Thinking out of the Box... Problem It s a richer problem than we ever imagined

Size: px
Start display at page:

Download "Thinking out of the Box... Problem It s a richer problem than we ever imagined"

Transcription

1 From the Mthemtics Techer, Vol. 95, No. 8, pges Wlter Dodge (not pictured) nd Steve Viktor Thinking out of the Bo... Problem It s richer problem thn we ever imgined The bo problem hs been stndrd optimiztion eercise in lmost every clculus tetbook since Leibniz nd Newton invented clculus. With the cpbility of technology in the form of grphing clcultors, this eercise hs recently become stndrd fre erlier in the mthemtics curriculum. We even find it in middle school curricul s nice hnds-on eercise in dt nlysis. With some vritions in the numericl dimensions of the pper, the problem is similr to the following: Given rectngulr sheet of pper 8.5 inches 11 inches, form bo by cutting congruent squres from ech corner, folding up the sides, nd tping them to form bo without top. To mke bo with mimum cpcity, how lrge should the squre cutouts from the corners of the originl pper be? See figure 1. rough determintion of the cutout size tht results in bo with mimum cpcity. In course prior to clculus, students might be sked to write the function of tht describes the volume of ny bo in which the length of the side of the squre cutout is denoted by. This function is s follows: v() = (8.5 2)(11 2) Depending on the course nd on the technology vilble, students cn gther dt from this function or grph it over the intervl [0, 4.25] nd thus determine the bo of mimum volume, tht is, the bsolute mimum point of the dt or of the grph over the given intervl. A grph of this function, drwn with TI-83 grphing clcultor, is given in figure 2. Grphing clcultors hve llowed the bo problem to become stndrd fre erlier in the curriculum 8.5" 11" Fig. 1 Fig. 2 In beginning clculus course, students could use symbolic mnipultor or tke the derivtive of the volume function by hnd, set it equl to 0, solve, nd thus determine the -vlue tht gives the bo of mimum volume: In middle school setting, groups of students re often given sheets of pper nd sked to cut uniform squres from the corners, cutting differentsized squres for ech sheet. They then fold up the sides to mke vriety of boes of different sizes nd fill these boes with something, such s popcorn, nd mesure or count the mount needed to fill the boes. In this mnner, students obtin Wlter Dodge, tught mthemtics t New Trier High School in Winnetk, IL He is currently mthemtics contest writer for the North Suburbn Mthemtics Legue of the Chicgo Are. Steve Viktor, is the mthemtics deprtment chir t New Trier High School, Winnetk, IL He is frequent speker nd is especilly interested in the Advnced Plcement clculus progrm. 568 MATHEMATICS TEACHER Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3 - Techers Teching with

2 or v() = (8.5 2)(11 2) v() = v'() = = Discrding the solution tht is not in the prcticl intervl [0, 4.25] yields n pproimte solution of = inches. We cn verify this -vlue yielding the bsolute mimum by testing the endpoints where the volume is zero nd the volume t = tht is positive. The previously described eperiences usully comprise the totl eposure tht student, or for tht mtter, techer, hs with this problem tht my lso led to interesting eplortions for lgebr nd geometry students. This rticle revels further questions tht cn be investigted from this simply stted problem. We ssigned some of these questions to our AP clculus students s projects to complete outside of clss. We hve hd fun developing some of the lter questions on our own nd pln to use them with students in the future. Question 1 If we lwys strt with squre sheet of pper, does common reltionship eist between the length of the side of this squre nd the length of the side of the smller squres tht re cut out from ech corner? We sked students to eperiment with severl squre sheets of pper of different sizes, gther dt, try to find generl reltionship, nd then prove tht generl reltionship. In the interests of spce, we give only the generl solution for squre sheet of pper tht mesures units by units, s shown in figure 3. The following work could lso be done using symbol mnipultor. Let v() = ( 2) 2, where is in the intervl [0, /2]. Then v() = , v'() = , 0 = We see tht it fctors, so nd 0 = (6 )(2 ) =. 6 The other solution obviously yields minimum volume. The solution mkes students relize tht is the vrible for differentition nd tht, lthough vrible, is constnt with respect to the differentition, tht is, it is one of those very useful fied but still vrible vribles. This concept is lso precursor of multivrite clculus. In ddition, we obtin very simple generl result, which sys tht to find the bo of mimum volume strting with ny squre sheet of pper, we simply mke the squre cutouts t ech corner with side length tht is one-sith tht of the side length of the originl squre. Question 2 If we strt with squre but think dynmiclly of incresing one side of tht squre to form lrger nd lrger rectngulr sheets of pper while still keeping the djcent side of fied length, how does the side of the squre cut out from the corners of this pper to form the bo of mimum volume vry s this dynmic side becomes lrger nd lrger? For emple, we consider sheets of pper of the following sizes: 6 inches 6 inches, 6 inches The concept is precursor of multivrite clculus Fig. 3 Photogrph by Richrd Nelson; ll rights reserved Vol. 95, No. 8 November Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3 - Techers Teching with

