UofUMath1050-1Spring2004 Young-SeonLee ... · UofUMath1050-1Spring2004 Young-SeonLee....

U of U Math 1050-1 Spring 2004

Young-Seon Lee.

WeBWorK assignment number 1.

due 1/29/04 at 9:00 PM.

The main purpose of this first WeBWorK set is toreview some prerequisites for this class and to helpyou familiarize yourself with WeBWorK.Here are some hints on how to use WeBWorK ef-

fectively:� After first logging into WeBWorK changeyour password.� Find out how to print a hard copy on the com-puter system that you are going to use. Con-tact me if you have any problems. Print a hardcopy of this assignment.� Get to work on this set right away and answerthese questions well before the deadline. Thatway you won’t run out of time if you can’ssolve a problem right away, and it will giveyou a chance to to figure out what’s wrong ifan answer is not accepted.� The primary purpose of the WeBWorK as-signments in this class is to give you the op-portunity to learn, for example by having in-stant feedback on your active solution of rel-evant problems. Make the best of it!

Procrastination is hazardous!

Peter Alfeld, JWB 127, 581-6842.

1.(10 pts) This first question is just an exercise inentering answers into WeBWorK. It also gives youan opportunity to experiment with entering differentarithmetic and algebraic expressions into WeBWorKand seeing what WeBWorK really thinks you are do-ing (as opposed to what you believe it should think).Notice the buttons on this page and try them out be-fore moving to the next problem. Use the ”Back”Button on your browser to get back here whenneeded.

� ”Prob. List” gets you back to the list of allproblems in this set.� ”Next” gets you to the next question in thisset.� ”Submit Answer” submits your answer asyou might expect, but there may be otherways to do so. Specifically, in this problem,there is only one question. In that case youcan submit your answer by typing it into theanswer window and then pressing ”Return”(or ”Enter”) on your keyboard. But even inthis case, you can also type the answer andclick on the ”submit” button. There is noharm in submitting an answer even if you arenot quite sure that it’s correct, since if it isnot you have an unlimited number of addi-tional tries. On the other hand, it is usuallymore efficient to print your own private prob-lem set, work out the answers in a quiet envi-ronment like your home, and then sit down infront of a computer and enter your answers. Ifsome are wrong you can try to fix them rightat the computer, or you may want to go backand work on them quietly elsewhere beforereturning to the computer.� Pressing on the ”Preview Answer” Buttonmakes WeBWorK display what it thinks youentered in the answer window. After using”Preview” you can modify your answer anduse a ”Preview Again” button.� ”typeset” denotes the ordinary display modeon your workstation, but ”formatted text” isa little faster. Very occasionally the outputfrom ”typeset” gets corrupted. In that caseyou can send me an email and I will fix it.You can also see the intended output, if notquite as nicely, using ”formated text”. Theoptions of plain ”text” and ”typeset2” are notuseful.� ”Logout” terminates this WeBWorK sessionfor you. You can of course log back in andcontinue.� ”Feedback” enables you to send a message tome. If you use this way of sending e-mailI receive information about your WeBWorKstate, in addition to your actual message.

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� The ”Help” Button transports you to an offi-cial WeBWorK help page that has more infor-mation than this first problem.� ”Problem Sets” transports you back to thepage where you can select a certain problemset. When you do this particular problem inthis first set, there is only one set, but eventu-ally there will be 13 of them.� For all problems in this course you will beable to see the Answers to the problems af-ter the due date. Go to a problem, clickon ”show correct answers”, and then click on”submit answer”. You can also download andprint a hard copy with the answers showing.These answers are the precise strings againstwhich WeBWorK compares your answer. Ifthe answer is an algebraic expression your an-swer needs to be equivalent to the WeBWorKanswer, but it may be in a different form. Forexample if WeBWorK thinks the answer is2 � a, it is OK for you to type a �

a instead. IfWeBWorK expects a numerical answer thenyou can usually enter it as an arithmetic ex-pression (like 1 � 7 instead of � 142857), andusually WeBWorK will expect your answerto be within one tenth of one percent of whatit thinks the answer is.� Most of the problems (including this one) inthis course will also have solutions attachedthat you can see after the due date by click-ing on ”show solutions” followed by ”sub-mit answers”. The solutions are text typedby your instructor that gives more informa-tion than the ”answers”, and in particular of-ten explains how the answers can be obtained.

Now for the meat of this problem. Notice that the an-swer window is extra large so you can try the thingssuggested above.Type the number 3 here:

.Try entering other expressions and use the pre-

view button to see what WeBWorK thinks you en-

tered. Return to this problem to try out things

when you get stuck somewhere else.

Here are some good examples to try. Check them allout using the Preview button. (In later questions you

will get to use what you learn here.) Never mind thatyou may have already answered the correct answer 3.Once you get credit for an answer it won’t be takenaway by trying other answers.a/2b versus a/2/b versus a/(2b)a/b+c versus a/(b+c)a+b**2 versus (a+b)**2sqrt a+b versus sqrt(a+b)4/3 pi r**2 versus (4/3) pi r**2 (In other words, ifyou are not sure use parentheses freely.)Note: WeBWorK will not usually let you enter al-

gebraic expressions when the answer is a number, andit will only let you use certain variables when the an-swer is in fact an algebraic expression. So the abovewindow, and the opportunity for experimentationthat it offers is unique. Make good use of it!Presumably this has been your first encounter withWeBWorK. Come back here to try things out andto refresh your memory if you get stuck somewheredown the line.

2.(10 pts) The purpose of this exercise is to illus-trate further the use of the buttons on this page andto show you the most common way in which WeB-WorK processes partially correct problems. Try en-tering incorrect answers in the answer fields below,to see what happens. (This time WeBWorK will re-ject algebraic expressions since I told it to expect anumerical answer.)Type the number 4 here: .Type the number 5 here: .

3.(10 pts) WeBWorK will usually consider a nu-merical answer to be correct if it is within one tenthof percent of the answer that it has been given. How-ever, you do not usually have to enter a decimal ap-proximation. WeBWorK will do simple arithmetic

for you. The answer in this problem is 27. You can

enter this number in various forms, e.g., 2/7, 4/14,1/(7/2), 0.285714. Try it below. Also try to enterrough approximations like 0.3, 0.29, 0.286, 0.2857,0.28571 and see which WeBWorK will accept.Occasionally, WeBWorK will insist on an answer be-ing a decimal expression. If so that expectation willbe stated in the problem.27 � .

4.(10 pts) You will find that your interaction withWeBWorK usually goes pretty smoothly. Suppose,

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however, you are sure that you have the right answer,but WeBWorK will not accept it. In that case youmay of course actually have the wrong answer. Inrare cases, WeBWorK may have been told the wronganswer. (If that happens send me a message and youwill receive an extra point in our One Point Contest.)A frequently occurring third possibility is that you areexpressing your answer in a way that is inconsistentwith the rule of arithmetic precedence:� A missing operator means multiplication.� Functions are evaluated first.� Multiplication and Division come before Ad-

dition and Subtraction.� Operations of the same level of precedenceare carried out from left to right.� Expressions in parentheses are evaluated first,and so parentheses overrule the precedingrules.� It’s OK to have unnecessary parentheses (forexample, if you are not sure WeBWorK willunderstand what you mean).

In this and the next few problems, you will beasked to enter algebraic expressions. Keep in mindthe above rules, try alternatives, and if WeBWorKwill not accept your answer, use the preview buttonto see what WeBWorK thinks you are saying. It maydiffer from what you think you are saying.a � bc � .

5.(10 pts)a � bc � d � .

6.(10 pts) Remember that a missing operatormeans multiplication. Thus you can enter the productof c and d as cd or c*d. Try it:cd � .Now entera � bcd � .

If you get stuck use the preview button!

7.(10 pts) It is useful to understand howWeBWorKdetermines whether you entered a correct algebraicexpressions. Algebra is driven by the fact that ex-pressions can be changed to equivalent ones. For ex-ample, the second binomial formula states�

a � b 2 � a2 � 2ab �b2

for all real and complex numbers a and b. You canindicate a power with a double asterisk, but how doyou know whether to enter the expression (a-b)**2or a**2 - 2*a*b + b**2?The answer is that it does not matter. The code defin-ing the WeBWorK problem contains a particular ex-pression that’s the “correct” answer. However, WeB-WorK evaluates your expression for some randomvalues of the variables, and it evaluates its expres-sion at the same random numbers. Then it comparesthe two sets of values and if they agree it deems youranswer correct. (I once heard it said that all of math-ematics is driven by the creative confusion of neces-sary and sufficient conditions. This must be an in-stance of this drive.)Try entering both alternatives below. Also try enter-ing something that’s actually wrong, like a**2-b**2.�a � b 2 � .

8.(10 pts) Enter here the expression

11a

� 1b

Enter here the expression

a�b

�1

1� 1a � b

9.(10 pts)Enter here the expression

ab

� cd

ef

� gh

�10.(10 pts) To enter square roots you can use the

function sqrt. For example, to enter the square rootof 2 you can type sqrt(2).2 � . (Try the exact value and a numerical

approximation.)Enter here the expression

a

11.(10 pts) Enter here the expressiona

�b


aa

�b

3


a�b

a�b

12.(10 pts)Enter here the expression�

x2�y2


x�x2

�y2


x�y�

x2�y2

13.(10 pts)Enter here the expression� b �

b2 � 4ac2a

(This is of course the celebrated quadratic formula.)Use the preview button if you get stuck!

14.(10 pts)For each of the WeBWorK phrases below enter a

T (true) if the two given phrases describe the samealgebraic expression and an F (false) otherwise. Oneway you can decide whether the phrases are equiva-lent is to substitute specific values for a, b, etc. If youget two different results the two phrases are certainlynot equivalent. If you get the same values there issmall chance this happened accidentally for just thatchoice of particular values. In any case, pay closeattention to when these phrases are equivalent andwhen they are not, it will help you tremendously withfuture WeBWorK assignments.a

�b b

�a

a�b

�c a

� �b

�c

a � b � c a � �b � c

15.(10 pts)More of the same.

a�b2

�a

�b 2

a2�b2

�a

�b 2

a � b � c a � �b � c

a � b � c a � �b � c

16.(10 pts) The following question reviews the ter-minology of the number system. See section A.1 fordetails.

Match the statements defined below with the let-ters labeling particular numbers. Use all the letters.Of course a natural number is also a rational num-ber, for example. However, there is only one correctmatching that uses all five letters A through E.

1. x is a natural number2. x is an irrational number3. x is an integer4. x is neither positive nor negative5. x is a rational number

A. x � 12B. x � � 17C. x � 0D. x � 17

12E. x � π

17.(10 pts) This question reviews the terminol-ogy for the four basic arithmetic operations: adding(sum), subtracting (difference), multiplying (prod-uct), and dividing (quotient). Don’t use words like“plussing” or “timesing”, they are juvenile, and youhave moved beyond them.Match the phrases given below with the letters la-

beling the algebraic expression.You must get all of the answers correct to receivecredit.

1. The difference of x and x2 divided by the sumof x and x2

2. The sum of x and 83. The product of the sum of x and 2 and the sumof x2 and 2

4. The quotient of x and the sum of x and 85. The sum of x and 2, all squared

A. xx � 8

B. x�8

C.�x

�2 �x2

�2

D.�x

�2 2

E. x � x2x � x2

18.(10 pts) Indicate whether the following state-ments are True (T) or False (F).

1. The difference of two natural numbers is al-ways a natural number.

2. The product of two natural numbers is alwaysa natural number.

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3. The quotient of two natural numbers is al-ways a natural number.

4. The ratio of two natural numbers is alwayspositive

5. The quotient of two natural numbers is al-ways a rational number

6. The difference of two natural numbers is al-ways an integer.

7. The sum of two natural numbers is always anatural number.


1. The difference of two integers is always a nat-ural number.

2. The sum of two integers is always an integer.3. The quotient of two integers is always a ra-tional number (provided the denominator isnon-zero).

4. The ratio of two integers is always positive5. The difference of two integers is always aninteger.

6. The product of two integers is always an inte-ger.

7. The quotient of two integers is always an in-teger (provided the denominator is non-zero).


1. The sum of two rational numbers is always arational number.

2. The quotient of two rational numbers is al-ways a rational number (provided the denom-inator is non-zero).

3. The ratio of two rational numbers is alwayspositive

4. The difference of two rational numbers is al-ways a rational number.

5. The quotient of two rational numbers is al-ways a real number (provided the denomina-tor is non-zero).

6. The difference of two rational numbers is al-ways a natural number.

7. The product of two rational numbers is al-ways a rational number.


1. The product of two real numbers is always areal number.

2. The quotient of two real numbers is always arational number (provided the denominator isnon-zero).

3. The difference of two real numbers is alwaysan irrational number.

4. The ratio of two real numbers is never zero.5. The quotient of two real numbers is alwaysa real number (provided the denominator isnon-zero).

6. The difference of two real numbers is alwaysa real number.

7. The sum of two real numbers is always a realnumber.

22.(10 pts) Remember that we multiply powerswith the same base by adding the exponents, we di-vide them by subtracting the exponents, and we takea power to a power by multiplying the exponents.

x4y3

x7y6 � 2 � xaybwhere a � and b � .

23.(10 pts) �x1

8 y2

3

x4

5 y1

7 � 1

3 � xaybwhere a � and b � .

