the formula

41251505.0)( 2 += x x xK 100 300 x

where K x( ) is the total annual cost in crowns and x is the number of floor ball sticks the company

requests per order

Use differentiation to decide how many floor ball sticks the company should purchase per order so that the annual cost will be as small as possible. (3p)

Suggested Solutions:The minimum cost is associatedwith the point where the deriva-

tive of the function is zero.Therefore, first we may differen-tiate the function, then solve

0)( = xK for x . Then we mayconstruct the necessary

)( _ )( _ xK xK x table to examineif the value obtained is that of the minimum point.

41251505.0)( 2 += x x xK 100 300 x 15205.0)(41251505.0)( 122 =+= x xK x x xK [1/0] 1510.0)( = x xK

15010.0

1501510.00)( ==== x x x xK [1/0] Answer: 150= x

30004125)150(15)150(05.0)150( 2 =+=K [1/0] Answer: kr K 3000)150( = 10.0)(1510.0)( == xK x xK 10.0)( = xK

x 110= x 150= x 200= x 1510.0)( = x xK 04)110( =K

positive 41251505.0)( 2 += x x xK kr K 3000)150( =

minimum

10.0)(=

xK Positive

2700

3000

3300

3600

3900

4200

100 120 140 160 180 200 220 240 260 280 300x

K ( x )

41251505.0)( 2 += x x xK

kr K 3000)150( =


4/35



4. A person buys shares for kr 00020 . Five years later she sells them for kr 72056 .

What is the corresponding yearly percent increase in her capital? (3P)

Suggested Solutions:Assuming exponential growth, with annual increase rate a :

00020720560002072056 55 == aakr [1/0]

( ) 232.1836.2836.2 51

5 == aaa [1/0] Answer: The share is increased at the rate %2.23 per year. [1/0]

5. A sports club has 500 members. The committee plans to build a club house. Asthe issue is of such importance to the club, the committee will only build the clubhouse if a majority of the clubs assembled members can be expected to supportthe plans.Therefore a meeting was arranged for themembers. Unfortunately only 185 membersattended. Of these members, 125 wanted theclub house to be built and the rest did not.As so many members did not participate in themeeting, the committee did a supplementarysurvey. They rang 75 randomly chosen memberswho had not attended the meeting. Of these, 26

answered yes and 49 no.Do you think that the committee should decide to build the club house?Take into consideration the result of both the meeting and the supplementarysurvey. (3p)

Suggested Solutions:Of those who attended the meeting 60125185 = voted no.

315185500 = 315 of the members did not attend the meeting.Survey:

Yes: 35.07526

=

No: 65.07549

= [1/0]

If we assume the rest of club members follow the same pattern:

Yes: %4747.0500234234315

7526125 ===+

No: %5353.0500265265315

754960 ===+ [1/0]

Answer: The committee probably should not build the club house. If those who did not attend the meeting have the same opinion as those

who randomly were chosen to answer the question, then only 47%are for the building the club, and the rest, i.e. a majority of 53% areagainst it. [1/0]


5/35



6. The functions f , g and h have the following properties 3)2( = f and = f ( )0 1 g is increasing for all values of x and g ( )0 1= =h ( )2 0 and h( )0 1=

The graphs of each of the functions f, g, and h are amongst the graphs below.Decide which graph corresponds to each function.Only an answer is required.

Suggested Solutions:The figure A corresponds to ( ) x f . It is the only function whose slope is 1 at 0= x .The figure D corresponds to ( ) xg . It is the only function which is increasing for all val-ues of x , and whose value is 1 at 0= x .The figure E corresponds to ( ) xh . The function ( ) xh has a local maximum at 2= x ,and its value is 1 at 0= x .

A B

C D

E

y

x

1

1

x

y

1

1

1

x 1

y

y

1x

1

y

x

1

1


6/35



7. Many people interested in films think that the price of cinema tickets hasincreased a lot over the last few years.

Using the table below, compare the price of a cinema ticket with the change in theConsumer Price Index during the period 1992-1995. You should consider eachyear during the period under question, and your comparison should include bothcalculations and comments. (3p)

The information in the table was provided by the Statistics Central Office.

CPI = consumer price index.Suggested solution:In order to compare the increase in the movie ticket, we may changethe base year to 1992 for both the index as well as the price of ticket.The increase in the price of movie ticket were all time 1992-1995 be-low the index! Note also the fact that the price of ticket did notchange from 1994 to 1995!

Year 1992 Baseyear

1993 1994 1995

Average price of cinematicket (crowns each)

1006060 =

103

6080.61 =

106

6080.63 =

106

6080.63 =

CPI (Base year 1980)

1009.2349.234

=

1049.2343.244

=

5.1069.2344.250

=

256.0

1109.234

256=

8. In the equation ( ) 000500.95000160 x = where x denotes the time in years.

a) Formulate a problem that can be solved using this equation. (2p) b) Solve the equation and give an answer to the problem you have formulated

Suggested solution:a. Terese bought a car for kr 000160 , the devaluation of the type of

car which she bought is %5 per year. What is the value of Terese10 years after the purchase? [0/1]

b. ( ) kr 80095kr 798950.95000160P 1010 ==

Answer: The car worth kr 80095 ten years after its purchase. [0/1]

Year 1992 1993 1994 1995Average price of cinematicket (crowns each) 60.00 61.80 63.80 63.80CPI (Base year 1980) 234.9 244.3 250.4 256.0


7/35



9. A student wants to find the gradient of the curve y x= 3 at x = 2 . As he cannot differentiate y x= 3 he can not solve the problem by calculating the valueof the derivative at x = 2 .

Therefore he determines an approximate value of the derivative by calculating therate of change quotient

9.11.29.131.23

Write down a new rate of change quotient which should give a better approxima-tion to the derivative.