3 This question is much tougher but lso much more rewrding for the persevering student 8 inches, 6 inches 10 inches, 6 inches 12 inches, nd so on. We know tht for the 6 inch 6 inch squre, we cut out 1 inch 1 inch squres. Is the length of the cutout for the 6 inch 8 inch squre more thn 1 inch, less thn 1 inch, or still 1 inch? Wht hppens to the cutout length s the vrible side of the rectngle gets longer nd longer? Does limiting vlue eist? If so, wht is it? We sk our students to eperiment by solving severl concrete emples nd obtining pttern, then generlizing, nd finlly mybe even proving their generliztions. This question is much tougher thn the first one but lso much more rewrding for the persevering student. In this rticle, we offer only flvor of the totl eperience. We ssume tht the originl squre sheet of pper is units units nd tht the side denoted by b is the one tht is incresing in size. We net wnt to find the vlue of for ny vlues of nd b tht yield the bo of mimum volume, s shown in figure 4. b This eqution solves ny bo problem, given the dimensions of the originl sheet of pper, nd b. When b =, it gives, s it should, the solution found in question 1. To get n ide of the solution to the queries given in question 2, we sked our students to fi = 6 nd then consider s function of only b. The result is (b) = (6 + b) 36` `6b`+`b[. We net use grphing clcultor to mke grph of s function of b. On TI-83 clcultor, Y1 ssumes the role of, nd X ssumes the role of b. Therefore, the -is represents the length of the rectngle whose djcent side is 6, nd the y-is represents the cutout size for the corner squres tht yields the bo with mimum volume: y 1 = (6 + ) 36` `6`+`[. with window of : [0, 100] nd y: [0, 3]. See figure 5. Fig. 5 Fig. 4 Agin, symbolic mnipultor cn be used to do the following clcultions. Let v() = ( 2)(b 2), where is in the intervl [0, /2]. Then v() = 4 3 (2 + 2b) 2 + b, v'() = ( + b) + b, 0 = ( + b) + b. Solving by using the qudrtic formul yields We notice tht s the side b increses beyond 6, the cutout size, for the bo with mimum volume lso increses; but limiting vlue, tht is, horizontl symptote for the grph, does seem to eist. Using the tble feture of the TI-83 in Ask mode (through TBLSET) nd trying higher nd higher vlues for b, we find tht the limit for b seems to be 1.5 units. See figure 6. We did sk students to try couple of other fied vlues for so tht they might see generl pttern. If students do so, they see tht the cutout = ( + b) ± [` `b`+`b[. For ny vlue of b >, the solution using the positive root is greter thn or equl to /2, so it is not the miml solution tht we desire. Hence, the miml solution is given by = ( + b) [` `b`+`b[. Fig MATHEMATICS TEACHER Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3 - Techers Teching with