24.(10 pts) The world’s educational system makesyou (and millions of fellow sufferers) study mathe-matics because it enables you to solve word prob-lems. This is why you will find word problems inevery home work and every exam of this class. Stu-dents tend to dislike word problems because there isan additional layer of difficulty: you have to trans-late the words into mathematics before you can ap-ply mathematics. But keep in mind that math classesare the only kind of classes where some problems arenot word problems. The word problems in this classwill require only commons sense and and knowledge(or else we will discuss the subject involved in classbefore the word problems are assigned).We start gently. You are five feet tall, and walking

in the forest. You cast a shadow of length 11 feet.5

You rest under a tree with a shadow of length 88 feet.The height of the tree is feet.Many problems in this class, like this particular

one, will come with hints and solutions. After theset closes you can go back to it, click on “show solu-tions”, and then click on “submit answer” once again.This will show you the solution. If the problem hashint (like this one) a new button labeled “show hint”will appear after you submit your first answer. Click-ing on this button and submitting an answer againwill show you the hint. Try it on this problem even ifyou have already solved it.

25.(10 pts) Light travels through vacuum at aspeed of approximately 186,000 miles per second.The distance of the sun from earth is approximately93,000,000 miles. Thus it takes minutes forlight to travel from the sun to earth.Proxima Centauri, the closest known star other

than the Sun, has a distance of approximately 2 � 5 �1013 miles. It takes sun light yearsto reach Proxima Centauri. (Assume that a year has365.25 days.)

26.(10 pts) One of the things we’ll learn how to dobetter in this class is counting. Here’s a clever warmup problem, taken from the delightful book “Prob-lems for Mathematicians, Young and Old” by Paul R.Halmos.

Suppose that 839 tennis players want to play an elim-ination tournament. That means: they pair up, at ran-dom, for each round; if the number of players beforethe round begins is odd, one of them, chosen at ran-dom, sits out that round. The winners of each round,and the odd one who sat it out (if there was an oddone), play in the next round, till, finally, there is onlyone winner, the champion. What is the total numberof matches to be played altogether, in all the roundsof the tournament?Your answer: .

27.(10 pts) The following problem foreshadows(but of course does not require) another one of ourtopics, logarithms:Supposing the same situation as before, how manyrounds need to be played?Your answer: .

28.(10 pts) Suppose in a certain season the Jazz av-erage 98 points in their first three games. How manypoints do they have to score in their fourth game toaverage 100 points in their first four games?Your answer: .

29.(10 pts) One pipe can fill a tank three times asfast as another pipe. If together the two pipes can fillthe tank in 36 minutes, how many minutes would ittake the slower pipe to fill the tank by itself?Your answer: .

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c�UR

6

U of U Math 1050-1 Fall 2004

Young-Seon Lee.


due 2/4/04 at 9:00 PM.

The second home work set reviews some of theprerequisites for our class.The first 11 problems address some common alge-

braic errors. You are asked if certain proposed identi-ties are true or false. Those that are false are nonethe-less often believed to be true by many people. To getthe most benefit from these problems, look carefullyif you find that something you believed to be true ac-tually isn’t. Figure out why not, and remember yourmistake in the future!Problems 12 through 17 are meant to shed insight

on our standard approach to equation solving: keepapplying the same operation on both sides of theequation until you have the variable by itself.Problems 18 through 22 present a few typical

forms of equations that you should be able to solvewithout difficulties.Problems 32 and 33 illustrate the fact that all ra-

tional numbers can be written as repeating decimalexpressions, and all repeating decimal expressionsrepresent a rational number. Problem 34 is one ofthose extra challenging problems promised in yoursyllabus.The remaining problems in this set are simple word

problems that usually lead to a single equation in oneunknown that you can solve. (You may be able tofind alternative, and perhaps simpler, ways, for someof these problems.)

1.(10 pts) Many times in this class we will useidentities, i.e., statements of equality that are true forall values of the variables involved.For example,

x�y � y �

x

for all real (or complex) numbers x and y. (This par-ticular identity is called the commutative law of addi-tion. It does not matter in what sequence we add twonumbers.) In the next few problems, enter the string

”=” if the identity holds and the string ”N” (or ”n”) ifit does not. In all cases, omit the quotation marks.To get the hang of this we start with the above iden-tity:x

�y y

�x.

(Enter an equality sign.)Here is an example of a non-identity (enter an uppercase N):x

�y x � y.

Many algebraic mistakes are caused by equating al-gebraic expressions that are not identical.The next few problems illustrate widely used identi-ties and frequently believed non-identities. You couldconceivably get credit for these problems by rush-ing through the various possibilities, but you shouldpause each time that the answer is not what you ex-pect and figure out why you were wrong and what isreally going on. The benefits from those moments ofthoughtfulness will be significant and long lasting!Unless stated otherwise, identities need to hold for allreal numbers. This includes positive and negativereal numbers. We don’t consider complex numbers.

2.(10 pts) Enter ”=” if the proposed identity holds,and ”N” otherwise.

1x

� 1y

1x � y .

1x

� 1y

x � yxy.


x � 1x � 2 1

2.

ax � b a

x

� ab.

x

ab

bx

a.

x

ab

x

ab.


1

x

ay

b

xy

ab.

x

ay

b

ab

xy.

x

ay

b

bx

ay.

x

ay

b

ax

by.

5.(10 pts) This problem concerns the appropriateuse of a slash to denote division, more than a typi-cal identity (which is independent of notational issueslike using a fraction bar instead of a slash).

Enter ”=” if the proposed identity holds, and ”N” oth-erwise.

1 � x 1x.

1 � 3x 13x.

1 � 3x x3.


a � �x � b a � x � b.

a � �x � b a

�x

�b.

a � �x � b a � x �

b.

7.(10 pts) Enter ”=” if the proposed identity holds,and ”N” otherwise.�12a � �

12b � 1

2ab.�

12a � �

12b � ab

2.�

12a � �

12b � ab

4.�

12a � �

12b � 1

4ab.

8.(10 pts) Enter ”=” if the proposed identity holds,and ”N” otherwise.�x

�3 2 x2

�9.�

x�3 2 x2

�6.

x2 � 9x � 3 x

�3.

x2�9 x

�3.

9.(10 pts) Enter ”=” if the proposed identity holds,and ”N” otherwise.�a

�b 2 a2

�b2.�

a�b 2 a2

�2ab

�b2.�

a � b 2 a2 � b2.�a � b 2 a2 � 2ab �

b2.

a2 � b2 �a � b �

a�b .

10.(10 pts) Enter ”=” if the proposed identityholds, and ”N” otherwise.�x2 � 3 x5.�x2 � 3 x6.

x2x3 x5.

x2x3 x6.

11.(10 pts) This problem is a bit subtle. Thinkabout it!Enter ”=” if the proposed identity holds, and ”N” oth-erwise.x2 x.x2 � x � .12.(10 pts) Solving an equationmeans figuring out

which values of the variable make the equation true.The basic approach to equation solving consists ofapplying the same operation on both sides of theequation until we come up with another equation that

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has the variable by itself on one side and an expres-sion not containing that variable on the other.

For example, in the (very simple) equation

x�3 � 5

we subtract 3 on both sides and obtain the equation

x � 2 �The latter equation tells us the solution, and we ver-ify that x � 2 does indeed solve the original equationby substituting 2 for x in x

�3 � 5. Since 2 � 3 does

equal 5 we have indeed found the solution.

The preceding two paragraphs are deceptively sim-ple, but they describe one of the key ideas of algebra.

The following problems explore a subtlety of the con-cept of ”applying the same operation on both sides ofthe equation”. In the above example, the two equa-tions involved are equivalent, i.e., one implies theother. If x � 2 then x � 3 � 5. Conversely, if x � 3 � 5then x must be 2, there is no other solution. Some-times, however, doing the same thing on both sides ofan equation creates a new equation that is not equiv-alent to the old one.

For example, if x � 3, then squaring on both sidesgives x2 � 9. It is true that x � 3 implies that x2 � 9.But the other direction does not hold, if x2 � 9 thenx may be � 3 since the square of � 3 also equals 9.The process of squaring introduces a extraneous (orsometimes called spurious) solution. The existenceof such solutions is a major reason why you alwayscheck your answer.

In this and the next few problems you are asked todecide whether two equations are equivalent, one im-plies the other, or neither implies the other. En-ter (without the quotation marks) ” �� ” if the leftequation implies the right, ” � �� ” if the right equa-tion implies the left, ” � �� ” if either equation im-plies the other, and ”¿¡” if neither equation impliesthe other. An equation equation A implies an equa-tion B if B is true for all variables for which A is true.

For example,

x�3 � 5 � �� x � 2x � 2 �� x2 � 4

x2�6 � 10 � �� x � 2

x�3 � 5 � � x2 � 9

Use these statements in the following items:

x�3 � 5 x � 2.

x � 2 x2 � 4.x2�6 � 10 x � 2.

x�3 � 5 x2 � 9.13.(10 pts) Enter (without the quotation marks)

” �� ” if the left equation implies the right, ” � �� ”if the right equation implies the left, ” � �� ” if ei-ther equation implies the other, and ”¿¡” if neitherequation implies the other.

5x�3 � 2x � 4 3x

�3 � � 4. (Subtract 2x on

both sides.)

3x�3 � � 4 3x � � 7. (Subtract 3 on both

sides.)

3x � � 7 x � � 73 . (Divide by 3 on both sides.)Note: this problem illustrates the logic of solving asimple linear equation.

14.(10 pts)This problem illustrates the logic involved in solvinga quadratic equation by completing the square. (Thisparticular equation can also be factored, but it’s beenchosen so as to keep the arithmetic simple.)

Enter (without the quotation marks) ” �� ” if theleft equation implies the right, ” � �� ” if the rightequation implies the left, ” � �� ” if either equationimplies the other, and ”¿¡” if neither equation impliesthe other.

x2�2x � 8 � 0 x2

�2x�1 � 9. (add 9 on both

sides).

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x2�2x�1 � 9 �

x�1 2 � 9. (Apply

the binomial formula x2�2ax

�a2 � � x � a 2, with

a � 3, on both sides.)x2�2x�1 � 9 x

�1 � 3. (one possibility).

x2�2x�1 � 9 x

�1 � � 3. (the other

possibility).Thus the solutions of x2

�2x � 8 � 0 are x � � 4 and

x � 2.15.(10 pts) Enter (without the quotation marks)

” �� ” if the left equation implies the right, ” � �� ”if the right equation implies the left, ” � �� ” if ei-ther equation implies the other, and ”¿¡” if neitherequation implies the other.� x � 2 � � � 1 � 2x � x

�2 � 1 � 2x.

x�2 � 1 � 2x x � � 13 .� x � 2 � � � 1 � 2x � � � x � 2 � 1 � 2x.� � x � 2 � 1 � 2x x � 3.

The above considerations suggest that the originalequation � x � 2 � � � 1 � 2x �has the two solutions x � 3 and x � � 13 . Substitutingin the original equation shows that this is indeed thecase.

16.(10 pts)In this problem, assume that x is positive (so we cantake its square root).

Enter (without the quotation marks) ” �� ” if theleft equation implies the right, ” � �� ” if the rightequation implies the left, ” � �� ” if either equationimplies the other, and ”¿¡” if neither equation impliesthe other.

x � 5 x � 6 � 0 x2 � 5x � 6 � 0.x � 5 x � 6 � 0 5

x � x � 6.

5x � x � 6 25x � x2 � 12x � 36.

This last equation is an ordinary quadratic equa-tion. Solve it to answer the remaining question: Thesmaller solution of x � 5 x � 6 � 0 is x � , andthe larger is x � .

17.(10 pts) The last problem in this group illus-trates the concept of an extraneous solution. Theequation involves absolute values, but to illustrate theconcept we solve it in an unusual way: by squar-ing instead of considering two cases for the absolutevalue.

Enter (without the quotation marks) ” �� ” if theleft equation implies the right, ” � �� ” if the rightequation implies the left, ” � �� ” if either equationimplies the other, and ”¿¡” if neither equation impliesthe other.� x � 1 � � 2x � 1 �

x�1 2 � � 2x � 1 2.

(Square on both sides. The square of the absolutevalue is the square of what is inside that absolutevalue.)�x�1 2 � � 2x � 1 2 x2

�2x�1 � 4x2 � 4x � 1.

(Apply the Distributive Law, or, more specifically, thebinomial formulas.)

x2�2x�1 � 4x2 � 4x � 1 3x2 � 6x � 0.

(Subtract�x2�2x�1 on both sides and switch

sides.)

3x2 � 6x � 0 x�x � 2 � 0.

(Factor, and divide by 3 on both sides.)

x�x � 2 � 0 x � 0.(one possible solution.)

x�x � 2 � 0 x � 2.(the other possible solution.)

Only one of the two proposed solutions satisfies theoriginal equation � x � 1 � � 2x � 1. It is x � .

18.(10 pts) The next few problems present a se-quence of typical equations and ask you to solvethem. You need to understand the techniques in-volved and gain facility in using them. So you should

4

do these problems by hand rather than use an ad-vanced calculator or a symbol manipulation languagelike Maple. I recommend you enter answers as exactarithmetic expressions rather than decimal approxi-mations. So, for example, type 3/7 instead of 0.42857or sqrt(3)/2 instead of 0.86602.

The solution of the equation�x � 1 �

x � 2 � �x

�2 �x � 3

is x � .

19.(10 pts) The solution of the equation

1

2x�3 � 1

x � 1is x � .


x � 1x

�3 � x �

2

x � 1is x � .

21.(10 pts) The quadratic equation

x2 � 2x � 1 � 0has two real solutions. The smaller one is x � .

22.(10 pts) This is essentially problem 193 on pageA58 of the textbook. Two planes leave simultane-ously from Salt Lake City International, one flyingdue north and one due east. The northbound planeis flying 50 miles per hour faster than the east boundplane. After 2 hours the planes are 1063 miles apart.The speed of the slower plane is miles per hour.Round your answer to the nearest mile per hour. As-sume the earth is flat—this is wildly inaccurate overa distance of 1063 miles!