An answer consisting of just the new rate of change quotient is required. (2p)

Suggested solution:( ) ( )

0002.09999.130001.232

9999.10001.29999.130001.232

=

= f f [0/1]


8/35



10. The health clinic at a school with 700 students would like to know to whatextent the pupils take part in sporting activities. Therefore they carried out asurvey and reported the results in the following form.

Lisa is writing a special project about to what extent school pupils take part insporting activities and wants to use the school health clinics conclusion.

Therefore she wants to find out if the survey was carried out properly but sherealises that the report does not contain sufficient information for her to be able todo it. Therefore she decides that she will not be able to make a good evaluation of the quality of the survey.Which parts of the research does Lisa need to know more about in order to be ableto evaluate the quality of the research?Why are these things important? (4p)

Suggested solution:It is not mentioned if the students were chosen randomly, and if theywere representative group reflecting the composition of the studentbody, for example, sex, ethnic background, religion, health, .The decline 1 is not reported.It is very important to report the decline group, otherwise the survey

is not representative.

1 Decline: bortfall

Extent of pupils participation in sporting activities Aim: The aim of this survey is to determine to what extent

pupils in the school take part in sporting activitiesMethod: We asked 100 of the schools pupils to fill in the following

questionnaire1. Are you a girl or a boy? Girl Boy 2. How many hours a week do you take part in sporting

activities? None at all Less than At least

4 h/week 4 h/week

Results: Not everyone filled in the questionnaire, but the results for those that did respond are given by the table below

Conclusion: 13% of the schools boys and 12% of the schools girls donot participate in any sporting activities at all


9/35



11. A friend of yours, who is studying the same mathematics course as you, comes toyou and says, "I dont understand a thing about this differentiation."Help your friend by explaining what a derivative is.Explain in as much detail as you can and in as many ways as you can.You should neither derive nor write down the differentiation rules. (4p)

Suggested solution:Differentiation is used in the following circumstances: We may use differ-entiation to report howfast a function ischanging. If the graph of thefunction is given to findthe instantaneous rateof change (i.e. how fast

the function is changing,we may draw thetangent of the functionat the given point. Theslope of the tangent isthe derivative of thefunction at the point.

The first order ap-proximation is choosing two points on the curve of the function, in

the close vicinity of the point of interest. The difference between thevalues of the function in the neighbourhood of the interest divided bythe increment used gives the approximate value of the derivative.

For example, if only the graph of the function is given, as in thefigure above, which representing number of bacteria in a culturesample, and the goal is to find how fast bacteria in the sample isgrowing 25 hour after the culture is set. WE may draw a tangentat the point of interest, i.e. at ht 24= . The find the slope of theline, by making a right-angle triangle, and by dividing the length

of the vertical side of the triangle to the horizontal length:hbacteriak /305.29

203010305

=

=


10/35



12. For a given medicine to have the desired effect, a patient must have mg15 of themedicine in his body. If the total amount of the medicine is given in one dose, there is a risk of seriousside effects. Therefore the patient is given small doses at hourly intervals. After 10 such equally sized doses the medication is stopped and the patient should thenhave 15 mg of the medicine in his body.How large should the dose be if it is known that the medicine starts to work immediately and that %16 of it is broken down by the body per hour? [0/4/]

Suggested Solutions:Lets denote the hourly does by x . Ten such does are given athourly intervals, but %16 of themedicine is broken down by thebody per hour. The total accu-mulative medicine in the bodyafter 10=n ten such equally sizeddoses reaches the desired valueof mg15 . Therefore, the problemmay be treated as a geometricseries where:

10=n , 84.016.01 ==k ,mgS 1510 =

( )k

k xk xk xk x xS

=++++=

1

1...10

9210

( )( ) mg x 1584.01

84.01 10=

( )( )( )( ) mgmg xmg

x 9087.284.011516.015

16.084.01

10

10

=

==

Answer: mg x 9.2

0

3

6

9

12

15

18

1 2 3 4 5 6 7 8 9 10 11

n: does number

A c c u m u

l a t e d m e

d i c i n e

i n t h e

b o

d y

Does Medicine (mg) Accumulatedn left in the body Medicine (mg)

1 0,60564 0,605642 0,72100 1,326633 0,85833 2,184964 1,02182 3,206785 1,21645 4,423246 1,44816 5,871407 1,72400 7,595398 2,05238 9,647779 2,44331 12,09108

10 2,90870 15,000Total 15,000


11/35



13. The following is known about the function ( ) x f f ( )7 3= and 2.1)(8.0 x f for 7 9 x .

Determine the largest possible value of f ( )9 . (3p)Suggested Solutions:Largest value of f ( )9 is

22.1)7()9( += f f L [0/1] 4.23)9( += L f [0/1]

4.5)9( = L f [0/1/] 28.0)7()9( += f f S

6.13)9( +=S f 6.4)9( =S f

Answer: The largest value of f ( )9 is 4.5)9( = L f , and its smallest valueis 6.4)9( =S f


12/35



NATIONELLT PROV IMATEMATIK

KURS CVREN 1996

Breddningsdel

Anvisningar

Provperiod Vecka 18 - 22 1996.

Provtid Enligt beslut vid skolan men minst 100 minuter (under normallektionstid).

Hjlpmedel Enligt lokalt beslut vid skolan.

Provmaterialet Provmaterialet inlmnas tillsammans med dina lsningar.

Skriv ditt namn, komvux/gymnasieprogram och fdelsedatum p de papper du lmnar in.

Provet Breddningsdelen innehller tre alternativa uppgifter varavdu

vljer en uppgift. Frgorna i uppgiften kan vara sdana att du sjlv mste ta stll-ning till de mjliga tolkningarna. Du skall redovisa de utgngs- punkter som ligger till grund fr dina berkningar och slutsatser.

ven en pbrjad icke slutfrd redovisning kan ge underlag fr positiv bedmning.