4 vlue lwys seems to pproch /4, where is the dimension of the fied side of the sheet of pper. A nice symbolic-mnipultor eercise for students using TI-89, TI-92, or the like is to hve it ctully evlute the limit. In this sitution, technology mkes this limit ccessible to gret number of students who would not be ble to do the limit clcultions by hnd: lim = lim ( + b) [` `b`+`b[. b b Some more mthemticlly ble students cn evlute this limit nlyticlly by rtionlizing the numertor s follows: lim ( + b) [` `b`+`b[ b = lim ( + b) [` `b`+`b[ ( + b)+ [` `b`+`b[ b ( + b)+ [` `b`+`b[ 3b = lim b 6( + b + [` `b`+`b[ ) = lim b b b[ b = 4 We know tht for ny size b sheet of pper, where b, the cutout size,, for the length of the side of the squre cut from ech corner to yield the bo of mimum volume lwys stisfies the inequlity <. 6 4 The squre sheet of pper uses the smllest cutout size; nd the more elongted the pper is, the closer the cutout size should be to /4. Question 3 For ny rectngulr sheet of pper tht mesures b, does reltionship eist between the lterl re nd the re of the bse for the bo of miml volume found by cutting congruent squres of side length from the corners of the pper? If so, wht is this reltionship? This question ws one tht we hd not immeditely considered. Only lter did we begin to pursue the reltionship between the lterl re nd the re of the bse of the bo with miml volume. This question becomes fundmentl in the rest of our work. Agin, students should eperiment before seeking the forml result; however, this result is quite esy in its generl form. We ssume tht the originl sheet of pper is b with b, nd the cutout-squre side length is gin denoted by. Where is in the intervl [0, /2], v() = ( 2)(b 2). Rther thn epnding the term on the right out to obtin polynomil, we cn tke the derivtive in this form using the product rule. One of us hd done so initilly nd noted tht the first term ws the re of the bse of the bo nd wondered whether the remining term hd ny physicl significnce: v'() = ( 2)(b 2) + [( 2)( 2) + (b 2)( 2)] Rewriting this result in slightly different form yields the following: v'() = ( 2)(b 2) [2( 2) + 2(b 2)] 0 = ( 2)(b 2) [2( 2) + 2(b 2)] We notice tht ( 2)(b 2) is the re of the bse of our bo nd tht [2( 2) + 2(b 2)] is the lterl re of the bo. Hence, when we hve the bo of miml volume, the re of the bse minus the lterl re equls 0. Therefore, the bo of miml volume is lwys the bo tht hs the property tht the lterl re is the sme s the re of the bse. This result gives us n esy wy to verify whether ny open bo previously constructed from rectngulr sheet of pper is indeed bo with miml volume. We simply mesure the length, width, nd height of the bo nd then clculte the bse re nd the lterl re. If the two results re equl, the bo is the bo with miml volume; otherwise, it is not. If we hd relied only on symbolic mnipultor, we might not hve been ble to see this reltionship. A symbolic mnipultor gives only the symbolic form tht hs been progrmmed into it. A different form often gives one better insight into generliztion. In this sitution, writing the eqution in our specil symbolic form enbled us to clerly see the reltionship. So fr, we hve just been using rectngulr sheets of pper nd hve been cutting squres out t ech corner to form bo. No reson dicttes tht the piece of pper must be rectngulr. Question 4 ) If the piece of pper tht we strt with is n equilterl tringle, how do we cut out the corners so tht we cn then fold up the sides nd hve bo tht hs n equilterl tringle for bse? b) Once we hve solved prt (), wht is the reltionship between the side of the originl equilterl tringle nd the height,, of the lterl sides of the bo formed in prt () tht gives the bo of mimum volume? We no longer cut squres with side length of out of the corners. Insted, we drw in the ngle Technology mkes this limit ccessible to gret number of students Vol. 95, No. 8 November Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3 - Techers Teching with