23.(10 pts) This is essentially problem 140 on pageA69 of the textbook. Your department sends its copy-ing to the photocopy center of your company. Thecenter bills your department $0.09 per page. Youhave investigated the possibility of buying a depart-mental copier for $3200. With your own copier, thecost per page would be $0.02. The expected life ofthe copier is 6 years. You figure that you must makeat least copies per year to justify buying thecopier.Your answer should be a natural number. The param-eters in this problem differ for each student. While

I am confident that WeBWorK rounds its result cor-rectly, there may be a subtle round off effect thatmakes your answer differ from ww’s answer. So ifww does not accept your answer at first, try one moreor one less than your computed answer.[Of course this question totally ignores the conve-nience of having your own copier. Also, the num-ber of copies is likely to go up, and so may the pricecharged by the copy center. But it’s just a textbookquestion!]

24.(10 pts) Your backyard is 24 feet longer than itis wide. Its area is 8881 square feet. Its width isfeet.

25.(10 pts) You invest $720 at 7% annual interest.Assume all of your interest is paid at the end of eachyear.After one year your investment has grown to $ .After two years your investment has grown to $ .After 10 years your investment has grown to $ .Enter your answers rounded to the nearest cent.

26.(10 pts) You have 600 micrograms of a certainradioactive material. In the course of each year 8 per-cent of the material decays (and turns into somethinginert and benign).After one year you have micrograms of thematerial left..After two years you have micrograms.After 10 years you have micrograms.WeBWorK will accept your answer if it is within onetenth of one percent of what it believes to be the trueanswer. You are safe if you enter the number of mi-crograms with 2 digits beyond the decimal point. Youcan also enter an arithmetic expression.

27.(10 pts) According to theUS Bureau of the Census, the world population in 1950 was 2,555,360,972.In 2003 it was 6,302,486,693. (These areestimated midyear figures . The table given by the Bureaualso lists projected population figures for 2004-2050.)Based on these figures, assuming that the populationincreased by the same percentage each year from1950 to 2003, that yearly increase waspercent. The corresponding figures for the US arepopulations of 152,271,000 in 1950 and 290,343,000in 2003. This corresponds to an annual growth rateof percent.

5

28.(10 pts) The next four (simple) problems aretaken from the 2000 edition of the ARCO GREGMAT Math Review. They can be answered bysetting up and solving single equations in one un-known. (You may find an easier way for some ofthem, though.)Eddie earns $12 per hour for a normal 8-hour work-

day and 32times this rate for hours worked overtime.

On a certain day, Eddie earned $141 and workedhours and minutes.[Actual GRE tests are of course multiple choice.]

29.(10 pts) Tony is now three years older thanHoward. Five years ago the sum of their ages was59. Now, Tony is years old.

30.(10 pts) One can fitcubes, each having a surface area of 54 square inches,exactly inside a larger cube having a surface area of216 square inches.

31.(10 pts) In a promotional sale, Roxanne boughtone item for its regular price and a second item for$1. She paid $20 for both items. What fraction of theprice of the first item is the price of the second item?

32.(10 pts) The fraction

z � 41333can be written as a repeating decimal z �0 � abcabcabc �� wherea � , b � , and c � .

33.(10 pts) The repeating decimal

z � 0 � 142857142857142857 ��can be written as a fraction z � � .Make sure you simplify your fraction before you en-ter it.34.(10 pts) This is one of the more challenging

problems promised in your syllabus. It looks prettybewildering but it can be solved by a straightforwardapplication of one of the main principles we discussin this class every day. Suppose

x � �� 1 � 1�"!1�"#1� 1� ��

where the square roots go on forever. You may as-sume (and it is true) that this expression actually de-fines a positive number x. What is it? Enter it as adecimal or radical expression:


6


Young-Seon Lee.


due 2/11/04 at 9:00 PM.

This problem set continues the review of old mate-rial and leads to the new subjects that we will discussin this semester. All of the material needed for thisset will have been covered in class by the time yousee this set.The focus of the set is the solution of linear, qua-

dratic, and radical equations. The first few exercises,however, are concerned with the simplification of ra-tional expressions.

1.(10 pts) This and the next few problems are exer-cises in writing rational expressions in standard form.You are asked to enter the numerator and denomina-tor as polynomials. For example, since we carry outdivisions by increasing length of the fraction bar (asdiscussed in class),

ab

c � a

bc $and so you would enter a as the numerator and bc asthe denominator. Try it now:ab

c � � .

2.(10 pts)abc

� � .

3.(10 pts)abcd

� � .

4.(10 pts)a

b

c

d � � .

abc

d

� � .

5.(10 pts)

1� 1x

1 � 1x

� � .

(Enter your answer in such a form that the leadingcoefficient in the numerator equals 1.)

6.(10 pts) The next few problems reinforce yourmastery of exponents. Remember that you multiplypowers with the same base by adding the exponents,you divide them by subtracting the exponents, andyou take a power to a power by multiplying the expo-nents.We begin by reviewing some basic identities.

aman � azwhere z � .�

am n � azwhere z � .

7.(10 pts)

x3

5 x2

5 � xzwhere z � .%

x3

5 & 25 � xzz � .

8.(10 pts) If

2x � 23 � 27then x � .

9.(10 pts) If �2x 3 � 27

then x � .

10.(10 pts) If2x

23 � 27then x � .

11.(10 pts) If

23

2x � 27then x � .

12.(10 pts) The next few questions continue the re-view of equation solving. Some of these equationscontain parameters.The solution of the equation

3x�12 � 6

is x � .The solution of the equation

ax�b � c

is x � . (You may assume that a '� 0).1

13.(10 pts) The equation

x2�3x � 4 � 0

has two solutions. The smaller is x � .The equation

ax2�bx

�c � 0

may have two real solutions, one real solution, or aconjugate complex pair of solutions. Assuming thereare two real solutions, and a � 0, the smaller is x �

. (This is the celebrated quadratic formula.)


x � 1 � xx

has two solutions. The larger is x � .


x � 1 � x

has one solution. It is x � .

16.(10 pts) The equation� x � 1 � � � 2x � 1 � � 2has two solution. The smaller isx � , and the larger isx � .

17.(10 pts) The next few problems deal with inter-est earning investments. If your money earns p per-cent interest per year then at the end of the year yourmoney is multiplied with the factor

�1

� p100 � . (The

interest is paid annually—at this stage we ignore sub-tleties like paying interest every month.)You invest $700 at 5 % annual interest. After ten

years you have $ in the Bank.

18.(10 pts) You invest $8000. Your banker tellsyou that your investment will double in exactly 6years. At your next party you tell your friends thatyou invested money (you modestly omit the amount)at percent annual interest. (Enteryour percentage with at least three digits beyond thedecimal point.)

19.(10 pts) At the same party, your friend tells youthat her banker told her that her investment will triplein 12 years. Her annual interest rate is percent.

20.(10 pts) There’s a pond in your village. Some-body plants a water lily in it that doubles its size everyday. While you are off on a vacation the lily grows

by a factor 512. Your vacation was dayslong. (Note: this problem foreshadows one of ourmajor topics in this class, i.e., logarithms. It doesnot require logarithms for its solution, however.)

21.(10 pts) The remaining problems of this setall lead to quadratic equations that you are asked tosolve. WeBWorK will accept numerical answers, butyou should be able to produce exact answers, for ex-ample using square roots. In other words, don’t justkey the equation into your calculator and transfer thenumerical values.The equation

x2 � 3x � 10 � 0has two solutions. The smaller is , and the larger

.


x2 � 6x � 9 � 0has two solutions. The smaller is , and the larger

.


2x2 � 6x � 9 � 0has two solutions. The smaller is , and the larger

.


2x � 1x

�2

� x �1

x � 1 � 2 � 0has two solutions. The smaller is , and the larger

.


x4 � 5x2 �6 � 0

has four solutions. Enter them in increasing se-quence:

, , , .


x � 5 x

�6 � 0

has two solutions. The smaller is, and the larger is ,

27.(10 pts) The Hypotenuse of an isosceles righttriangle is 5cm long. Its sides are cm long.

2

Now suppose the Hypotenuse has length h. Then thelength of the other two sides is . (Of course youranswer will depend on h).

28.(10 pts) You are working in a room whose longside runs south to north. The room is 23 feet long, and18 feet wide. You have an exquisitely accurate laserrange finder that tells you that the distance from thelower southwest corner to the upper northeast corner

of the room is1078 feet. The height of the room is

feet.29.(10 pts) This delightful problem is taken from a

collection of 200 ”problem solving cards” designedfor eighth grade. It’s the hardest problem of thatset, but of course you are well beyond eighth grade!

Consult the original card here . The problem state-ment below does not say that the tank is lying on oneof its three sides, but it’s clear from the picture on thecard.A fuel storage tank is in the shape of an equilateraltriangular prism. The prism is 8 feet high and 10 feetlong. When the tank is half-full, the depth of the fuelis feet.

30.(10 pts) In your summer home you have a fueltank that is in the shape of a pyramid sitting on itsbase. Its height is h. The depth of the fuel when thetank is half full equals . (Of course your answerwill depend on h.)


3


Young-Seon Lee.


due 2/18/04 at 9:00 PM.

The first ten problems in this set explore the con-nection between inequalities and intervals. The bulkof the set, problems 11–28, deal with graphs of equa-tions, including straight lines and circles. A majortheme is the correspondence between the algebra andsymmetry of the graph. The last three questions il-lustrate two simple ideas of data analysis, linear andexponential interpolation, concepts that fit nicely intoour current investigation. All of the material in thisset has been covered in class, except that we have notyet discussed functions. It was convenient to phrasethe last three problems in terms of functions, andso this concept enters those problems in a peripheralway. You know what a function is from Math 1010,and know that the graph of a function f is the graphof the equation y � f �

x . However, we will discussfunctions in more depth starting nex week.

1.(10 pts) The first few problems are exercises insolving inequalities and using interval notation, asdescribed in sections A1 and A6 of the textbook. Forexample, the set of inequalities� 1 � x ( 3describes the interval

� � 1 $ 3 ) Enter that interval bytyping a round opening parenthesis, the number -1,the number 3, and a square closing parenthesis in thefollowing four answer fields:

, .In this problem, WeBWorK will tell for each answerindividually if it is correct, so you can check whatyou are doing. In the following problems you willneed to get all answers right before getting credit.

2.(10 pts) The inequalities

5 ( x � 12describe the interval:

, .

3.(10 pts) The inequality� x � 8 � � 6describes the interval:

, .

4.(10 pts) The inequality� x � 12 � ( 6describes the interval:

, .

5.(10 pts) The inequality

x2 � 1describes the interval:

, .


x2 ( 25describes the interval:

, .

7.(10 pts) The inequality�x

�1 2 ( 25

describes the interval:, .


x2�3x � 1

describes the interval:, .

9.(10 pts) The domain of the expression�36 � 4x2

is the interval, .


2x � 7x � 5 * 3

describes the interval, .

11.(10 pts) The next few problems involve theCartesian coordinate system and the graphs of equa-tions. The textbook jumps right into these top-ics, if you need a more basic review you may findthis web page handy.

The distance between the two points� � 1 $ 2 and�

3 $ 4 is .1

12.(10 pts) From your camp ground you walk 1

mile east,2 miles northeast, and two miles south.

The distance between you and the camp ground isnow miles. (Assume the world is flat.)

13.(10 pts)The circle in the graph above is described by the

equation

�x � h 2 � �

y � k 2 � r2whereh � ,k � , andr � .The graph is not very detailed, but all the answers

in this question are integers. The screen version of thegraph is larger than the printed version In this prob-lem, WeBWorK will tell you for each partial answerwhether it us correct or not.

14.(10 pts)The circle in the graph above is described by the

equation

�x � h 2 � �

y � k 2 � r2whereh � ,k � , andr � .

15.(10 pts)

� A.2

� B.

� C.Match the graphs above with the functions below.(Enter A, B, and C, as appropriate.) Clicking on theabove graphs will show you a larger version of thesame graph.y � x �

1 ,x2

�y2 � 1 , and

y � x2 .

16.(10 pts)

� A.

� B.

� C.Match the graphs above with the functions below.(Enter A, B, and C, as appropriate.) Clicking on theabove graphs will show you a larger version of thesame graph.y � x3 � x ,y2 � x �

11 , andy � x4 � 4x2 .

17.(10 pts)Recall our discussion of symmetry in class. The

graph G of an equation is symmetric with respect tothe x-axis if

�x $ � y is on G whenever �

x $ y is. It issymmetric with respect to the y axis if

� � x $ y is onG whenever

�x $ y is. And it’s symmetric with the re-

spect to the origin if� � x $ � y is on G whenever �

x $ y is.For example, the graph of y � x2 is symmetric with

respect to the y axis since if we replace x with � x weobtain the same y. Similarly, the graph of x � y2 issymmetric with respect to the x-axis. The graph ofy � x3 is symmetric with respect to the origin becausewe get � y when we replace x with � x. On the other

3

hand, the equation y � x � 1 is not symmetric in anyof these three senses.Below, enter x if the graph of the given equation issymmetric with respect to the x-axis, y if it is sym-metric with respect to the y axis, o (lower case O) ifit is symmetric with respect to the origin, and n (forNone) if is has none of these three symmetries.� y � x � 1� y � x3� y � x2� x � y218.(10 pts) Below, enter x if the graph of the given

equation is symmetric with respect to the x-axis, y if itis symmetric with respect to the y axis, o (lower caseO) if it is symmetric with respect to the origin, and n(for None) if is has none of these three symmetries.� y � x4� y � x2 � 1� y � x2 � x� y � � 1 � x2 319.(10 pts) Below, enter x if the graph of the given

equation is symmetric with respect to the x-axis, y if itis symmetric with respect to the y axis, o (lower caseO) if it is symmetric with respect to the origin, and n(for None) if is has none of these three symmetries.� y � x3 � x� y � � x3 � 1 2� y � 1

1 � x2� y � x1 � x2 .