Till varje uppgift finns en beskrivning av vad lraren kan ta hn-syn till vid bedmning av ditt arbete.

Om ngot r oklart frga din lrare.

Arbetsformer Ansvarig lrare informerar om de arbetsformer som gller fr breddningsdelen i provet.

Redovisning av uppgifterna sker individuellt.


13/35



1 SALE OF ICE CREAMNote that Statistics is no longer part of math C. Ice Cream Ltd. is a small family business which manufactures ice cream in a town inmid Sweden and sells it throughout the whole country. The company has developed twonew types of ice cream which are called Lingonberry Cone and Cloud Berry Cone. IceCream Ltd. will start to manufacture and sell one of these. The companys management,that is, the mother and father, hope that the new product will interest both those who buy a lot of ice cream and those who have not bought much ice cream before.

In order to be able to choose which ice cream to manufacture, two pieces of marketresearch were carried out. Two family members, David and Elin, did their own randomsampling with the aim of discovering which type of ice cream would sell better. Thenthey presented their surveys to the company. You can see their reports on the following

two pages.As David and Elin reached different conclusions and furthermore the companymanagement was doubtful about parts of the surveys, they decided to engage you as aconsultant.

Your task is the following: Carry out a critical review of the surveys David and Elin have done.

The company management was doubtful about the way in which David used theresults of his missing data investigation.Do a new calculation and decide for yourself which ice cream you think theyshould manufacture.

Plan your own enquiry with the aim of finding out which of the two types of icecream will sell better.

When evaluating your work, the teacher will take into consideration: how much justified criticism, negative as well as positive, you are able to

give of the two surveys. if you perform correct calculations and draw likely conclusions.

how well you plan and present your investigation.


14/35



Davids ReportAim: To determine which type of ice cream we should manufacture.

Sample: I took a sample by starting from a register of all the 189 gymnasium andadult education classes in the district. I then picked out the first three peopleon the class list of each class. These people received a letter which invitedthem to take part in a tasting of the ice creams.

Method: The tasting took place at each of the schools. All the tasters were given oneice cream of each type to eat. Afterwards they received a short questionnaire(see below). Of the 567 who were invited, 300 attended the tasting.

Result: LINGONSTRUT HJORTRONSTRUT Male Female Male Female

Yes 148 101 Yes 135 96 No 29 22 No 42 27

Missing2 data: 267 of those invited did not go to the tasting. I chose 50 of theseand visited them at the schools between two lessons. They tasted theice creams and filled in the questionnaire.

The results from the questionnaires were as follows:

Result: LINGONSTRUT HJORTRONSTRUT Male Female Male Female

Yes 12 20 Yes 17 27 No 8 10 No 3 3

Conclusion: A total of 148 + 101+ 12 + 20 = 281 people liked the LingonberryCone and 135 + 96 + 17 + 27 = 275 people liked the Cloud BerryHjortronstrut. Eftersom det r fler som tycker om Lingonstrut s blir slutsatsen att vi br tillverka Lingonstrut.

2 Bortfall

Tasting Questionnaire Cross the correct alternative:

1. Are you female or male? Female Male

2. Do you like the Lingonberry Cone? Yes No

3. Do you like the Cloud Berry Cone? Yes No

Thank you for your participation!


15/35



Elins ReportMethod: I could not do a complete survey of the whole of the Swedish

population so I took a random sample.

Sample: I sent 25 Lingonberry Cones and 25 Cloud Berry Cones to one of thekiosks which sells our other varieties of ice cream. The kiosk owner promised to offer every fifth ice cream customer a free ice cream of each sort on the condition that they ate them immediately andanswered a question. Not all the ice cream customers accepted theoffer but after a while 25 people had tasted both types of ice cream.

Procedure: The tasters were given a questionnaire which consisted of thefollowing question:Do you prefer the Lingonberry Cone to the Cloud Berry Cone?

Result: Yes 10

No 15Missing Data: As the people who did not want to take part in the survey were

replaced by the next person in the queue, I did not have any missingdata.

Conclusion: As the NO-answers were in the majority, we should manufacture theCloud Berry Cone.


16/35



2 SAVING FOR THE FUTURE The Swedish pension system has changed and in the future there will not be as muchstate pension as before. To compensate for this shortfall in income, some start to save

for their pension while they are still young. There are different ways to save for a pension. One way is IPS (Individual Pension Sav-ing). It can function so that you save a regular amount which then increases in value.When you become a pensioner, you can take out money over a number of years and paynormal income tax.

With IPS, you can decide to save in a share account. When you forecast the amount of money you can take out when you become a pensioner, you can use the average growthrate per year and assume that it is relevant during the whole saving period. The banksusually assume that the average yearly growth rate lies between 7% and 12%.

Assume that you will start saving for a pension.Your saving shall be such that you deposit a fixed amount into a share account once ayear. This continues every year until you start to take out your pension. After you become a pensioner, you withdraw money once per year. You will withdraw the sameamount every year that you take out your pension.

You will decide: how large an amount you will save every year how many years you will save

when you shall start withdrawing your pension (however theearliest is when you are 55) how many years you will draw your pension (at least 5 years)

You must also make your own assumptions about how large the average growth ratewill be. How large will the amount that you have in your share account be when you start

to draw your pension? What is the smallest amount you must save each year so that your savings will

have grown to 2 million crowns when you start to draw your pension?Investigate the different possibilities

How large a yearly pension will you have from these savings?

When evaluating your work, the teacher will take into consideration: how many of the parts yo have solved. if your choices are reasonable. if your calculations are correct. how well you perform your investigation. how clear and complete your report is.


17/35



SAVING FOR THE FUTURE3 [4/4/] The Swedish pension system has changed and in the future there will not be as muchstate pension as before. To compensate for this shortfall in income, some start to save

for their pension while they are still young.There are different ways to save for a pension. One way is IPS (Individual Pension Sav-ing).It can function so that you save a regular amount which then increases in value.When you become a pensioner, you can take out money over a number of years and paynormal income tax.