5 bisectors of the three ngles nd mrk off the sme distnce on ech one. We then connect the three endpoints of these ngle-bisector segments. We cn see the bse of our solid in figure 7. Finlly, we drw in the si perpendiculr segments from these points to the originl three sides of the pper. The length of these perpendiculr segments is denoted by. Hence, becomes the height of our bo. Thus, for the equilterl tringle, we cut out congruent kites from ech corner. Ech kite hs two opposite right ngles nd the 60-degree ngle from the originl equilterl tringle pper. to follow ny simple pttern. We lso tried to obtin the result for regulr hegonl sheet of pper nd the generl regulr n-gon sheet of pper. We used the formul A = (1/2)p for the re of the bse of the bo when writing the volume function. During these clcultions, we finlly rrived t better wy to look t our results. We decided to compre the height of the bo of mimum volume with the pothem of the regulr polygonl sheet of pper, not with the side, s we hd been doing. Becuse the pothem is hlf the side of the squre, the mimum bo occurs when the cutout size is one-third of the pothem, or = s 6 = 2 `6` 3 s 3 =, 3 where is the pothem of the squre nd s is the side. We net look t the equilterl tringle, s shown in figure 8. Fig. 7 Compre the height with the pothem of the regulr polygonl sheet of pper To nswer prt (b) of question 4, we need to write formul for the volume in terms of the originl side, s, of the equilterl tringulr pper nd. We cn immeditely write it in generl, but students should do few concrete emples first. We use the fct tht the re of n equilterl tringle is given by Then A = (side)2 3. ````4```` v() = (s 2 3) 2 3, `4` where is in the intervl 0, s 3. `6` We net tke the derivtive, set it equl to 0, nd solve for. Using symbolic mnipultor yields = s 3/18 or = s 3/6. Becuse the ltter result obviously gives minimum volume, our mimum occurs when = s 3/18. This result is not s stisfying s we hd hoped. So fr, we hve worked with two regulr polygons, the squre nd the equilterl tringle. For the squre, the result ws = s/6. We were hoping for some simple reltionship tht would give us the result quickly for ll regulr polygons. The work becme quite tedious, nd the result did not seem 3 Fig. 8 From the digrm, we see tht s = 2 3, so tht = s 3 `18` = ```18``` =. 3 For both the squre nd the equilterl tringle, the height of the bo of mimum volume is onethird the length of the pothem of the originl sheet of pper. Question 5 For ny regulr n-gon sheet of pper, if congruent kites re cut from the corners nd then the sides re folded up to form bo with similr n-gon for bse, wht is the reltionship between the height 572 MATHEMATICS TEACHER Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3 - Techers Teching with