20.(10 pts) Below, enter x if the graph of the givenequation is symmetric with respect to the x-axis, y if itis symmetric with respect to the y axis, o (lower caseO) if it is symmetric with respect to the origin, and n(for None) if is has none of these three symmetries.� y � 1� x � 1� y � � x �� y � 1

1 �,+ x + .21.(10 pts) Below, enter x if the graph of the given

equation is symmetric with respect to the x-axis, y if itis symmetric with respect to the y axis, o (lower caseO) if it is symmetric with respect to the origin, and n(for None) if is has none of these three symmetries.

� y � 1x� x � 1y� y �.-- 1x --� y � 2 � x41 � x222.(10 pts) Suppose you have a circle with radius

r. Then its diameter is , its circumference is, and its area is . All your answers should be interms of r. Use pi to denote π.

23.(10 pts) Suppose you have a circle with diame-ter d. Then its radius is , its circumference is, and its area is . All your answers should be interms of d. Use pi to denote π.

24.(10 pts) Consider the line passing through thepoints

�2 $ 6 and � � 2 $ 5 . It can be written in slope-

intercept form asy � x

�.

25.(10 pts) Consider the line of slope � 2 passingthrough the point

�1 $ 1 It can be written in slope-

intercept form asy � x

�.

26.(10 pts) Consider the line with x intercept 2 andy-intercept � 2. It can be written in slope interceptform asy � x

�.

27.(10 pts) Consider two lines. One has slope 2and y intercept -3. The other passes through thepoints

�1 $ 2 and � � 2 $ 3 . The two lines intercept in

the point� $ .

28.(10 pts) A line can be defined in general formby the equation

Ax�By � C

where A, B andC are constants. You may assume thatboth A and B are non-zero. Then the slope of the lineis , its x intercept is , and its y intercept is

. Of courseyour answers will depend on A, B, andC. Rememberthat mathematics, and WeBWorK, are case sensitive,so make sure to use upper case letters A, B and C.

29.(10 pts) The next three problems explore sometechniques of data analysis.According to the US Bureau of the Census ,

the world population in the year 1950 was A �2555360972, and in 2000 it was B � 6079006982 �

4

We’ll use A and B so we don’t have to keep writingthose large and idiosyncratic numbers. We usuallyuse y and x in the equation of a line, but in this andthe following problem let’s use N and t instead. tstands for time and N for the size of the population.If

N�t � mt �

b

such that N�1950 � A and N �

2000 � B, thenm � and and b � .

Suppose you want to estimate the population in 1975.To that end you compute N

�1975 =

. (Round your answers to the nearest integer. Theprocess illustrated in this problem is called linear in-terpolation.)

30.(10 pts) Using the same notation as before, let

M�t � %

1� p

100& t � 1950

A

where p � is chosen such thatM�2000 � B.

(This function might be referred to somewhat grandlyas an exponential model). Again you want to estimatethe population in 1975. So you compute M

�1975 =

. (Round your answers to the nearest integer.)

31.(10 pts) The Bureau of the Census estimates theworld population in 1975 to be 4087344760. Thisdiffers from your linear prediction by percentand the prediction based on the exponential model by

percent. (I’d be interested to hear your thoughtson the effectiveness of those two models.)


5


Young-Seon Lee.


due 2/25/04 at 9:00 PM.

The first five problems in this set illustrate how toapproach a difficult problem by building a hierarchyof easier problems. The remainder of the problemsfocus on functions, how to evaluate them, the con-cept of domain and range, combining functions, andpossible symmetries.

1.(10 pts) The first five problems in this set weremotivated by the feedback I received on problem 22(that airplane problem...) of set 2. I thought of it asa routine problem, but it seems to have been quitedifficult for a good number of people. These firstfive problems illustrate amajor principle of problemsolving:If a problem is hard simplify it and first solve

the simpler problem.

Put a little more loosely: If at first you don’t suc-ceed, do something easier.

To appreciate this lesson, before you start on thisproblem, try problem 5 on this set. You’ll see it’stricky. But it will be a piece of cake after you solvethe first four problems.So if you find yourself again (in this class, or much

beyond it) facing a hard problem, attack it by build-ing a sequence of easier problems, the easiest beingvery easy, that lead to the difficult problem and, in theend, make it easy to solve the hard problem.

You and your friend part at an intersection. Youdrive off north at 50 mph, and your friend drives eastat 50mph. After three hours the distance between youand your friend is miles.

2.(10 pts) This problem is like the first, except thespeeds are different.You and your friend part at an intersection. You

drive off north at 50 mph, and your friend drives eastat 60mph. After three hours the distance between youand your friend is miles.

3.(10 pts) In this problem the two speeds are dif-ferent and unknown. This is much like the airplaneproblem you solved on set 2.You and your friend part at an intersection. You

drive off north at a constant speed, and your frienddrives east at a speed that is 10 mph higher. After3 hours the distance between you and your friend is285.04 miles. You have been driving at mph.(Round to the nearest mile).

4.(10 pts) This problem differs from the precedingone in that the departure times are different. In build-ing a hierarchy of problems it’s a good idea to changeone ingredient at a time.You and your friend part at an intersection. Your

friend drives away north at a constant speed. Youlinger at the intersection for an hour, and then driveoff due east at a speed that is 11 miles per hour fasterthan your friend’s speed. 2 hours after your friend’sdeparture the distance between the two of you is 106miles. Your friend is traveling at a speed ofmiles per hours.

5.(10 pts) Now we are ready. Apply to this prob-lem what you’ve learned in the preceding problem.You and your friend part at an intersection. Your

friend drives away north at a constant speed. Youlinger at the intersection for an hour, and then driveoff due east at a speed that is 11 miles per hourfaster than your friend’s speed. Two hours after yourfriend’s departure the distance between the two ofyou is 193 miles. 4 hours after your friend’s depar-ture your distance from your friend ismiles. (Remember that ww expects your answer to bewithin one tenth of one percent of the true answer.)

6.(10 pts) The remainder of the problems in this setare focused on the concept of a function.Consider the function f defined by

f�x � x2 �

3x�1

Then � f �1 �� f � � 3 �� f �23

� �1

7.(10 pts) Consider the function f defined by

f�x � x2 � 3x � 5

Then � f � 1 �� f � � 3 �� f � 23� �

8.(10 pts) Consider the function f defined by

f�x � x � 3x � 5

Then the domain of f is the set of all real numbersexcept x � .

9.(10 pts) The answers to the next few questionsare intervals. Enter the intervals as you did in the firstfew problems of the last set. For example, the do-

main of the function f�x �

x2 � 25 is the interval/ � 5 $ 5 ) . Enter that interval by typing a square openingbracket, the number -5, the number 5, and a squareclosing bracket, in the following four answer fields:

, .Some intervals may have ∞ (infinity) in them. Enterit as the string infinity. Enter negative infinity as-infinity.For example, the domain of the function g

�x � x2 is� � ∞ $ ∞ . Enter it in these four answer fields:

, .The range of g is

/0 $ ∞ . Enter it here:

, .In this problem, WeBWorK will tell for each answerindividually if it is correct, so you can check whatyou are doing. In the following problems you willneed to get all answers right before getting credit.

10.(10 pts) The domain of the function

f�x � �

16 � x2is the interval

, ,and its range is

, .


f�x � �

4x � x2is the interval

, ,

and its range is, .


f�x �

4 � x21 � x2

is the interval, .


f�x � x � 4x � 2

is the set of all real number exceptand its range is the set of all numbers except

,


f�x � x2

4 � x2is the set of all real numbers with two exceptions. Thesmaller exception isx � , and the larger isx � .The range of f is the set of all real numbers exceptthose in the interval

, .

15.(10 pts) The next few problems focus on theevaluation of functions, at numbers, and at math-eamtical expressions.For example, if

f�x � 2x � 1 $

then

f�1 � 2 � 1 � 1 � 1 $f�t � 2t � 1 $

f�x2 � � 2x2 � 1 $

and

f�f�x � � 2 f � x 0� 1 � 2 � 2x � 1 0� 1 � 4x � 3 �

Enter these answers in the answerfields on this page:� f � 1 = ,� f � t = ,� f � x2 = , and� f � f � x � = .2

16.(10 pts) Suppose

f�x � 3x � 2 �

Then � f � 1 = ,� f � t = ,� f � x2 = , and� f � f � x � = .

17.(10 pts) Suppose

f�x � x

x�1�

Then � f � 1 = ,� f � t = ,� f � x2 = , and� f � f � x � = .

18.(10 pts) Suppose

f�x � x2 � x �

Then � f � 1 = ,� f � x2 = , and� f � f � x � = .

19.(10 pts) The next few problems are focused onthe combination of functions: Suppose f and g aretwo functions. Then we define� �

f�g � x � f � x � g � x $� �

f � g � x � f � x 0� g � x $� �f g � x � �

f � g � x � f � x 1� g � x $� �f 2 g � x � �

f � g � x � %fg& � x � f 3 x 4

g 3 x 4 $� �f 5 g � x � f � g � x � $ and� �g 5 f � x � g � f � x �6�

Suppose

f�x � 2x � 1

and

g�x � x � 2 �

Then � �f�g � x � ,� �

f � g � x � ,� �f g � x � ,� �f � g � x � ,� �f 5 g � x � , and� �g 5 f � x � .

20.(10 pts) Suppose

f�x � x2 � 1

andg�x � x � 1 �

Then � �f�g � x � ,� �

f � g � x � ,� �f g � x � ,� �f � g � x � ,� �f 5 g � x � , and� �g 5 f � x � .

21.(10 pts) Suppose

f�x � x

x�1

and

g�x � x � 1x �

Then��f 5 g 7� �

g 5 f � � x � .

22.(10 pts) Suppose

f�x � x � 1x

and

g�x � 1

1 � x �Then�f 5 g � x � , and�g 5 f � x � . Remarkable!

23.(10 pts) The next few problems deal with thequestion of whether an equation defines one of thevariables as a function of the other. You can answerall of these questions by seeing if the equation can besolved for one variable in terms of the other.For example, the equation

y � x2 � 0can be rewritten as

y � x2and so any choice of x uniquely determines y. We canthink of y as being given by the function

y � f � x � x2 �On the other hand, for a particular value of y, e.g.,y � 1, there are two values of x, i.e., x �98 1, and so

3

the equation y � x2 � 0 does not define x as a functionof y. Similarly, by switching the roles of x and y wesee that the equation

y2 � x � 0defines x as a function of y, but not y as a function ofx.By contrast, the equation

3x�4y � 5

determines x as a function of y and also determines yas a function of x, while the equation

x2�y2 � 1

defines neither variable as a function of the other.You can verify all of the above statements by apply-ing the vertical and horizontal line tests.For the following equations, enter� y if the equation defines y as a function of x

but not vice versa,� x if the equation defines x as a function of ybut not vice versa,� b (for ”both”) if the equation defines x as afunction of y and also y as a function of x,and� n (for ”neither”) if the equation defines nei-ther variable as a function of the other.

These equations are the same as discussed above, soyou can check that you understand how to answer thisand the following questions:� y � x2 � 0 .� y2 � x � 0 .� 3x �

4y�5 � 0 .� x2 �

y2 � 1 .

24.(10 pts) As in the preceding problem, enter x,y, b, or n, as appropriate.� y � x3 � 0 .� y �

x � 0 .� x � 5 .

� y � 2 .

25.(10 pts) As in the preceding problem, enter x,y, b, or n, as appropriate.� �

x�1 2 � �

y � 4 2 � 49 .� x � y � 0 .� y � 3x � 6 .� y � 1x � y .

26.(10 pts) As in the preceding problem, enter x,y, b, or n, as appropriate.� yx2 � 1 .� x3 �

y2 � 0 .� y � x � 2 .� x5 �y3 � 0 .

27.(10 pts) What do you think?As in the preceding problem, enter x, y, b, or n, as

appropriate.y � 2x .

28.(10 pts)For the following functions, enter E if they are

even,O if they are odd, and N if they are neither evennor odd.� f �

x � x2 .� f �x � x3 .� f �x � x2 �

x3 .

29.(10 pts)For the following functions, enter E if they are

even,O if they are odd, and N if they are neither evennor odd.� f �

x � � x � .� f �x � 2 .� f �x � x3 � x

3 � x2 .� f �x � � x � x .

30.(10 pts) Enter below the definition of a functionthat is both even and odd:f

�x � .


4


Young-Seon Lee.


due 3/3/04 at 9:00 PM.

This set is focused on functions and their graphs.The graphs will show up in black and white on yourhard copy, but when you go to the problem onlineyou will see them in color. In some quations you areaksed to identify shifts or scalings. They are all inte-gers, so you will be able to tell what they are from thegiven graphs. The solutions will tell you the Maplecommands that generated the graph.

1.(10 pts)The Figure above shows the graph of f

�x � � x � .

You can enter � x � into WeBWorK by typing abs(x).Try it now:The Figure in this problem shows the graph off

�x � .

2.(10 pts)The Figure in this problem shows the graph off

�x � .


�x � .

1


�x � .

5.(10 pts)Match the colors of the graphs in this Figure with

the functions given below. Enter y for yellow, b forblue, r red, p for purple, and g for green, as appropri-ate. � f

�x � x2� f

�x � �

x � 1 2� f�x � �

x�1 2� f

�x � x2 � 1� f

�x � x2 �

1

6.(10 pts)Match the colors of the graphs in this Figure with

the functions given below. Enter y for yellow, b forblue, r red, p for purple, and g for green, as appropri-ate. � f

�x � x3� f

�x � �

x � 1 3� f�x � �

x�1 3� f

�x � x3 � 1� f

�x � x3 �

1

7.(10 pts)The Figure above shows the graph of the mystery

function y � f �x .

In the Figure below, match the colors of the graphsin this Figure with the functions given. Enter y foryellow, b for blue, r red,and g for green, as appropri-ate.