With IPS, you can decide to save in a share account. When you forecast the amount of money you can take out when you become a pensioner, you can use the average growthrate per year and assume that it is relevant during the whole saving period. The banksusually assume that the average yearly growth rate lies between 7% and 12%.

Assume that you will start saving for a pension.

Your saving shall be such that you deposit a fixed amount into a share account once ayear. This continues every year until you start to take out your pension. After you be-comea pensioner, you withdraw money once per year. You will withdraw the same amountevery year that you take out your pension.

You will decide: how large an amount you will save every year how many years you will save when you shall start withdrawing your pension (however the earliest is when

youare 55)

how many years you will draw your pension (at least 5 years)

You must also make your own assumptions about how large the average growth ratewill be.

How large will the amount that you have in your share account be when youstartto draw your pension?

What is the smallest amount you must save each year so that your savings willhave grown to 2 million crowns when you start to draw your pension?Investigate the different possibilities

How large a yearly pension will you have from these savings?

Suggested solution:Main Case:Assume I start saving in my IPS at my 25 th birthday. I will save 500kr/month, i.e. year kr /6000 . I will start withdrawing my retirement onmy 65th birthday, and will withdraw once a year on my birthdays un- 3 Final MaCNVCO06


18/35



til my 75 th birthday. The goal is to calculate the fixed amount of money I may withdraw from my IPS for ten years. Assume the unre-alistic interest of 10% per year:From my 25 th birthday to my 64 th birthday I deposit year kr /6000 for40 years and the saving grows at the annual rate of

%10. At my 64 th

birthday I have saved kr 5556552 :

( ) ( ) ( ) ( )[ ] C kr =

=+++ 5556552

110.1110.1600010.16000...10.1600010.160006000

40392

I will withdraw year kr x / from this money an the rest will grow at theannual rate of %10 . There will be the total of 11 withdraws.

( ) ( ) ( ) ( ) ( )[ ]110.1

110.110.1...10.110.110.1555655211

10211

=+++=

x x x x xkr

( ) kr x 757660953116706.18 =

year kr kr

x /85840853.18

7576609==

The table below confirms the calculations above:In the simple EXCEL program below you are supposed to start saving at your 25th birthday andstart withdrawing at your 65th birthday. The amount of withdraw per year is constant,the last withdraw is on your 75th birthday. You may decide on:

Anual saving in CrownsInterest rate in percent

n Birth day saving n Birth day saving1 25 6 000 21 45 384 015 yearly saving 6000 Crowns2 26 12 600 22 46 428 416 Start saving on your 25 birthday

3 27 19 860 23 47 477 258 Start withdrawl on your 65 birthday4 28 27 846 24 48 530 984 Interest rate (%) 10 percent5 29 36 631 25 49 590 082 The growth rate 1,16 30 46 294 26 50 655 091 yearlly withdraw 408 857,6 Crowns7 31 56 923 27 51 726 6008 32 68 615 28 52 805 260 n Birth day saving after9 33 81 477 29 53 891 786 withdrawal

10 34 95 625 30 54 986 964 1 65 2 512 25311 35 111 187 31 55 1 091 661 2 66 2 354 62112 36 128 306 32 56 1 206 827 3 67 2 181 22513 37 147 136 33 57 1 333 509 4 68 1 990 49014 38 167 850 34 58 1 472 860 5 69 1 780 68215 39 190 635 35 59 1 626 146 6 70 1 549 892

16 40 215 698 36 60 1 794 761 7 71 1 296 02417 41 243 268 37 61 1 980 237 8 72 1 016 76818 42 273 595 38 62 2 184 261 9 73 709 58819 43 306 955 39 63 2 408 687 10 74 371 68920 44 343 650 40 64 2 655 555 11 75 0,0000


19/35



Alternative II:On the other hand, if the total saving at the beginning of my re-tirement is to be two million Swedish Crowns, assuming the identi-cal assumptions as above, except of course, for the annual saving,the problems character is changed to the following:

Alternative II.1: ykr x / : annual saving yn 40= : total number of years saving 10.1=k : 10% annual interest rate Skr S 000000240 = : total saving at the beginning of the retire-

ment

( ) ( ) ( ) ( )[ ] Skr x x x x x 0000002110.1

110.110.1...10.110.140

392 =

=+++

( )[ ] ySkr Skr

x /5194110.1

00020040

=

=

Answer: In order to save two million Swedish Crown at the begin-ning of my retirement, I must save at least ySkr x /5194= annu-ally at the fixed annual interest rate of 10% for 40 years.

Of course there could be a few variation of this problem as well:

Alternative II.2:

07.1=k : 7% annual interest rate ykr x / : annual saving yn 40= : total number of years saving Skr S 000000240 = : total saving at the beginning of the retire-

ment

( ) ( ) ( ) ( )[ ] Skr x x x x x 0000002107.1

107.107.1...07.107.140

392 =

=+++

( )[ ] ySkr Skr

x /01810107.1000140

40=

=


Alternative II.3:On the other hand if we assume 10% interest rate but instead sav-ing for 20 years instead, i.e.:

10.1=k : 10% annual interest rate ykr x / : annual saving


20/35



yn 20= : total number of years saving: Start Saving on the 45 th birthday and start to withdraw on the 65 th birthday

Skr S 000000240 = : total saving at the beginning of the retire-ment

( ) ( ) ( ) ( )[ ] Skr x x x x x 0000002110.1 110.107.1...07.107.120

192 =

=+++

( )[ ] ySkr ySkr Skr

x /000350/192349110.1

00020020

=

=

Answer: In order to save two million Swedish Crown at the begin-ning of my retirement, I must save at least ySkr x /000350 an-nually at the fixed annual interest rate of 10% for 20 years.