6 of the bo nd the pothem of the originl sheet of pper for the bo of mimum volume? We hve n ide tht the nswer might be = /3, where is the height of the bo nd is the pothem of the originl n-gon sheet of pper. To prove this result, perhps we should write the re of the n-gon s function of the pothem,, of the originl sheet of pper. For the squre. v() = (2 2) 2 = 4( ) 2. We hve lredy shown tht the mimum occurs when = /3. Thus, if we tke the derivtive of v() nd set it equl to 0, our result will be = /3. For the equilterl tringle. We know tht s = 2 3, so tht v() = ( ) 2 3 `4` = 3 3( ) 2. We notice tht this formul differs only by constnt fctor from tht of the volume of the squre, so the derivtive hs the sme roots. Hence, we gin see tht = /3 is the correct solution. The generl regulr n-gon. From figure 9 nd using the fct tht the re of ny regulr n-gon is given by A = (1/2)p, where is the pothem nd p is the perimeter, we know tht the volume of the bo is v() = 1 ( )n 2 tn π 2 tn π ; 2 n n v() = n tn π ( )2. n We notice tht this result is just constnt times the formul for the volume of the squre; hence, gin the bo with mimum volume occurs when, the height of the bo, is chosen such tht = /3, where is the pothem of the originl sheet of pper. When we hve the generl regulr n-gon volume formul, students cn go bck nd try vlues of n = 3 nd n = 4 nd verify tht these vlues re the ect constnt fctors tht we determined erlier when we did these problems seprtely. Question 6 Wht is the reltionship between the lterl re nd the re of the bse of the bo of mimum volume constructed from regulr n-gon sheet of pper? This reltionship is reltively esy one to determine from the fct tht the mimum volume bo hs = /3. Then the re of the bse is 1 ( )p = 1 p = 1 2 p 2 `3` = p 3 = p, which is the lterl re. Hence, the bo with mimum volume mde from ny regulr n-gon shped sheet of pper hs lterl re equl to the re of its bse. SUMMARY This clssic problem hs much more to offer thn wht ppers in most tetbooks. Along the wy in our questioning, we used gret del of high school mthemtics, nd we hve discovered interesting geometric nd lgebric reltionships. Finlly, we hope tht when reders see clssic problem, they will think beyond tht problem nd try to find interesting mthemticl generliztions lurking in the bckground. FOR DISCUSSION WITH STUDENTS AND COLLEAGUES We pose further questions tht re relted to the content of this rticle. We hve eplored nswers to the first three of these questions nd would be interested in seeing whether reders gree with our results nd seeing how they obtined their results. Etension 1 For ny given sheet of pper tht mesures b, where b, if we lwys mke the length,, of the side of the cutout squre, so tht = /5, how fr Tn n n Tn n Fig. 9 A section of regulr n-gon The formul differs only by constnt fctor from tht of the squre, so the derivtive hs the sme roots Vol. 95, No. 8 November Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3 - Techers Teching with

7 will we be from the bo with mimum volume? From prcticl stndpoint, for ll typicl boes tht would normlly be mnufctured, we re relly sking whether we could tell the production stff to lwys cut the squres t the corner of length = /5, where is the shortest dimension of the originl sheet of pper, nd not be too fr from the bo with mimum volume. Etension 2 For the regulr n-gon sitution, wht hppens in the limit s the number of sides of the regulr n-gon pproches infinity? Wht reltionships do we obtin for cylinder? Etension 3 We hve seen tht with regulr n-gon pper nd with rectngulr pper, the bo with miml volume occurs when the lterl re of the bo nd the re of the bse of the bo re equl. Do we continue to obtin this result if we re given ny conve polygon s the originl sheet of pper? The reder should either furnish proof or counteremple. Etension 4 Wht other etensions of this problem should students consider? Etension 5 Wht re pproprite uses of computer lgebr systems nd other technology in these etensions? Wht cn students lern bout pproprite technology use from these bo problems? Etension 6 The techer cn choose nother clssic problem from the mthemtics curriculum. Wht etension questions could be used with tht problem? How might students respond to these questions? Wht mthemtics would they lern or use in the solutions? BIBLIOGRAPHY Dodge, Wlter, Kthleen Goto, nd Philip Mllinson. Soundoff! I Would Consider the Following to Be Proof.... Mthemtics Techer 91 (November 1998): Dossey, John, et l. Focus on Advnced Algebr. Reding, Mss.: Addison-Wesley Publishing Compny, Finney, Ross L., George B. Thoms Jr., Frnklin D. Demn, nd Bert K. Wits. Clculus: Grphicl, Numericl, Algebric. Reding, Mss.: Addison- Wesley Publishing Compny, Stewrt, Jmes. Clculus. Pcific Grove, Clif.: Brooks/Cole Publishing Compny, MATHEMATICS TEACHER Copyright 2002 by the Ntionl Council of Techers of Mthemtics, Inc. All rights reserved. For use ssocited with Tes Instruments T3 - Techers Teching with

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply

More information

Reasoning to Solve Equations and Inequalities

Reasoning to Solve Equations and Inequalities Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

More information

Integration. 148 Chapter 7 Integration

Integration. 148 Chapter 7 Integration 48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but

More information

Graphs on Logarithmic and Semilogarithmic Paper

Graphs on Logarithmic and Semilogarithmic Paper 0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

More information

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )

Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( ) Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +

More information

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions. Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

More information

Factoring Polynomials

Factoring Polynomials Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles

More information

AREA OF A SURFACE OF REVOLUTION

AREA OF A SURFACE OF REVOLUTION AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.