2

� y � f � x � 2� y � f � x 0� 2� y � f � x � 2 � y � f � x � 2

8.(10 pts)Let

f�x � x � 1 and g

�x � 1 � 2x �

In the Figure above, match the colors of the graphsin this Figure with the functions given. Enter y foryellow, b for (dark) blue, r for red, p for purple, c forcyan (light blue), and g for green, as appropriate.� y � f � x � y � g � x � y � �

f�g � x �� y � �

f � g � x �� y � �f g � x �

� y � %fg& �x

9.(10 pts)Let


�x � 2x � 1 �

In the Figure above, match the colors of the graphsin this Figure with the functions given. Enter y foryellow, b for (dark) blue, r for red, p for purple, c forcyan (light blue), and g for green, as appropriate.� y � f � x � y � g � x � y � �

f�g � x �� y � �

f � g � x �� y � �f g � x �� y � %fg& �x

10.(10 pts)3

Let


�x � 2x � 1 �

In the Figure above, match the colors of the graphsin this Figure with the functions given. Enter y foryellow, b for blue, r for red, and g for green, as ap-propriate.� y � f � x � y � g � x � y � �

f 5 g � x �� y � �g 5 f � x �

11.(10 pts)Let


�x � � x � 1 �

In the Figure above, match the colors of the graphsin this Figure with the functions given. Enter y foryellow, b for blue, r for red, and g for green, as ap-propriate.� y � f � x � y � g � x � y � �

f 5 g � x �� y � �g 5 f � x �

12.(10 pts)Let

f�x � 1

1 � x and g�x � x � 1x �

In the Figure above, match the colors of the graphsin this Figure with the functions given. Enter y foryellow, r for red, and g for green, as appropriate.� y � f � x � y � g � x � y � �

f 5 g � x �� y � �g 5 f � x �

13.(10 pts) The next few problems are exercises infunction composition.Let

f�x � 2x � 1 and g

�x � x2 � 1

Then� �f 5 f � x � .� �g 5 g � x � .� �f 5 g � x � .� �g 5 f � x � .

14.(10 pts)Let

f�x � x2 � 1 and g

�x � x2 � 1

Then� �f 5 f � x � .� �f 5 g � x � .� �g 5 f � x � .

15.(10 pts) Let

f�x � 3x � 4 and g

�x � 2x2 � x

4

Then� �f 5 f � x � .� �g 5 g � x � .� �f 5 g � x � .� �g 5 f � x � .

16.(10 pts) Let

f�x � 2x � 1 and g

�x � 3x � 1

Then� �f 5 f 5 f � x � .� �g 5 g 5 g � x � .� ��f 5 g 0� �

g 5 f � � x � .� ��f�g :5 � f � g � � x � .

17.(10 pts) Let

f�x � x2 � 2x � 3 and g

�x � 4x3 � 5x2 � 4x � 1

For brevity let’s write deg�p for the degree of a poly-

nomial p. So in the above example, deg�f � 2 and

deg�g � 3. You can answer the following questions

without actually computing the indicated functions.� deg � f � g .� deg � f � g .� deg � f g .� deg � f 5 g .� deg � g 5 f .

18.(10 pts) This is a more general version of theprevious problem. Suppose f is a polynomial of de-gree m, and g is a polynomial of degree n. For sim-plicity, let’s also suppose that

m � n �(Ponder what the complications would be if we didnot make that assumption.)� deg � f � g .� deg � f � g .� deg � f g .� deg � f 5 g .� deg � g 5 f .

19.(10 pts) The next few problems concern inversefunctions.Suppose

f�x � 3x � 4 �

Thenf � 1 � x � .

20.(10 pts) Suppose

f�x �

x � 4 �Thenf � 1 � x � .

21.(10 pts) Suppose

f�x � x � 1x � 2 �

Thenf � 1 � x � .

22.(10 pts) Suppose


�x � 2x � 5 �

Then � �f 5 g � x � .� �f 5 g � 1 � x � .� �f � 1 5 g � 1 (x) = .� �g � 1 5 f � 1 (x) = .

23.(10 pts) Suppose

f�x � x8 � 3 and g

�x � x3 �

Then � �f 5 g � x � .� �f 5 g � 1 � x � .� �f � 1 5 g � 1 � x = .� �g � 1 5 f � 1 � x = .

(You can enter a cube root by typing, for example,xˆ (1/3) . Note the parentheses around the exponent.)

24.(10 pts) This problem encapsulates the lessonfrom the preceding two problems.Suppose f and g are invertible functions, and the

indicated compositions can be carried out. Then�f 5 g 5 g � 1 5 f � 1 � x � .

5

25.(10 pts)The Figure above shows the graph of

f�x � x3 � 3x �

The answers below are all integers.The graph f has a relative maximum at x �

of f�x � .

It has a relative minimum at x � of f�x = .

The graph is decreasing in the interval/

, ) .


f�x � 3x2 � x3 �

1 �The answers below are all integers.The graph f has a relative maximum at x �

of f�x � .

It has a relative minimum at x � of f�x = .

The graph is increasing in the interval/

, ) .


f�x � x55 � x4

2� x33

�x2 �

The answers below are all integers.The graph of f shows relative maxima and

relative minima,for a total of relative extrema.The graph is increasing on the bounded interval

/, ) .Note: a bounded interval is one of finite length.

28.(10 pts)This problem illustrates that we can make state-

ments about functions without knowing how they aredefined. The Figure shows for functions that formtwo pairs of functions and their inverses. Enter theletters b for blue, g for green, r for red, and y for yel-low below, as appropriate.

6

� The inverse of the blue function is .� The inverse of the red function is .� The inverse of the green function is .� The inverse of the yellow function is .

29.(10 pts) The last two problems in this set arereasonably challenging word problems that lead to aquadratic equation. If you solve the next problem firstyou can solve this one by applying a formula. On theother hand, you may prefer first to use specific num-bers instead of variables, in which case you should dothis problem first.You’ll need the following information. Suppose

you release an object and let it fall. After t seconds itreaches a depth of

d � � gt22feet

where on earth g � 32 feet per second squared. Inother words, the speed of the falling object increasesby 32 feet per second every second.This equation holds only if we ignore air resis-

tance, and if we assume that the pull of gravity is

constant. The former assumption is grossly unreal-istic, the latter is reasonable close to the surface ofthe earth.Sound travels through the air at a constant speed c.Here is the problems You drop a rock into a deep

well. You can’t see the rock’s impact at the bottom,but you hear it after 7 seconds. The depth of the wellis feet.The time that passes after you drop the rock has

two components: the time it takes the rock to reachthe bottom of the well, and the time that it takes thesound of the impact to travel back to you. Assumethe speed of sound is 1100 feet per second.

30.(10 pts) Repeat the previous problem, but as-sume you are on a planet with a gravity constant g(instead of 32 feet per second squared) and a speedof sound c (instead of 1100 feet per second), and thatyou hear the impact of the rock after t seconds. Thenthe depth of the well is

feet, where your answer depends on c, g, and t. (Thusyou will have a formula that you can use to determinethe depth of a well on any planet and in any atmo-sphere.)


7


Young-Seon Lee.


due 3/10/04 at 9:00 PM.

The problem set covers the following topics:� Computing the inverse of a function.� Obtaining the equation of a parabola in stan-dard form via completing the square.� Possible differences between the graphs ofpolynomial and non-polynomial functions.� Synthetic Division� Finding zeros of polynomials by synthetic di-vision, after one root is known.� Complex Numbers� Solution of quadratic equations possibly hav-ing a conjugate complex pair of solutions.� Constructing quadratic factors when knowing2 zeros.� Solving polynomial equations by deflation(dividing by suitable linear or quadratic fac-tors).� Graphs of rational functions, particularlytheir asymptotes.

1.(10 pts) The first few problems ask you to com-pute the inverse of a function f . Remember that youdo this by solving the equation y � f �

x for x and thenswitching the roles of x and y.Suppose

f�x � 2x �

4 �Thenf � 1 �x � .

2.(10 pts) Suppose

f�x � � 5x � 7 �

Thenf � 1 �x � .

3.(10 pts) Suppose

f�x � ax �

b

where a '� 0. Thenf � 1 �x � .

4.(10 pts) Suppose

f�x � 1x

Thenf � 1 �x � .

Supposeg

�x � x

Theng � 1 �x � .

5.(10 pts) The next one is a little tricky.Suppose

f�x � � x � x

Thenf � 1 �x � if x * 0,

andf � 1 �x � if x ( 0.6.(10 pts) Suppose

f�x � x3

Thenf � 1 �x � .

7.(10 pts) Suppose

f�x � 2

x � 1Thenf � 1 �x � .

8.(10 pts) Suppose

f�x � 2x � 1

3x � 1 �Thenf � 1 �x � .

9.(10 pts) The next few problems ask you to findthe vertex of a parabola by completing the square.The vertex of the parabola defined by

f�x � x2 �

4x

is the point�, ).

10.(10 pts)The vertex of the parabola defined by

f�x � � 5x2 �

4x�1

is the point�, ).

1

11.(10 pts)The vertex of the parabola defined by

f�x � ax2 �

bx�c

is the point�, ).

12.(10 pts)The Figure above shows the graphs of five functions,distinguished by color. For every function, enter be-low the letter p if the graph may be that of a poly-nomial, and the letter n if it cannot be the graph of apolynomial. (Of course we assume that figure showsall the relevant aspects of the graph.)� : blue� : green� : purple (magenta)� : red� : yellow

13.(10 pts)Let

p�x � 2x3 �

3x2�4x

�5 �

As discussed in class, p�2 can be computed via syn-

thetic division by filling in for the letters a through gin the table

2 3 4 5b d f

a c e g

where� � a,� � b,

� � c,� � d,� � e,� � f ,� � g.Moreover, p

�2 � and p

�x can be written as

p�x � �

x � 2 q �x �r

whereq

�x � is a quadratic polynomial andr � is a real number.

14.(10 pts) Let

p�x � x3 � x2 �

2x � 5 �p

� � 3 can be computed via synthetic division by fill-ing in for the letters a through g in the table

1 � 1 2 � 5b d f

a c e g

where� � a,� � b,� � c,� � d,� � e,� � f ,� � g.Moreover, p

� � 3 � and p�x can be written as

p�x � �

x�3 q �x �r

whereq

�x � is a quadratic polynomial andr � is a real number.

15.(10 pts)Let

p�x � x3 � 3x2 � 2x �

1 �p

� � 2 can be computed via synthetic division by fill-ing in for the letters a through g in the table

1 � 3 � 2 1b d f

a c e g

where� � a,� � b,2

� � c,� � d,� � e,� � f ,� � g.Moreover, p

� � 2 � and p�x can be written as

p�x � �

x�2 q � x � r

whereq�x � is a quadratic polynomial andr � is a real number.

16.(10 pts)Let

p�x � 6x3 � 5x2 � 2x � 1 �

It is easy to check that p�1 � 0, i.e., 1 is a zero of

p. p has two more real zeros. The smaller is x �, and the larger is x � .

17.(10 pts)Let

p�x � 21x4 � 11x3 � 23x2 � 11x � 2 �

It is easy to check that p�1 � p � � 1 � 0, i.e., 1

and � 1 are zeros of p. p has two more real zeros.The smaller is x � , and the larger is x � .

18.(10 pts) Let

u � 2 � 3i and v � � 3 � i �Write the following complex numbers in the standardform of a complex number.� u � v � �

i.� u � v � �i.� uv � �i.� u

v � �i.� v

u � �i.� v2 � �i.� v3 � �i.� v4 � �i.

19.(10 pts) There are only two complex arithmeticproblems on this set, since you have seen complexnumbers before. This one is a little involved, how-ever. You may want to make judicious use of a calcu-lator or a symbol manipulation language like Maple.Of course there is a reason for the fancy real andimaginary parts, and your final expression for z5 will

tell you that reason. Remember that WeBWorK ex-pects your answers to be within one tenth of one per-cent of the answer that it has been given.Let

z � � 1 � 5

4

� 2�5� 5

4i �

Enter the following powers of z in the standard formof a complex number:� z � �

i.� z2 � �i.� z3 � �i.� z4 � �i.� z5 � �i.

20.(10 pts) The next three problems once againconcern quadratic equations. Enter them in the formu+/-v S where ”S” is the letter ”i” if the solution is aconjugate complex pair, and it is the digit ”1” if thesolutions are real.For example, if the solution is 3/2 +/- sqrt(5) i, say,you would enter ”3/2”, ”sqrt(5)”, and ”i”. If the solu-tion is 2 +/- 1 you would enter ”2”, ”1”, and ”1”. Ifthere is only one solution, say 3, you would enter it abit ponderously as ”3”, ”0”, ”1”.The solution of the quadratic equation

x2�2x

�2 � 0

is x � � �;� .The solution of the quadratic equation

x2�2x � 2 � 0


x2�2x

�1 � 0

is x � � �;� .

21.(10 pts)The solution of the quadratic equation

x2�5x

�10 � 0


x2�5x � 10 � 0

is x � � �;� .

22.(10 pts)3

The solution of the quadratic equation

3x2�5x

�10 � 0


3x2�5x � 10 � 0

is x � � �;� .

23.(10 pts)Suppose the solutions of a quadratic equation p

�x �

0 are 3 and � 5. Then that quadratic equation can bewritten as p

�x � x2 � x � � 0 �

24.(10 pts)Suppose the solutions of a quadratic equation p

�x �

0 are x � 2 8 i. Then that quadratic equation can bewritten as p

�x � x2 � x

� � 0 �25.(10 pts)

The equation

x4 � 9x3 � 31x2 � 49x � 30 � 0has the conjugate complex pair of solutions x � 2 8 i.The equation also has two real solutions. The smalleris , and the larger is .