Alternative II.4:

The last but not the least alternative is assume 10% interest rateand an annual payment of for example ySkr /00010 and then cal-culate the number of years necessary to accumulate the desiredtwo million Swedish Crowns at the beginning of my retirement,i.e.:

10.1=k : 10% annual interest rate ySkr x /00010= : annual saving ?=n : total number of years saving

Skr Sn 0000002=

: total saving at the beginning of the retirement ( ) ( ) ( ) ( )[ ] Skr

nn 0000002

110.1110.10001010.100010...10.10001010.10001000010 12 =

=+++

( )[ ] 200

110.1110.1

=

n

( ) 20110.1 =n ( ) 2110.1 =n

( ) 21log10.1log =n 21log10.1log =n

yn 32943.3110.1log

21log==


Alternative III:

On the other hand if my desire is to have the possibility of with-draw of at least ySkr /000500 for eleven years starting on my 65 th birthday and ending on my 75 th birthday. Assuming again a fixedannual interest rate of 10% and assuming that I start my saving


21/35



at the 25 th birthday we may calculate the minimum amount annualsaving for 40 years:

( ) ( ) ( ) ( )[ ] ( )[ ]110.1

110.100050010.1...10.110.1100050010.1S11

1021140

=+++= kr

kr 5312473S40 = Now the problem is changed to a problem similar to a problemsimilar to the Alternative II.2, i.e.:

10.1=k : 10% annual interest rate ykr x / : annual saving yn 40= : total number of years saving Skr 5312473S40 = : total saving at the beginning of the retire-

ment

( ) ( ) ( ) ( )[ ] Skr x x x x x 5312473110.1 110.110.1...10.110.140

392 =

=+++

( )[ ] ySkr Skr

x /37573110.1

75332440

=

=

Answer: In order to be able to withdraw a minimum of ySkr /000500 for eleven years starting on my 65 th birthday and

ending on my 75 th birthday. I must save at least ySkr x /37573= annually at the fixed annual interest rate of 10% for 40 years.


22/35



3Functions

Draw a graph of a function. The derivative of the function assumes the value 2where the graph cuts the y-axis. It is up to you to decide the appearance of theremainder of the graph.

Draw the graph of two quadratic functions. Both the functions should havederivatives which assume the value 0 where the graphs cut the y-axis. It is up toyou to decide the appearance of the remainder of the graph.Also give the equations of the functions.

Determine the conditions which must be satisfied by the coefficientsa, b and c inthe quadratic function ( ) cbxax x f ++= 2 so that the graph of the function has alocal maximum on the positive y-axis.

Find two quadratic functions which have graphs with a local minimum at the point (1,0).

Find a third degree function ( ) d cxbxax x f +++= 23 which has a graph with alocal maximum on the y-axis and a local minimum at the point (1,0).

Determine the conditions which must be satisfied by the coefficientsa,b,c and d in the third degree function ( ) d cxbxax x f +++= 23 so that the graph of thefunction has a local maximum on the y-axis and a local minimum at the point(1,0).

When evaluating your work, the teacher will take into consideration:

how many of the parts you have solved. if your calculations are correct. how well you have motivated and presented your solutions.


23/35



3 Functions [4/6/]

Draw a graph of a function. The derivative of the function assumes the value 2where the graph cuts the y-axis. It is up to you to decide the appearance of theremainder of the graph.

Draw the graph of two quadratic functions. Both the functions should havederivatives which assume the value 0 where the graphs cut the y-axis. It is up toyou to decide the appearance of the remainder of the graph.Also give the equations of the functions.

Determine the conditions which must be satisfied by the coefficientsa, b and cin the quadratic function ( ) cbxax x f ++= 2 so that the graph of the functionhas a local maximum on the positive y-axis.


Find a third degree function ( ) d cxbxax x f +++= 23 which has a graph with alocal maximum on the y-axis and a local minimum at the point (1,0).

Determine the conditions which must be satisfied by the coefficientsa,b,c and d in the third degree function ( ) d cxbxax x f +++= 23 so that the graph of thefunction has a local maximum on the y-axis and a local minimum at the point(1,0).


24/35



Suggested Solutions: Functions Draw a graph of a function. The derivative of the function assumes the value 2

where the graph cuts the y-axis. It is up to you to decide the appearance of theremainder of the graph.

Suggested Solutions:In the figure below ( ) 523 2 += x x x f and its derivative ( ) 26 += x x f areplotted. The derivative of the function cuts the y-axis at 2= y . [1/0]

-6

-4

-2

0

2

4

-2 -1 0 1 2

x

f , f '

( ) 523 2 += x x x f

( ) 26 += x x f

Draw the graph of two quadratic functions. Both the functions should have

derivatives which assume the value0 where the graphs cut the y-axis. It is up to

you to decide the appearance of the remainder of the graph.Also give the equations of the functions.Suggested Solutions:In the figure below function ( ) 42 = x x f and its derivative ( ) x x f 2= ,as well as the function ( ) 63 2 += x xg and its derivative ( ) x xg 6= areplotted. Both derivatives pass through the origin ( )0,0 .[1/0]

-6

-4

-2

0

2

4

6

8

-3 -2 -1 0 1 2 3

x

f , f '

, g ,

g '

( ) 42 =

x x f

( ) x x f 2=

( ) 63 2 += x xg( ) x xg 6=


25/35



Determine the conditions which must be satisfied by the coefficientsa, b and cin the quadratic function ( ) cbxax x f ++= 2 so that the graph of the functionhas a local maximum on the positive y-axis.


In order for the function ( ) cbxax x f ++= 2 to have a local maximum onthe y-axis, i.e. at 0= x the following conditions must be simultane-ously satisfied:

1. The derivative of the function ( ) bax x f += 2 must be zero at 0= x ,i.e. ( ) ( ) 00020 ==+= bba f .

2. The derivative of the function ( ) ax x f 2= must be positive for 0< x .If < 0a ( ) 002 >= xax x f .

4. The function must be positive at 0= x . i.e.: ( ) 000 >> c f .

Answer: 0c . Therefore, if 0a ,

0=b , and a positive real number c . Both have a global maximum onthe y-axis.