More information

Experiment 6: Friction

Experiment 6: Friction Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht

More information

10.6 Applications of Quadratic Equations

10.6 Applications of Quadratic Equations 10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,

More information

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

More information

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

More information

SPECIAL PRODUCTS AND FACTORIZATION

SPECIAL PRODUCTS AND FACTORIZATION MODULE - Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come

More information

Section 7-4 Translation of Axes

Section 7-4 Translation of Axes 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the

More information

Algebra Review. How well do you remember your algebra?

Algebra Review. How well do you remember your algebra? Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then

More information

9 CONTINUOUS DISTRIBUTIONS

9 CONTINUOUS DISTRIBUTIONS 9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete

More information

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1 PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

More information

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.

Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers. 2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this

More information

4.11 Inner Product Spaces

4.11 Inner Product Spaces 314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces

More information

Integration by Substitution

Integration by Substitution Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is

More information

Section 5-4 Trigonometric Functions

Section 5-4 Trigonometric Functions 5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

More information

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES

LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of

More information

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding

Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding 1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde

More information

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.

5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one. 5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued

More information

Math 135 Circles and Completing the Square Examples

Math 135 Circles and Completing the Square Examples Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for

More information

6.2 Volumes of Revolution: The Disk Method

6.2 Volumes of Revolution: The Disk Method mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of

More information

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn 33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

More information

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity

Babylonian Method of Computing the Square Root: Justifications Based on Fuzzy Techniques and on Computational Complexity Bbylonin Method of Computing the Squre Root: Justifictions Bsed on Fuzzy Techniques nd on Computtionl Complexity Olg Koshelev Deprtment of Mthemtics Eduction University of Texs t El Pso 500 W. University

More information

Lecture 5. Inner Product

Lecture 5. Inner Product Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right

More information

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials

Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic

More information

Regular Sets and Expressions

Regular Sets and Expressions Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite

More information

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100

Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100 hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by

More information

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY

PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive

More information

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS

RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is

More information

Unit 6: Exponents and Radicals

Unit 6: Exponents and Radicals Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

More information

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes

9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors

More information

Introduction to Integration Part 2: The Definite Integral

Introduction to Integration Part 2: The Definite Integral Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the

More information

Distributions. (corresponding to the cumulative distribution function for the discrete case).

Distributions. (corresponding to the cumulative distribution function for the discrete case). Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive

More information

Basic Analysis of Autarky and Free Trade Models

Basic Analysis of Autarky and Free Trade Models Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently

More information

and thus, they are similar. If k = 3 then the Jordan form of both matrices is

and thus, they are similar. If k = 3 then the Jordan form of both matrices is Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If

More information

Or more simply put, when adding or subtracting quantities, their uncertainties add.

Or more simply put, when adding or subtracting quantities, their uncertainties add. Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re

More information

Lectures 8 and 9 1 Rectangular waveguides

Lectures 8 and 9 1 Rectangular waveguides 1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves

More information

Solving BAMO Problems

Solving BAMO Problems Solving BAMO Problems Tom Dvis tomrdvis@erthlink.net http://www.geometer.org/mthcircles Februry 20, 2000 Abstrct Strtegies for solving problems in the BAMO contest (the By Are Mthemticl Olympid). Only

More information

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style

15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time

More information

EQUATIONS OF LINES AND PLANES

EQUATIONS OF LINES AND PLANES EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint

More information

Physics 43 Homework Set 9 Chapter 40 Key

Physics 43 Homework Set 9 Chapter 40 Key Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x

More information

Helicopter Theme and Variations

Helicopter Theme and Variations Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike

More information

CHAPTER 11 Numerical Differentiation and Integration

CHAPTER 11 Numerical Differentiation and Integration CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods

More information

Binary Representation of Numbers Autar Kaw

Binary Representation of Numbers Autar Kaw Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy

More information

Econ 4721 Money and Banking Problem Set 2 Answer Key

Econ 4721 Money and Banking Problem Set 2 Answer Key Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in

More information

Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Mean and the Pythagorean Theorem Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

More information

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324

A.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324 A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................