26.(10 pts)One solution of the the equation

p�x � x4 � 9x3 � 30x2 � 33x � 13 � 0

is x � 3 � 2i. The equation also has two real solu-tions. The smaller is , and the larger is.

27.(10 pts)Two zeros of the polynomial

p�x � x4 � 4x3 � 24x2 � 20x � 7

are 1 and 7. The other two zeros are real, but irra-tional.The smaller is , and the larger is .

28.(10 pts)Match the graphs shown above with the functionslisted below. Enter ”r” for red, ”g” for green, and”y” for yellow.� : f

�x � x

x � 1 �� : f�x � x2

5 3 x � 1 4 �� : f�x � x

x2 � 1 �

29.(10 pts)Match the graphs shown above with the functionslisted below. Enter ”r” for red, ”g” for green, ”p”for purple, ”b” for blue, and ”y” for yellow.� :

�x � 1x �� : f

�x � 1

x2 � 1 �� : f�x � x

x2 � 1 �� : f�x � x2

x2 � 1 �� : f�x � x3

x2 � 1 �4

30.(10 pts) This is a warmup for the next problem.For the following functions, use ”x” to indicate that

the x-axis is an asymptote, ”h” to indicate a horizon-tal asymptote other than the x axis, ”v” to indicate avertical asymptote, ”s” to indicate a slanted asymp-tote, and ”n” the lack of an asymptote. If the graphof a function has several types of asymptotes indicatethem all in alphabetical order.For example, the function

f�x � x3

x2 � 1has a slanted asymptote since the degree of the nu-merator is one more than the degree of the denom-inator, and it also has two vertical asymptotes (atx �98 1). So you would enter ”sv” (without the dou-ble quotation marks. The graph of

f�x � 1x

has vertical asymptote (the y-axis) and the x-axis asan asymptote, so you would enter ”vx”. On the otherhand, the graph of

f�x � x3

x2�1

has only a slanted asymptote, so you would enter just”s”. � f

�x � 1x �� f

�x � x3

x2 � 1 �� f�x � x3

x2 � 1 �To make this clear the following picture shows thegraphs involved:

It may not be clear from the picture that the green

graph (of f�x � x3

x2 � 1 has a slanted asymptote, tomake this clearer the Figure also contains the (red)graph of its asymptote defined by the equation y � x.The yellow graph is the graph of f

�x � x3

x2 � 1 , theblue graph is the graph of f

�x � 1x .

31.(10 pts) For the following functions, use ”x” toindicate that the x-axis is an asymptote, ”h” to in-dicate a horizontal asymptote other than the x axis,”v” to indicate a vertical asymptote, ”s” to indicatea slanted asymptote, and ”n” the lack of an asymp-tote. If the graph of a function has several types ofasymptotes indicate them all in alphabetical order.� f

�x � x

x � 1 �� f�x � x

x2 � 1 �� f�x � x � 1

x2 � 1 �� f�x � x4

x2 � 1 �32.(10 pts) For the following functions, use ”x” to

indicate that the x-axis is an asymptote, ”h” to in-dicate a horizontal asymptote other than the x axis,”v” to indicate a vertical asymptote, ”s” to indicatea slanted asymptote, and ”n” the lack of an asymp-tote. If the graph of a function has several types ofasymptotes indicate them all in alphabetical order.� f

�x � x3 x � 1 4 2 �� f�x � x23 x � 1 4 2 �� f�x � x33 x � 1 4 2 �

5

� f�x � x2

x2 � 1 �33.(10 pts) Problem 72 on page 177 of the text-

book is an audacious question to ask in a CollegeAlgebra textbook. As stated, the answers, the widthand height of that printed page, are irrational num-bers, and unless I missed something there is no wayto solve this problem using the tools we have dis-cussed in class, or that are covered in the textbook.(However, the question is straightforward as a Calcu-lus problem.)This WeBWorK problem is an adaptation of the text-book problem which is such that the answers are nice

numbers. You can find them, for example, by plottingand closely examining the graph of a suitable rationalfunction.A rectangular page is designed to contain 72

square inches of print. The margins at the top andbottom of the page are each 4 inches deep. The mar-gins on each side are 2 inches wide. The dimensionsof page are such that the least possible amount of pa-per is used.Thus the width of the page is inches, its

height is inches, its total area issquare inches. There’s a whole lot of white space onthat page, its minimum area not withstanding!


6


Young-Seon Lee.


due 3/24/04 at 9:00 PM.

Note that this set is open one week longer thanusual because of the Spring Break.The focus of this home work is on exponential

functions. The last few problems also involve log-arithms. WebWorK understands the letter e for thebase of the natural logarithm. You can write e**x orexp(x) for the natural exponential fucntion

f�x � ex

and ln(x) or log(x) for the natural logarithm. There isno notation in WeBWorK for the common logarithmwith base 10.

1.(10 pts)The Figure above shows the graphs of four exponen-tial functions, listed below. Match the functions withthe colors, using b for blue, r for red, g for green, andy for yellow.� : f

�x � 2x �� : f

�x � 3x �� : f

�x � �

12 � x �� : f

�x � �

13

� x �

2.(10 pts)The lessons we learned about shifting graphs apply toexponential functions just as they apply to any otherfunctions.

The Figure above shows the graphs of five exponen-tial functions, listed below. Match the functions withthe colors, using b for blue, r for red, g for green, pfor purple, and y for yellow.� : f

�x � 2x �� : f

�x � 2x �

1 �� : f�x � 2x � 1 �� : f

�x � 2x � 1 �� : f

�x � 2x � 1 �

3.(10 pts)The Figure above shows the graphs of four exponen-tial functions, listed below. Match the functions with

1

the colors, using b for blue, r for red, g for green, andy for yellow.� : f

�x � 2x �� : f

�x � � 2x �� : f

�x � 2 � x �� : f

�x � � 2 � x �

4.(10 pts)The Figure above shows the graphs of four exponen-tial functions, listed below. Match the functions withthe colors, using b for blue, r for red, g for green, andy for yellow.� : f

�x � 2x < 4 �� : f

�x � 2x < 2 �� : f

�x � 2x �� : f

�x � 22x �

5.(10 pts)

The Figure above shows the graphs of four exponen-tial functions, listed below. Match the functions withthe colors, using b for blue, r for red, g for green, pfor purple, and y for yellow.� : f

�x � 1 � 2 � 4x �� : f

�x � 1 � 2 � 2x �� : f

�x � 1 � 2 � x �� : f

�x � 1 � 2 � x < 2 �� : f

�x � 1 � 2 � x < 4 �

6.(10 pts) The next few problems are simple wordproblems that illustrate the key aspect of exponentialgrowth.In any time interval of a given length, the func-

tion value is multiplied by the same factor.

The first problem reviews a discussion we had inclass.You invest money a at certain annual interest ratep. Interest is paid once a year. We have discussedthis problem before, and will now examine it in moredepth. It is customary to used instead of the percent-age the rate r define by

r � p

100�� Every year you multiply your current capital

by a factor (where you answer of coursedepends on r).� In 2 years you multiply your initial capitalwith .� In 3 years you multiply your initial capitalwith .� In t years you multiply your initial capitalwith (where your answer depends on rand t.� Supposing that your initial deposit is I, after tyears, your balance isf

�t = (where your answer dependson r, t, and I. (Remember that variables arecase sensitive, you do need to use the properupper or lower case for each variable.)

7.(10 pts) Suppose in the year 2004 you invest$1000.- at an interest rate p which is such that yourcapital doubles every 6 years. That interest rate ispercent.� During the period 2004-2010 your money

will grow by a factor .

2

� During the period 2005-2011 your moneywill grow by a factor .� During the period 2007-2019 your moneywill grow by a factor .� During the period 2017-2035 your moneywill grow by a factor .� During the period 2010-2025 your moneywill grow by a factor .� When you retire in 50 years your investmentwill have grown to $ .

8.(10 pts) Suppose you have a population (of somespecies, it doesn’t matter which) that, over a long pe-riod, grows at a rate that causes it to double its sizeevery 30 years. That rate is percent. Sup-pose that at present, in the year 2004, that populationis 1,000,000.� The population will be in the year 2034.� The population was in the year 1974.� It was in the year 1944.� In the year t the population is f � t �

. (Your answer will depend on t. Make sureyou have the term t � 2004 somewhere in anexponent.

9.(10 pts) Carbon-10 is a radioactive isotope with ahalf life of 19.255 seconds. Suppose you start with 1gram of carbon-10. Then the amount f

�t of carbon-

10 left after t seconds isf�t � grams.

10.(10 pts) Suppose f is a function of the form

f�x � abx

for some constants a and b. The doubling time for fis 1, i.e.,

f�x�1 � 2 f � x

for all x.Then f

�x � .

11.(10 pts) Suppose f is a function of the form

f�x � abx

for some constants a and b. The doubling time for fis 3, i.e.,

f�x�3 � 2 f � x

for all x.Then f

�x � .

12.(10 pts) Suppose f�x gives the amount of a ra-

dioactive isotope with a half life of 5 years, after xyears. f

�x is a function of the form

f�x � abx

for some constants a and b.Then f

�x � .

13.(10 pts)Suppose f

�t is a function of the form

f�t � abt

for some constants a and b. Suppose t measures timein years, and

�t describes the decay of a radioactive

isotope with a half life of T years.Then f

�t � .

14.(10 pts) Suppose

A�m �

1� 1m � m �

As discussed in class, A�m approaches

e � 2 � 71828183 ��as m increases without bounds. This exercise illus-trates that fact. You’ll need to enter numerical valuesof e � A � m that are correct to 4 digits (after the ini-tial zeros). You can compute them with your calcu-lator, and you may find it convenient to use scientificformat. Note how the difference of e and A

�m ap-

proaches zero, and A�m approaches e as m grows.

If you are curious you can see the first 10,000 digits of ehere. � e � A � 1 � .� e � A � 2 � .� e � A � 10 � .� e � A � 100 � .� e � A � 1 $ 000 � .� e � A � 10 $ 000 � .� e � A � 100 $ 000 � .� e � A � 1 $ 000 $ 000 � .

3

15.(10 pts)The Figure above shows the graphs of five functions,listed below. Match the functions with the colors, us-ing b for blue, r for red, g for green, p for purple, andy for yellow.� : f

�x � x �� : f

�x � ex �� : f�x � 5x �� : f

�x � ln � x 6�� : f

�x � log5 � x 6�

16.(10 pts)The lessons we learned about shifting graphs apply tologarithmic functions just as they apply to any otherfunctions.

The Figure above shows the graphs of five logarith-mic functions, listed below. Match the functions withthe colors, using b for blue, r for red, g for green, pfor purple, and y for yellow.� : f

�x � ln � x 6�� : f

�x � ln � x � 1 =�� : f�x � ln � x � 1 =�� : f�x � ln � x 7� 1 �� : f�x � ln � x � 1 �

17.(10 pts) Before calculators were widely avail-able people used logarithm tables to simplify multi-plication and division, using the fact that the loga-rithm of a product equals the sum of the logarithmsand the logarithm of the quotient equals the differ-ence of the logarithms.These tables were constructed by computing labo-

riously and carefully the logarithms of a few selectednumbers and then combining the logarithms using therules just mentioned.This exercise suggest how the process may haveworked.Let

L�x � loga � x

where we don’t know the base a. However, we doknow that

L�2 � 0 � 39051 and L � 3 � 0 � 61895 �

Use this information to compute� L � 6 � .� L � 9 � .� L � 12 � .� L � 23� � .� L � 310 � � .� L � 1 � .

18.(10 pts) Currently, the largest known primenumber is

P � 220 > 996 > 011 � 1 �It has decimal digits.


4


Young-Seon Lee.


due 3/31/04 at 9:00 PM.

This problem set is focused on logarithms and ex-ponentials. It is open a week longer than usual be-cause of the Spring Break, but it has the usual al-lotment of problems. Questions 18, 21, and 22 re-quire numerical answers. In those cases ww will re-ject arithmetic expressions.The next home work set, hw 10, will open on March26.1.(10 pts) You can compute the following loga-

rithms with your basic knowledge of powers.For example, since

23 � 8we know that

log2 8 � 3 �� log5 25 � .� log6 36 � .� log3 27 � .� log10 10 $ 000 � .� log10 0 � 001 � .� logπ1 � .

2.(10 pts) More logarithms:� log2 4 � .� log9 81 � .� log2 1024 � .� log3 81 � .

3.(10 pts)Let

L�x � loga �

x where we don’t know the base a. However, we doknow that

L�2 � 0 � 37659 and L �

3 � 0 � 59689 �Use this information to compute� L �

4 � .

� L �a2 � .� L �a3 � .� L �65 � .

4.(10 pts)Let

L�x � loga �

x where we don’t know the base a. However, we doknow that

L�2 � 0 � 35886 �

Then a � .

5.(10 pts) Complete the following list. Enter thelogarithms as decimal expressions with at least fourdigits of accuracy.� log10 10 � .� ln10 � .� log2 10 � .� log5 10 � .� log1 < 2 10 � .

6.(10 pts) Complete the following list. Enter thelogarithms as decimal expressions with at least fourdigits of accuracy.� log10 e � .� lne � .� log2 e � .� log5 e � .� log1 < 2 e � .

7.(10 pts) You are investing money at 9.5 percentannual interest, compounded continuously. It willtake you years to double your investment.Note: This is a bit unrealistic. I don’t know of anyplace where an interest rate is compounded contin-uously. (But of course it’s just an exercise.) Alsoremember that ww expects your answer to be withinone tenth of one percent, so you will need to enter anumber of years with at least three digits beyond thedecimal point.

8.(10 pts) Let t denote timemeasured in years, witht � 0 corresponding to the year 2004. Let P be theworld population at the beginning of this year. If theworld population p

�t were to obey the formulap

�t � Pe0 ? 012t

1

then the annual growth rate would be percent,and it would take years for the population todouble.Again, remember to enter your answer with a total ofabout four digits.