-4

-2

0

2

4

6

-3 -2 -1 0 1 2 3

x

f ( x

)

( ) 52 += x x f

( ) 25 2 += x x f


26/35





In order for a quadratic function ( ) cbxax x f ++= 2 to have a localminimum at ( )0,1 the following conditions must be simultaneouslysatisfied:

1. The derivative of the function, ( ) bax x f += 2 , must be zero at1= x . i.e.: ( ) ( ) abbaba f 2020121 ==+=+= . ab 2=

2. The derivative of the function ( ) aax x f 22 = must be negative for1< x . i.e.: ( ) 1022 a 3. The derivative of the function ( ) aax x f 22 = must be positive for

1> x , i.e.: ( ) 1022 >>= xaax x f . This condition is automaticallysatisfied if the condition above is true, i.e. if 0>a . 0>a

4. The function must be tangent to the x-axis at 1= x , i.e.: ( ) 01 = f

( ) ( ) ( ) ( ) accacaacaa f caxax x f ==+=+=+=+= 002012112 22 0>= ac Answer: Any quadratic function ( ) ( ) 012 22 >=+= a xaaaxax x f wouldhave a local minimum at ( )0,1 . Below the graphs of two functionsthat both satisfy the criteria above are illustrated: [1/1]

1=a ( ) ( )22 112 =+= x x x x f has a local minimum at ( )0,1 .5=a ( ) ( )22 155105 =+= x x x xg has a local minimum at ( )0,1 .

-1

0

1

2

3

4

5

-2 -1 0 1 2 3 4

x

f ( x

)

( ) 122 += x x x f

( ) 5105 2 += x x xg


27/35



Find a third degree function ( ) d cxbxax x f +++= 23 which has a graph with alocal maximum on the y-axis and a local minimum at the point (1, 0). Determine the conditions which must be satisfied by the coefficientsa, b, c and d in the third degree function ( ) d cxbxax x f +++= 23 so that the graph of the

function has a local maximum on the y-axis and a local minimum at the point (1, 0).Suggested Solutions:In order for a cubic function ( ) d cxbxax x f +++= 23 to have a localmaximum on the y-axis, i.e. at 0= x , and a local minimum at ( )0,1 the following conditions should simultaneously be fulfilled: [0/1/] 1. The derivative of the function, i.e. ( ) cbxax x f ++= 23 2 should be

zero at 0= x . i.e.: ( ) ( ) ( ) 0002030 2 ==++= ccba f ( ) bxax x f 23 2 += 0=c

The derivative of the function, i.e. ( ) bxax x f 23 2 += should be zero at

1= x . i.e.: ( ) ( ) ( ) abbaba f 230230121312 ==+=+= [0/1] ab 23

=

2. As illustrated in the table below, if 0>a all conditions stated in thetable, are simultaneously satisfied. [0/1]

3. If the function has a local minimum at ( )0,1 , it must be tangent tothe x-axis at 1= x : ( ) 01 = f . This condition will be satisfied if:

( ) ( ) ( ) ( ) =+=+=+= 02101

2311

23 2323 d ad aa f d axax x f ad

21

=

x 1= x 0= x 5.0= x 1= x 2= x ( ) ( )1333 2 == xaxaxax x f ( ) 01 > f ( ) 00 = f ( ) 05.0 f

( ) 021

23 23 >+= aaaxax x f max

min

( )21

23 23 += x x x f has a local max on the y-axis and a local min at ( )0,1 .

( ) 132 23 += x x xg has a local max on the y-axis and a local min at ( )0,1 :

-1

0

1

2

-1 0 1 2

x

f , g

( )21

23 23 += x x x f

( ) 13223

+= x x xg


28/35



Bedmningsanvisningar - tidsbunden del

Uppg. Bedmningsanvisningar Pong

1. Max: 2pa) Korrekt svar ( = f x x( ) 12 43 ) +1p b) Korrekt svar (92)

2. Max: 4pa) Korrekt svar ( =g x x( )

2) +1p

b) Korrekt svar ( =h x e x

( ) 63

) +1pc) Godtagbart svar ( x e= 2 ) +1p

d) Godtagbart svar (813 000) +1p

3. Max: 3pRedovisad godtagbar lsning (150 st) +1-3p

4. Max: 3pRedovisad godtagbar lsning (23,2%) +1-3p

5. Max: 3p( Nej, endast 234 av de 500 medlemmarna kan frvntas stdja ett stugbygge.)

Redovisat att de som varken var p mtet eller deltog i bortfallsundersk-ningen kan betraktas tycka som de som deltog i bortfallsunderskningen +1-2p

och i vrigt redovisat en godtagbar lsning +1p

6. Max: 3pKorrekt svar f A: +1pKorrekt svar g D: +1pKorrekt svar h E : +1p


29/35



Uppg. Bedmningsanvisningar Pong

7. Max: 3pRedovisade, relevanta och korrekta berkningar och en godtagbar ochrelevant kommentar som tar hnsyn till tv r +1-2p

Redovisade, relevanta och korrekta berkningar fr en jmfrelse som tar hnsyn till alla r under perioden 1992-1995 +1p

8. Max: 4pa) Formulerat ett problem som kan lsas med hjlp av ekvationen +1-2p b) Redovisad godtagbar lsning ( x = 23) +1-2p

9. Max. 2p

Godtagbart svar (t. ex. 3 2 01 3 12 01 1

, ,99, ,99

) +1-2p

10. Max: 4p Nmnt ngot om urvalet och motiverat varfr det r viktigt +1-2p(slutsatser om en population dragna utifrn en stickprovsunderskningbygger p att urvalet r representativt)

Nmnt ngot om bortfallet och motiverat varfr det r viktigt +1-2p(representativiteten i ett urval kan frstras om bortfallet inte hanteras rtt )

11. Max: 4pRedovisad godtagbar frklaring +1-4p

(Bedmda elevlsningar bifogas)

12. Max:4pRedovisad lsning som bygger p en geometrisk talfljdmed korrekt frsta element, frndringsfaktor och antal termer +1-3pmed korrekta berkningar och godtagbart svar (2,9 mg) +1p

13. Max: 3pRedovisad godtagbar lsning (5,4) +1-3p


30/35



Exempel p bedmda elevlsningar till uppgift 11

Vid bedmningen kan hnsyn tas till hur vl elevens beskrivning av derivata tar upp de-finitionen, grafisk tolkning och exemplifieringar av anvndning av derivata.