More information

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

More information

Review Problems for the Final of Math 121, Fall 2014

Review Problems for the Final of Math 121, Fall 2014 Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since

More information

PHY 140A: Solid State Physics. Solution to Homework #2

PHY 140A: Solid State Physics. Solution to Homework #2 PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: jixun@physics.ucl.edu Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.

More information

The Definite Integral

The Definite Integral Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know

More information

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report

DlNBVRGH + Sickness Absence Monitoring Report. Executive of the Council. Purpose of report DlNBVRGH + + THE CITY OF EDINBURGH COUNCIL Sickness Absence Monitoring Report Executive of the Council 8fh My 4 I.I...3 Purpose of report This report quntifies the mount of working time lost s result of

More information

MODULE 3. 0, y = 0 for all y

MODULE 3. 0, y = 0 for all y Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)

More information

Warm-up for Differential Calculus

Warm-up for Differential Calculus Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:

More information

The remaining two sides of the right triangle are called the legs of the right triangle.

The remaining two sides of the right triangle are called the legs of the right triangle. 10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

More information

Applications to Physics and Engineering

Applications to Physics and Engineering Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics

More information

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration

2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting

More information

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)

www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values) www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input

More information

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the

More information

Review guide for the final exam in Math 233

Review guide for the final exam in Math 233 Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered

More information

Exponential and Logarithmic Functions

Exponential and Logarithmic Functions Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define

More information

Vectors 2. 1. Recap of vectors

Vectors 2. 1. Recap of vectors Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms

More information

ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS

ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS CHAPTER ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS Although people tody re mking greter use of deciml frctions s they work with clcultors, computers, nd the metric system, common

More information

Unit 29: Inference for Two-Way Tables

Unit 29: Inference for Two-Way Tables Unit 29: Inference for Two-Wy Tbles Prerequisites Unit 13, Two-Wy Tbles is prerequisite for this unit. In ddition, students need some bckground in significnce tests, which ws introduced in Unit 25. Additionl

More information

19. The Fermat-Euler Prime Number Theorem

19. The Fermat-Euler Prime Number Theorem 19. The Fermt-Euler Prime Number Theorem Every prime number of the form 4n 1 cn be written s sum of two squres in only one wy (side from the order of the summnds). This fmous theorem ws discovered bout

More information

6 Energy Methods And The Energy of Waves MATH 22C

6 Energy Methods And The Energy of Waves MATH 22C 6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this

More information

COMPONENTS: COMBINED LOADING

COMPONENTS: COMBINED LOADING LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of

More information

DIFFERENTIATING UNDER THE INTEGRAL SIGN

DIFFERENTIATING UNDER THE INTEGRAL SIGN DIFFEENTIATING UNDE THE INTEGAL SIGN KEITH CONAD I hd lerned to do integrls by vrious methods shown in book tht my high school physics techer Mr. Bder hd given me. [It] showed how to differentite prmeters

More information

Week 7 - Perfect Competition and Monopoly

Week 7 - Perfect Competition and Monopoly Week 7 - Perfect Competition nd Monopoly Our im here is to compre the industry-wide response to chnges in demnd nd costs by monopolized industry nd by perfectly competitive one. We distinguish between

More information

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required

More information

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix.

addition, there are double entries for the symbols used to signify different parameters. These parameters are explained in this appendix. APPENDIX A: The ellipse August 15, 1997 Becuse of its importnce in both pproximting the erth s shpe nd describing stellite orbits, n informl discussion of the ellipse is presented in this ppendix. The

More information

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.

Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1. Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose

More information

The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus Section 5.4 Te Funmentl Teorem of Clculus Kiryl Tsiscnk Te Funmentl Teorem of Clculus EXAMPLE: If f is function wose grp is sown below n g() = f(t)t, fin te vlues of g(), g(), g(), g(3), g(4), n g(5).

More information

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.

LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes. LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.

More information

2012 Mathematics. Higher. Finalised Marking Instructions

2012 Mathematics. Higher. Finalised Marking Instructions 0 Mthemts Higher Finlised Mrking Instructions Scottish Quliftions Authority 0 The informtion in this publtion my be reproduced to support SQA quliftions only on non-commercil bsis. If it is to be used

More information

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.

Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a. Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles

More information

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation

FUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does

More information

Second-Degree Equations as Object of Learning

Second-Degree Equations as Object of Learning Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, 2008. Abstrct Second-Degree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson,

More information

Introduction. Teacher s lesson notes The notes and examples are useful for new teachers and can form the basis of lesson plans.

Introduction. Teacher s lesson notes The notes and examples are useful for new teachers and can form the basis of lesson plans. Introduction Introduction The Key Stge 3 Mthemtics series covers the new Ntionl Curriculum for Mthemtics (SCAA: The Ntionl Curriculum Orders, DFE, Jnury 1995, 0 11 270894 3). Detiled curriculum references

More information

MATH 150 HOMEWORK 4 SOLUTIONS

MATH 150 HOMEWORK 4 SOLUTIONS MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive

More information

The Velocity Factor of an Insulated Two-Wire Transmission Line

The Velocity Factor of an Insulated Two-Wire Transmission Line The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives

More information

Pure C4. Revision Notes

Pure C4. Revision Notes Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd

More information

Rotational Equilibrium: A Question of Balance

Rotational Equilibrium: A Question of Balance Prt of the IEEE Techer In-Service Progrm - Lesson Focus Demonstrte the concept of rottionl equilirium. Lesson Synopsis The Rottionl Equilirium ctivity encourges students to explore the sic concepts of

More information

3 The Utility Maximization Problem

3 The Utility Maximization Problem 3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best

More information

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999

Economics Letters 65 (1999) 9 15. macroeconomists. a b, Ruth A. Judson, Ann L. Owen. Received 11 December 1998; accepted 12 May 1999 Economics Letters 65 (1999) 9 15 Estimting dynmic pnel dt models: guide for q mcroeconomists b, * Ruth A. Judson, Ann L. Owen Federl Reserve Bord of Governors, 0th & C Sts., N.W. Wshington, D.C. 0551,

More information

4 Approximations. 4.1 Background. D. Levy

4 Approximations. 4.1 Background. D. Levy D. Levy 4 Approximtions 4.1 Bckground In this chpter we re interested in pproximtion problems. Generlly speking, strting from function f(x) we would like to find different function g(x) tht belongs to

More information

Numerical Methods of Approximating Definite Integrals

Numerical Methods of Approximating Definite Integrals 6 C H A P T E R Numericl Methods o Approimting Deinite Integrls 6. APPROXIMATING SUMS: L n, R n, T n, AND M n Introduction Not only cn we dierentite ll the bsic unctions we ve encountered, polynomils,

More information

0.1 Basic Set Theory and Interval Notation

0.1 Basic Set Theory and Interval Notation 0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined

More information

10 AREA AND VOLUME 1. Before you start. Objectives

10 AREA AND VOLUME 1. Before you start. Objectives 10 AREA AND VOLUME 1 The Tower of Pis is circulr bell tower. Construction begn in the 1170s, nd the tower strted lening lmost immeditely becuse of poor foundtion nd loose soil. It is 56.7 metres tll, with

More information

Physics 2102 Lecture 2. Physics 2102

Physics 2102 Lecture 2. Physics 2102 Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields Chrles-Augustin de Coulomb (1736-1806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other

More information

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example 2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel

More information

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful

Pentominoes. Pentominoes. Bruce Baguley Cascade Math Systems, LLC. The pentominoes are a simple-looking set of objects through which some powerful Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this

More information