9.(10 pts) Suppose

p�t � e � 0 ? 00001t

is the amount of a certain radioactive isotope after tyears. The half-life of your isotope is years.

10.(10 pts) (This is a modification of problem 82on page 224 of our textbook.)The relationship between a number of decibels β

and the intensity of a sound I (in watts per squaremeter) is

β � 10log10 I

10 � 12 � �The number of decibels for a sound with an intensityof 10 � 6 watts per square meter is .A sound with ten times that intensity hasdecibels.In general, a sound with ten times the intensity of asound with n decibels has decibels.

11.(10 pts) Remember from the last home work setthat the largest currently known prime number is

P � 220 > 996 > 011 � 1 �Previously you computed the number of decimal dig-its of P. Suppose you wish to express P in the binarysystem used by computers, where the base is 2 in-stead of 10, and the only possible digits are 0 and 1.The number of binary digits is .

12.(10 pts) Again, let

P � 220 > 996 > 011 � 1 �Suppose you have traveled to a faraway place

where the base of the number system is 37. (The in-habitants of that country are fond of prime numbers.)So it has 37 symbols in place of our 10 digits, butotherwise it works just like the decimal system. Ex-pressed in that system, P hasdigits.

13.(10 pts) The following few problems ask you tosolve exponential equations of increasing complex-ity. Your answer needs to be a decimal number withat least four digits. The first problems is easy:

The solution of the equation

5x � 100is x � .


3 � 5x � 25 � 100is x � .


3 � 43t � 1 � 5is t � .


22x � 5 � 2x � 6 � 0has two solutions. The smaller is and the largeris .


lnx � ln � x � 1 � 2has the solution x � .


ln�x�1 0� ln � x � 2 � lnx

has the solution x � .

19.(10 pts) For the following statements write T ifthey are true, and F if they are false. To avoid cum-bersome sentences it is tacitly understood that loga-rithms have the same base, and are applied to the ap-propriate terms. We also assume that all logarithmsare evaluated at positive numbers.For example, a more precise version of the first sen-tence would be: The logarithm with a certain base ofthe product of two positive numbers equals the sumof the logarithms with the same base of those twonumbers.� The logarithm of a product equals the

sum of the logarithms.� The logarithm of a quotient equals thedifference of the logarithms.� The logarithm of a power equals theproduct of the exponent and the logarithm ofthe base (of the power).

20.(10 pts) For the following statements, enter T ifthey are true and F if they are false.

2

� The logarithmwith a certain base is the inverse function ofthe exponential with the same base.� The quotient of two logarithms equalsthe logarithm of the difference.� The logarithm of a product equals theproduct of the logarithms.

21.(10 pts) For the following statements, enter Tif they are true and F if they are false. Logarithmsand exponentials can be defined for complex num-bers, and then can assume complex function values,but in the questions below concerning domain andrange, let’s consider only real numbers.� The natural exponential is the reciprocal

of the natural logarithm.� The natural exponential is the inverse ofthe natural logarithm.� The natural exponential is the negativeof the natural logarithm.� The domain of the natural logarithm isthe set of all positive numbers.� The domain of the natural logarithm isthe set of all real numbers.� The domain of the natural exponential isthe set of all positive numbers.� The domain of the natural exponential isthe set of all real numbers.

22.(10 pts) The next few questions ask you to com-pute the inverse of an exponential or a logarithmicfunction. Remember how we do this: we solve theequation y � f � x for x and then switch x and y.The purpose of this first problem is just to make

sure you understand theWeBWorK notation. You canenter the exponential with the natural base as exp(x)or eˆx and the natural logarithm as ln(x) or log(x).WeBWorK does not have a base 10 logarithm builtinto it, if you need a logarithm with a base other thane, let’s say a, then enter something like log(x)/log(a).The inverse of the function

f�x � ex

isf � 1 � x �The inverse of the function

f�x � lnx

isf � 1 � x �23.(10 pts) The inverse of the function

f�x � 10x


f�x � log10 x


f�x � ex � 1


f�x � ln � x � 1


f�x � �

lnx 7� 1isf � 1 � x �26.(10 pts) The inverse of the function

f�x � ln � x � 1 7� ln � 2x � 1

isf � 1 � x �27.(10 pts) The remaining problems are true/false

questions concerning logarithmic and exponentialidentities. You don’t need to memorize these, theyall flow from two facts:� Logarithms and Exponentials are inverses

of each other. (Of course they need to havethe same base.)� Exponential functions are just powers andlogarithms are just exponents.

For the following proposed identities enter T ifthey are true, and F if they are false. We assume thatthe expressions involved make sense. For exampleany base is positive and not equal to 1, and logarithmsare taken only of positive numbers.� loga

�uv � loga u � loga v �

3

� loga�u�v � �

logau � loga v 6�� loga�uv� � loga u � loga v �� loga

�u � v � loga u

loga v�� loga

�uv � �

logau � loga v 6�� loga�uv � u � loga v 6�� loga�uv � v � loga u 6�

28.(10 pts) For the following proposed identitiesenter T if they are true, and F if they are false.� eln 3 x � 1 4 � x � 1 �� e 3 lnx 4@� 1 � x

e��

ln��x � 1 � x � 2 � � ln � x � 1 � ln � x � 2 6�� lnx2 � 2lnx �

29.(10 pts) For the following proposed identitiesenter T if they are true, and F if they are false.� ln10x � x ln10

� log5�x � lnx

ln5�� ln10x � x

log10 e�� lnx2 � �

lnx 2 �� lnxln5 � log10 x

log10 5 � log5 � x 6�30.(10 pts) For the following proposed identities

enter T if they are true, and F if they are false.� lnex � x� ln% �ex 2 & � x2 �� ln

% �ex 2 & � 2x �� ln

1

ex � � � x �� ln�eu

�ev � u � v �� ln

�eu

�ev � uv �� ln

�eu

�ev � eu � v �


4

U of U Math 1050-1 Spring 2004

Young-Seon Lee.


due 4/7/04 at 9:00 PM.

This set illustrates applications of logarithms andexponential, particularly logistic growth, and loga-rithmic scales, and it introduces the subject of linearsystems of equations. There are not as many ques-tions as usual, but some of the problems may takeyou longer than usual, so as always you should getgoing on this set as soon as it opens.

1.(10 pts)As discussed in our textbook (p. 241) the logistic

curve is the graph of the function

f�x � a

1�be � rx

where a, b, and r are suitable parameters.This function may describe, for example, the ini-

tial rapid growth of a population, followed by a slow-down of the growth as resources become sparse.Since e � rx tends to zero as x tends to infinity f �

x approaches a as x tends to infinity, and at x � 0 theinitial population equals

f�0 � a

1�b

�The rate r is the usual growth rate that would prevailindefinitely in the presence of unlimited resources.Suppose a � 100, b � 9, and r � 0 � 01. Then the

initial population is and as time goes on thepopulation approaches but never quite reaches .

2.(10 pts) In this problem, all answers need to bedecimal expressions.You are growing bacteria. From your initial exper-

iments you know that these particular bacteria doublethe size of their population every hour in the presenceof unlimited resources. You start with a population of1 unit (like a million bacteria, say) and you let themgrow in an environment that allows a stable popula-tion of 100 units. You model the size of your popu-lation with a logistic curve. Accordingly, the size of

your population in your chosen units after x hours isgiven by

f�x � a

1�be � rx

wherea � ,b � ,r � , and the population after 8 hours isf

�8 � units.

3.(10 pts) This problem is similar to the precedingone except that you need to get all answers correct be-fore receiving credit. All answers need to be decimalexpressions.You are growing bacteria. From your initial exper-

iments you know that these particular bacteria dou-ble the size of their population every three hours inthe presence of unlimited resources. You start with apopulation of 1 unit and you let them grow in an envi-ronment that allows a stable population of 500 units.You model the size of your population with a logis-tic curve. Accordingly, the size of your population inyour chosen units after x hours is given by

f�x � a

1�be � rx

wherea � ,b � ,r � , and the population after 24 hours isf

�24 � units. After two days the population isf

�48 � units.

4.(10 pts) The brightness B of a star, called itsapparent magnitude m, is expressed on a logarith-mic scale:

B � a � mB0where

a � 5100 A 2 � 511886432

and B0 is a reference brightness. Notice the minussign in the exponent, the smaller m the brighter is theobject! For example, the star Vega has an apparentmagnitude close to zero. The dimmest stars visibleto the naked eye have an apparent magnitude close tosix. The brightest star in the sky, Sirius, has an appar-ent magnitude of � 1 � 4 and Venus may be as bright as� 4 � 4 (depending on its distance and phase).The full moon has an apparent magnitude of � 13.

It is times as bright as Sirius.1

The sun is 300,000 times as bright as the full moon,thus its apparent magnitude is .

5.(10 pts) As we discussed in class, on earth theratio of carbon 14 to carbon 12 t years after the deathof an organism is given by

R � 1

1012e � t < 8223 �

On your vacation you find a beautiful ancient wo-ven basket where the ratio of carbon 14 to carbon 12is 0 � 72 � 10 � 12. It is approximately years old.

6.(10 pts)On your next vacation in Siberia you come across

a recently excavated frozen mammoth where the ra-tio of carbon 14 to carbon 12 is 4 � 78 � 10 � 14. Thatmammoth died approximately years ago.

7.(10 pts)On the Richter Scale , the magnitude R of an

earthquake of intensity I is

R � log10 II0where I0 is a reference intensity.During the semester break, at 7:23am on March

19, 2003, an earthquake measuring 3.0 on the Richterscale occurred near the town of Nephi. An earthquakeof that magnitude is often felt, but rarely causes dam-age.By comparison, the earthquake that struck San Fran-cisco in 1906 measured 8.25 on the Richter scale. Itwas times as intensive as the recent Nephiearthquake.

8.(10 pts)A group of eight people goes to the movies. Tick-

ets are $6.- each for adults and $3 each for kids. To-gether they pay $33.- for the tickets. There areadults in that groupand kids.9.(10 pts)You buy a total of 4 pounds of two kinds of candy.

One is for your upcoming party and costs $2.- perpound. However, you also treat yourself to a fancykind that costs $10 per pound, to be enjoyed in soli-tude. You spend a total of $12.- and buy poundsof the gourmet candy.

10.(10 pts)The solution of the linear system

x�y � 7

3x � y � 1

isx � and y � .


x�y � 1

3x � y � 2

isx � and y � .


x�y

�z � 2

x�2y � z � � 1

x � 3y �2z � 7

isx � , y � , and z � .

Note: Keep a careful record of your calculationssince the next problem differs from this one only inthe right hand side, and you don’t want to redo yourcalculations.

13.(10 pts) This problem differs from the preced-ing one only in the right hand side. You have alreadydone many of the necessary calculations and don’tneed to redo them if you have good notes for yoursolution of the previous problem.The solution of the linear system

x�y

�z � 1

x�2y � z � 2

x � 3y �2z � 3

isx � , y � , and z � .

14.(10 pts) The next three problems go together.They illustrate an application of linear systems, aswell as a technique to approximate a complicatedfunction by a simpler one, in this case a polynomial.Suppose we are concerned with the function

f�x � 2x �

We can evaluate f at natural numbers x. Indeed,� f �1 � ,� f �2 � ,

2

� f � 3 � ,� f � 4 � ,

We are going to approximate f by three polynomials,of degrees 1, 2, and 3. Let’s call them p1, p2, and p3,respectively.p1 will be determined by the requirement that

p1�1 � f � 1 and p1

�4 � f � 4 =�

p2 will be determined by the requirement that

p2�1 � f � 1 $ p2 � 3 � f � 3 and p2

�4 � f � 4 6�

p3 will be determined by the requirement that

p3�1 � f � 1 $ p3 � 2 � f � 2 $ p3 � 3 � f � 3 and p3

�4 � f � 4 6�

Thus p1 is simply the linear function whose graphpasses through the points

�1 $ 2 and �

4 $ 16 .In fact,p1

�x � .p2 is the quadratic function whose graph intersects

the graph of f in the points�1 $ 2 , � 3 $ 8 , and �

4 $ 16 .p3 is the cubic function whose graph intersects thegraph of f in the points

�1 $ 2 , �

2 $ 4 , �3 $ 8 , and�

4 $ 16 .The idea is illustrated in this Figure:

The graph of f�x � 2x is shown in red. The graph

of p1 is green, that of p2 is yellow, and that of p3 isblue. The blue graph is mostly covered up by the redgraph, which indicates how closely the cubic polyno-mial p3 approximates the exponential f , particularlyin the interval

/1 $ 4 ) .

15.(10 pts) Next, computep2

�x � .

You may have a better idea, but one way to proceedis this. Write

p2�x � ax2 � bx � c �

You want to know a, b, and c. Writing the definingproperties of p2 we get three linear equations:

p2�1 � a

�b�c � 2 � f

�1

p2�3 � 9a

�3b

�c � 8 � f

�3

p2�4 � 16a

�4b

�c � 16 � f

�4

Just solve these equations for a, b, and c, and enterp2 above in standard form.

16.(10 pts) Now, for the clincher: Computep3

�x � .


3


Young-Seon Lee.


due 4/14/04 at 9:00 PM.

This set is focused on the solution of linear sys-tems. There are only 12 problems, but a couple ofthem are likely to take a significant amount of time.Recall our discussion in class, keep track of your cal-culations, guard against errors by keeping row sums,and proceed deliberately and correctly. When thereare several problems with the same coefficient ma-trix, organize your work so you have to process thecoefficient matrix only once.