Elev 1Eleven redovisar insikter i att derivata r frknippad med frndring och hastighet menvisar ingen kunskap om att det r den momentana frndringshastigheten det handlar om. 0p

Elev 2Eleven exemplifierar en anvndning av derivata ( hastigheten i en punkt). 1p

Elev 3Eleven redovisar en grafisk tolkning av derivata genom att nmna en grafs lutning. 1p

Elev 4Eleven redovisar goda insikter i hur derivata kan tolkas grafiskt.Detta sker dels genom att nmna grafers och tangenters lutning, dels genom flera exempel.Redovisningen r utfrlig. 2p

Elev 5Eleven redovisar en grafisk tolkning av derivata och ger ett exempel p anvndning.Elevens beskrivning av derivatans definition indikerar att eleven inte har full frstelsefr den. 2p

Elev 6Eleven redovisar utfrligt en grafisk tolkning av derivata. Detta sker bde genom en be-skrivning och med exempel om extrempunkter.Eleven kompletterar med att beskriva derivatan i ord (frndringshastighet).Eleven stller upp derivatans definition. Dock saknas en frklaring av definitionens in-nebrd. 3p

Elev 7Eleven visar goda insikter i begreppet derivata. Eleven beskriver definitionen, en gra-fisk tolkning och exempel p anvndning p ett klart och utfrligt stt. 4p


31/35



Bedmningsanvisningar breddningsdel

Uppgift 1 Glassfrsljning Vid bedmningen av elevernas arbete ska fljande aspekter beaktas:* Vilken grad av insikter i hur man planerar, analyserar och rapporterar en statistisk

underskning eleven visar.* Om eleven visar knnedom om, anvnder, freslr, diskuterar och vrderar olika

bearbetningsstrategier i en statistisk underskning med hnsyn till- enktfrgors relevans- metoder fr att gra stickprov- behandling av bortfall- genomfrandet av underskningen

* I vilken grad eleven visar tankegngen i sin skriftliga redovisning av sitt arbete.

Exempel p ett godknt elevarbeteEleven ger befogad kritik inom flera av ovannmnda delar av en stickprovsundersk-ning.

Eleven redovisar motiveringar och berkningar p ett sdant stt att tankegngen kanfljas.

Eleven visar, i sin plan fr den egna underskningen, kunskaper motsvarande de someleven tidigare visat i den kritiska granskningen.

Exempel p ett vl godknt elevarbete Eleven ger tskilligt med befogad kritik, bde positiv och negativ, samt freslr, disku-terar och vrderar bearbetningsstrategier inom flera ovan nmnda delar av en stick- provsunderskning.

Eleven gr en godtagbar bortfallsberkning.

Eleven gr en planering av den egna underskningen av sdan kvalit att eleven visar kunskaper motsvarande de eleven visar i sin kritiska granskning.

Eleven redovisar motiveringar och berkningar p ett sdant stt att en klar tankegngvisas.


32/35



Uppgift 2 Sparande fr framtiden

Vid bedmningen av elevernas arbete ska speciell hnsyn tas till kunskapsomrdena- berkningar av summor av geometriska talfljder - annuitetsberkningar.

Fljande aspekter ska beaktas: * Vilken grad av insikter eleven visar.* Vilken svrighetsgrad p problemstllningar eleven kan behandla* Om eleven visar frmga att utfra ndvndiga berkningar * Vilken grad av frmga att vrdera sina resultat eleven visar.* I vilken grad eleven visar tankegngen i sin skriftliga redovisning av sitt arbete.

Exempel p ett godknt elevarbete Eleven gr rimliga val av rsrntesats, rligt sparbelopp och tidsperiod fr sparandet.

Eleven berknar sparbeloppens vrde vid pensionsavgngen med hjlp av godtagbara berkningar.

Eleven berknar p ett godtagbart stt sparbeloppens vrde vid pensionsavgngen s attvrdet ligger i nrheten av 2 miljoner kronor.

Eleven redovisar sin berkningar s att tankegngen i den skriftliga redovisningen kanfljas.

Exempel p ett vl godknt elevarbeteEleven gr rimliga val av rsrntesats, rligt sparbelopp, tidsperiod fr sparandet ochtidsperiod fr uttagen.

Eleven berknar sparbeloppens vrde vid pensionsavgngen.

Eleven visar olika stt att spara ihop till 2 miljoner kronor. Sparbeloppen bestms s attderas vrde vid pensionsavgngen ligger i nrheten av 2 miljoner kronor.

Eleven berknar en rlig pension som blir nstan lika stor varje r.

Eleven redovisar sina antaganden, berkningar och vad som berknas p ett sdant sttatt eleven i den skriftliga redovisningen visar en klar tankegng.


33/35



Uppgift 3 Funktioner

Vid bedmningen av elevernas arbete ska speciell hnsyn tas till kursplanemlen - sambandet mellan en funktions graf och dess derivator av frsta och andra ord-ningen,- derivatans vrde d funktionen r given genom graf, tabeller eller formel.