1.(10 pts)Solve the following system:

x�y

�z � � 4

2x � y�3z � 13

3x�y � 4z � � 21

Keep track of your calculations, because in thenext problem you will have the same system, exceptthat the right hand side is different.The solution isx � , y � , z � .


x�y

�z � 10

2x � y�3z � 4

3x�y � 4z � 4

isx � , y � , z � .

3.(10 pts)The principle that you process the coefficient ma-

trix only once is so important that we need to practicesome more. This and the next two problems all havethe same coefficient matrix. You want to process itjust once, unless you enjoy retracing your steps.The solution of the linear system

r�

s�

t�

u � 4r

�2s

�3t

�4u � � 7

r � s�2t � u � � 7

r�2s � 3t

�2u � 27

isr � , s � , t � , u � .

4.(10 pts) The solution of the linear system

r�

s�

t�

u � � 7r

�2s

�3t

�4u � � 24

r � s�2t � u � � 8

r�2s � 3t

�2u � 4

isr � , s � , t � , u � .


r�

s�

t�

u � 5r

�2s

�3t

�4u � 9

r � s�2t � u � � 7

r�2s � 3t

�2u � 25

isr � , s � , t � , u � .

6.(10 pts) Let p be the quartic (degree 4) polyno-mial that satisfies

p�i � 2i $ i � 0 $ 1 $ 2 $ 3 $ 4 �

Thenp

�x � .

7.(10 pts) For the following linear systems enterthe letter U if the system has a unique solution, theletter N if it has no solution, and the letter I if it hasinfinitely many solutions.� .

3x�4y � 1

3x � 4y � 1� .

3x�4y � 1

6x�8y � 1� .

3x�4y � 1

6x�8y � 2

1


3x�4y

�5z � 1

3x � 4y�5z � 1

x�

y�

z � 1� .

3x�4y

�5z � 1

x�2y

�3z � 1

4x�6y

�8z � 2� .

3x�4y

�5z � 1

x�2y

�3z � 1

4x�6y

�8z � 4


x�2y

�3z � 1

4x�5y

�6z � 1

7x�8y

�9z � 1� .

x�y

�2z � 1

2x�y

�3z � 1

3x�y

�4z � 2� .

x�y

�2z � 1

2x�y

�3z � 1

3x�y

�5z � 2


x�

y�

z � 12x

�3y

�4z � 1

3x�5y

�7z � 1

� .

x�

y�

z � 12x

�3y

�4z � 1

3x�5y

�8z � 1� .

x�

y�

z � 12x

�3y

�4z � 1

3x�4y

�5z � 2

11.(10 pts) The last two problems on this set re-quire you to solve two systems of five equations infive unknowns. The coefficient matrices are the samein both cases. The most important part of these twoproblems is to organize your calculation so that youget everything right, and you don’t repeat work un-necessarily.You have boxes of five types. They are character-

ized by their length L, their widthW , their height H,their age A, and their price P, as given in the follow-ing table:

type L W H A P

I 1 1 1 1 1II 1 1 2 3 2III 1 2 3 2 2IV 1 2 4 1 2V 1 2 5 1 3

So for example, boxes of type IV measure 1 by 2 by4 feet, are a year old, and cost$2.- each.Suppose the sum of the lengths of your boxes is 19,

the sum of their widths 28, the sum of their heights54, the sum of their ages 38, and the total price ofthose boxes 39.You have� boxes of type I,� boxes of type II,� boxes of type III,� boxes of type IV,� boxes of type V.

12.(10 pts)This problem is the same as the preceding one, ex-

cept that the sum of the lengths of your boxes is 26,the sum of their widths 44, the sum of their heights90, the sum of their ages 38, and the total price ofthose boxes 56.You have

2

� boxes of type I,� boxes of type II,� boxes of type III,

� boxes of type IV,� boxes of type V.


3


Young-Seon Lee.


due 4/21/04 at 9:00 PM.

This home work let’s you practice the computationof determinants, matrix products, and of matrix in-verses. It is the last regular home work set of thissemester. The final home work set 13 will serve asa review of the entire seesmter and a preparation forthe final exam.

1.(10 pts)Let

A �CB � 8 42 1 D and B �CB 3 02 5 D �Then

AB �CB c11 c12c21 c22 Dwhere

c11 = ,c12 = ,c21 = ,c22 = ,

2.(10 pts) Let

A �EB 5 06 � 5 D and B �EB 6 � 5� 5 0 D �

Then

AB �CB c11 c12c21 c22 Dwhere

c11 = ,c12 = ,c21 = ,c22 = ,

3.(10 pts) Let

A � B 7 � 55 8 D and B � B 2 62 � 6 D �

Then

AB � B c11 c12c21 c22 D

wherec11 = ,c12 = ,c21 = ,c22 = ,

4.(10 pts) Let

A � B � 3 � 64 1 D and B � B 3 1� 5 9 D �Then

AB � BA �CB c11 c12c21 c22 Dwhere

c11 = ,c12 = ,c21 = ,c22 = ,

5.(10 pts) Let

A � B 3 66 � 7 D and B � B 0 56 4 D �

Then

AB � BA �CB c11 c12c21 c22 Dwhere

c11 = ,c12 = ,c21 = ,c22 = ,

6.(10 pts) The inverse of the matrix

A �CB 9 � 4� 2 � 9 Dis

A � 1 �CB b11 b12b21 b22 D $where

b11 = ,b12 = ,b21 = ,b22 = ,

7.(10 pts) The inverse of the matrix

A �CB 4 00 � 5 D

is1

A � 1 �CB b11 b12b21 b22 D $where

b11 = ,b12 = ,b21 = ,b22 = ,

8.(10 pts) The determinant of the matrix

A � F G 5 6 88 � 4 � 56 � 7 � 5 HI

is, and its inverse is

A � 1 � FG b11 b12 b13b21 b22 b23b31 b32 b33 HI $

whereb11 = ,b12 = ,b13 = ,b21 = ,b22 = ,b23 = ,b31 = ,b32 = ,b33 = .

9.(10 pts) Let

A � B � 8 3 83 � 6 9 D

and

B � FG � 4 � 55 � 3� 2 3 HI

Then

C � AB �CB c11 c12c21 c22 D $and

D � BA � FG d11 d12 d13d21 d22 d23d31 d32 d33 HI $

where

c11 = ,c12 = ,c21 = ,c22 = ,andd11 = ,d12 = ,d13 = ,d21 = ,d22 = ,d23 = ,d31 = ,d32 = ,d33 = .The determinant of D is .Talk to your friends about this last item. They havedifferent matrices. Maybe you’ll discover something.If you do drop me a note!

10.(10 pts) The determinant of the matrix

A � FJ JG � 1 7 0 � 62 7 0 00 � 9 0 03 � 2 � 3 � 8 HLKKI

is .11.(10 pts)The determinant of the matrix

A � FJJJJG0 3 � 2 0 0� 7 6 � 5 0 00 � 6 0 0 08 9 � 4 � 1 41 8 8 0 3

HLKKKKIis .12.(10 pts)And now for the grand finale: The determinant of

the matrix

A � FJJJJJJJJJJJG1 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 91 2 3 4 5 6 7 8 9

HLKKKKKKKKKKKI2

is .


3


Young-Seon Lee.


due 4/28/04 at 9:00 PM.

The purpose of this last assignment is to help youreview the entire semester and prepare for the finalexam. The problems on this homework only indicatethe general sort of subject area. There are no hintsor solutions. If you have difficulties with a particularproblem you should go back over your notes and thetextbook, and study the general area. For example, ifyou can’t solve the logarithmic equation in problem12, it is only of very limited usefulness to have some-one show you how to solve that particular problem.Instead study sections 3.3 and 3.4 of the textbook.Return to problem 12 only after you understand thosetwo sections.

1.(10 pts) Equations of straight lines and graphsof linear equations. Know how to obtain an equationof a straight line given two pieces of information, forexample, two points, or a point and slope. Interceptslead to special cases of points.Let L be the line through the points

�1 $ � 2 and� � 2 $ � 3 . The slope of L is . Using x

and y as the variables as usually, its equation in slopeintercept form is y � .The x-intercept of L is x � .

2.(10 pts) Quadratic Equations. Know how tosolve quadratic equations, by completing the square,or applying the quadratic formula.The equation

2x2�3x � 3 � 0

has two real solutions. The smaller is , and thelarger is .

3.(10 pts) Quadratic Equations. Quadratic Equa-tions don’t always look like such. To obtain one youmay have to carry out some sort of substitution ormanipulate expressions suitably.

The equation

2x � 5 x � 2 � 0has two real solutions. The smaller is , and thelarger is .The equation

2x4 � 5x2 � 2 � 0has two positive real solutions. The smaller is ,and the larger is .The equation

1

x � 1 � x

x�1

� 2 � 0has two real solutions. The smaller is , and thelarger is .

4.(10 pts) Functions. Understand the concepts ofdomain, range, and inverse of a function, and knowhow to evaluate and compose functions.Let

f�x � x � 5x � 2 �

Then the domain of f is the set of all real numbersexcept , and its range is the set of all realnumbers except .Moreover,f�2 � ,f�t�1 � , and

f�f�t2 � = .

This is hard to do in WeBWorK, but you should alsobe able to draw the graph of f , showing clearly allintercepts and asymptotes.

5.(10 pts) Linear Systems. Know how to solve alinear system.The solution of the linear system

2x�3y

�3z � 5

4x � 3y � z � 8� 2x � 6y � z � � 6isx � ,y � , andz � , and

6.(10 pts) Rules of Exponents. Understand howto combine powers.�

6y2 2 � 2y3 � 1 � aybwhere

1

a � andb � .

7.(10 pts) Simplifying Rational Expressions. Un-derstand how to manipulate rational expressions.They work just like fractions!

x � 5x2 � 25 � 3

x�5 � AB

where A and B are polynomials of degree as low aspossible and the leading coefficient of B is 1.A � andB � .

8.(10 pts) Graphing. Understand the interplay be-tween algebra and geometry when graphs are shiftedhorizontally and vertically.For example, the graph of

f�x � ex � 2 � 3

is the graph of the exponential y � ex shiftedunits horizontally and units vertically.(Distinguish left and right and up and down with theappropriate signs of the shift. You should also drawthe graphs of the original and the shifted exponentialin one coordinate system.)

9.(10 pts) More Graphing. Know how to graphrational functions and to compute asymptotes and in-tercepts. For example the graph of

f�x � 2x � 1x � 1

has a horizontal asymptote y � and a verticalasymptote x � .Its y intercept is y � and its x intercept is x �. You should also draw the graph.

10.(10 pts) Logarithm Rules. Understand therules for logarithms. You should be able to writelogarithms as sums, differences or multiples of log-arithms when appropriate, or expressions int terms ofseveral logarithms in terms of single logarithms. Forexample,log2 32

�a�1 � 4 � A � B log2 � C where A �

andB � are numbers andC � is an expression in terms of a.

11.(10 pts)More Logarithm Rules. Similarly

ln�x�1 0� ln � x2 � ln � C

where

C � is an expression in terms of x.

12.(10 pts) Logarithmic Equations. Understandhow to solve equations involving logarithms. For ex-ample, the solution of the equation

ln12 � ln � x � 1 � ln � x � 2 isx �13.(10 pts) Exponential Equations. Understand

how to solve equations involving exponentials. Forexample, the largest solution of the equation

2x2 � 5x � 9 � 8

isx �14.(10 pts) Matrices. Know how to add, subtract

and multiply matrices. For example, if

A �CB 1 23 4 D and B �CB 2 � 11 2 D $

then

AB � B2 �CB a bc d Dwherea � ,b � ,c � ,d � ,

15.(10 pts)Matrices Inverses. Know how to com-pute matrix inverses and use them to solve linear sys-tems. For example, if

A � B 1 2� 1 2 Dthen

A � 1 � B a bc d Dwherea � ,b � ,c � ,d � ,Use it to solve the linear system

x�y � 1� x � 2y � 2

You obtainx � ,y � ,

2

16.(10 pts)Exponentials. Go over past word prob-lems, particularly those involving exponentials andlogarithms.Suppose a population doubles every 20 years. Theneach year it increases by percent.

17.(10 pts) The Binomial Theorem. Let

p�x � �

2x � 1 5 � ax5 �bx4

�cx3

�dx2

�ex

�f �

Thena � ,b � ,c � ,d � ,e � , andf � .

18.(10 pts) Mathematical Induction. Study andunderstand mathematical induction. Do problems 11-14 on page 652 of the textbook. Then enter the phraseI did here: .19.(10 pts) Basic Counting. You have 3 pairs of

pants or skirts, 4 shirts or blouses, and 5 pairs ofshoes. You can use them to wear differentoutfits.You are a participant in a peace conference with10 participants. Everybody shakes everybody else’shand. There are handshakes altogether.A family of five is taking an extended vacation. Ev-ery day at lunch they stand in line at a cafeteria in

a different order than ever before. On the last day,however, they can’t help repeating a previous order.Their vacation lasted days.

20.(10 pts) You deposit $150 into an account at thebeginning of each month. The bank pays you 5% in-terest per year, compounded monthly. At the end of14 years, after 168 payments, your account contains$ .

21.(10 pts) More Equation Solving. I found thisdelightful problem on the internet. Unfortunately thelink to the original statement has since expired.One time, a few years back, I drove a boat up the

Mississippi river. My mom was with me, but she was

asleep the moment she stepped into the boat. A mile

up the river, her hat blew off into the water. It took

her five minutes to wake up and realize that her hat

was missing. She told me to turn the boat around and

head downstream to retrieve her hat. When we finally

reached the hat, it had floated to the dock where we

had left from. Now, I had kept the boat going at the

same speed the entire time, but what I want to know

is, how fast was the river flowing?

The speed of the river is miles per hour.At our annual problem solving contest in 2003, one

of our mathmajors solved this problem in 45 seconds.However, this last problem of the semester does comewith a solution. Look at it after the set closes.


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