Fljande aspekter ska beaktas: * Vilken grad av insikter eleven visar.* Vilken svrighetsgrad p problemstllningar eleven kan behandla.* I vilken grad eleven visar tankegngen i sin skriftliga redovisning av sitt arbete.* Kvalitn p de grafer och koordinatsystem eleven ritar.

Exempel p ett godknt elevarbete Eleven har med tydlighet ritat erforderligt koordinatsystem och grafen till en funktionsom uppfyller villkoret = f ( )0 2 .

Eleven har med tydlighet ritat erforderliga koordinatsystem och graferna till tv andra-gradsfunktioner som uppfyller villkoret = f ( )0 0 och har angett deras funktionsuttryck.

Eleven konstaterar atta < 0 och attc > 0 med motiveringen att maximipunkten ska ligga p den positiva y-axeln.

Exempel p ett vl godknt elevarbeteEleven har ritat ett korrekt och tydligt koordinatsystem och grafen till en funktion somuppfyller villkoret = f ( )0 2 .

Eleven har ritat korrekta och tydliga koordinatssystem och med tydlighet ritat grafernatill tv andragradsfunktioner som uppfyller villkoret = f c( )0 . Eleven har ocks angettderas funktionsuttryck.

Eleven bestmmer och motiverar de villkor som mste glla fr koefficienterna i ettandragradsuttryck fr att funktionens graf ska ha en lokal maximipunkt p den positiva

y - axeln. (a < 0, b = 0, c > 0)

Eleven bestmmer tv andragradsfunktioner med egenskapen att de har lokal minimi- punkt i punkten (1,0).

Eleven redovisar sina berkningar och vad som berknas p ett sdant stt att eleven iden skriftliga redovisningen visar en klar tankegng.


34/35



Exempel p bedmda elevarbeten

Uppgift 1 Glassfrsljning

(OBS! De bifogade elevlsningarna avser endast den kritiska granskningen av enktun-derskningen)

1. IG+ Eleven redovisar befogad kritik mot Davids frga och mot antalet personer i Elinsunderskningsgrupp. Han motiverar inte kritiken men freslr en bttre frga.Sammanfattningsvis r det vldigt lite befogad kritik.

2. G- Denna elev redovisar dremot befogad kritik inom de flesta delar av stickprovs-underskningen ven om det endast r lite kritik p varje del och det r ont ommotiveringar. Eleven kritiserar indirekt Davids rapport som inte ger informationom hur provsmakningen gick till och motiverar med hur ett felaktigt frfarandekan pverka underskningen. Eleven visar ocks vissa insikter i att ett stickprov br vara representativt d hon saknar information om lder p de som provsmaka-de. Hon riktar kritik mot Davids frga och frklarar varfr utan att tydligt utredade allvarliga konsekvenserna. Eleven gr ocks en rimlig vrdering av Davidsslutsats.

I vrigt har eleven svrigheter med att avlsa tabellerna d hon inte kan berknaantalet mn respektive kvinnor i Davids underskning. Kritiken mot att en var-fr-frga saknas r irrelevant.

3. VG- Denna elev redovisar tskilligt med befogad kritik inom de flesta delar av stick- provsunderskningen. Hon motiverer kritiken och frslag p frbttringar. Elevenvisar goda kunskaper om hur frgor br stllas och om iden bakom bortfallsunder-skningar. Hon tar upp aspekten plitlighet angende Elins underskning och gr rimliga vrderingar av bde Davids och Elins slutsatser.

Dremot redovisar hon inga insikter om urvalets respresentativitet och ingen posi-tiv kritik ges.

Uppgift 2 Sparande fr framtiden

4. G- Eleven gr rimliga val av sparbelopp etc. Eleven visar insikter i det aktuella kun-skapsomrdet och frmga att utfra ndvndiga berkningar ven om de inte r helt korrekta. Eleven gr inte ngon utredning av olika mjligheter att spara ihoptill 2 miljoner kr. I redovisningen kan tankegngen till strsta delen fljas. Hur eleven har kommit fram till det sparbelopp som anvnds i verifieringen av attsparbeloppens vrde blir 2 miljoner kr redovisas dock inte.


35/35


5. VG- Eleven gr rimliga val av sparbelopp etc. Eleven visar insikter i och frmga attutfra ndvndiga berkningar inom det aktuella kunskapsomrdet. Eleven gr eninte heltckande men strukturerad utredning. Eleven visar ytterligare insikter i deaktuella kunskapsomrdena genom att anvnda en framkomlig lsningsmetod fr berkning av sin rliga pension p den sista deluppgiften. Eleven lyckas inte heltlsa deluppgiften men fr kvar ett belopp som i sammanhanget inte r allt fr stort. Eleven visar i den skriftliga redovisningen en klar tankegng frutom atteleven inte redovisar hur den rliga pensionen i den tredje deluppgiften hittats.

Uppgift 3 Funktioner

6. G-

Eleven redovisar inte funktionsuttrycken fr andragradskurvorna men visar insik-ter i det aktuella kursplanemlet d grafer som uppfyller uppgiftskraven har ritats.Eleven skriver utan motivering atta ska vara negativ. Eleven har ritat korrektaoch tydliga koordinatsystem och tydliga grafer.

7. VG- Eleven ritar korrekta och tydliga koordinatsystem och grafer som uppfyller upp-giftskraven. Eleven motiverar (dock ej med hjlp av derivata) villkoren fr kon-stanterna a , b, c i den tredje deluppgiften. Eleven hittar ocks tv funktioner ochverifierar med hjlp av frsta- och andraderivata att de uppfyller kraven i den fjr-de deluppgiften. Dremot verifieras inte att x = 1 ger y = 0. I strre delen av redo-visningen visar eleven en klar tankegng. Information fattas dock om hur eleventagit fram funktionerna i den fjrde deluppgiften.

Documents

Suggested+Solution+NPMaC Vt96 Eng