Helicopter Theme and Variations

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Helicopter Theme and Variations"

Transcription

1 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 helicopter is dropped The mount of weight dded to the helicopter Whether the helicopter is dropped indoors or outdoors The mteril from which the helicopter is mde Which direction the rotors re bent (clockwise vs. counterclockwise) wilder vritions could include: Vritions on the bsic design of the helicopter Punching holes in the helicopter rotors Some possible response vribles re: How long the helicopter tkes to fll How fr the helicopter lnds from desired (point) trget Whether the helicopter hits desired (lrge) trget wilder vritions could include: Whether someone is ble to ctch the helicopter How ttrctive the flight of the helicopter is, subjectively judged by n observer Useful mterils for helicopter experiments: Helicopters of severl designs Stopwtches Mesuring tpes or meter sticks Pper clips (for vrying weight) or hevy binder clips for outdoor drops Msking tpe (to mrk drop heights on verticl wll nd to mrk trget) Pendulum (to locte point directly beneth drop) Crds, dice, or rndom digit tble A bsket or binder clip with long string ttched will gretly ese the return of helicopter from its lnding site to the pltform from which it ws dropped. NCSSM pge 1

2 Some prticulr experimentl designs re: One smple t: 1. Do we live in vcuum? If ir resistnce hs no effect on the fll of helicopter, then the durtion of its fll should be clculble from the physics lw 1 2 2h m h= 2 gt ; or, T =, where g = 9.8 g 2, h is the height of the fll in meters, nd s T is the time of the fll in seconds. Hve one person drop single helicopter 3 times from fixed height nd hve nother person time the descents. Clculte the theoreticl time to fll bsed on the ssumption of no ir resistnce, nd cll it 25m T. (E.g. from height of 5 meters, T = 9.8m s 2 1sec.) Using one-smple t test of significnce, test the hypothesis µ time = T vs. µ time > T. (Hopefully, the effect of ir resistnce cn be detected with very high level of significnce!) 2. How long does this helicopter tke to fll, on verge? One person drops single helicopter 3 times from fixed height. Estimte the men time for it to fll using one-smple t confidence intervl. In this cse, the popultion of inference is only drops by this prticulr dropper with his or her prticulr helicopter. In order to generlize to helicopter design (e.g., ll long-rotor helicopters) use rndom smple of 3 different long-rotor helicopters. Note: The extrpoltion to the popultion of ll humn droppers my be vlid, but it could only be done bsed on belief, not on sttisticl evidence. We often do this without thinking bout it. E.g., suppose tht we re testing long-rotor helicopters vs. short-rotor helicopters for some property, nd for convenience, we mke ll the long-rotors blue nd ll the short-rotors yellow. How do we know tht ny observed differences re due to rotor length nd not to color? We believe color hs no effect bsed on plusible resons for different helicopter behvior, but there is no sttisticl evidence tht the confounding vrible of color did not hve n effect. 3. How close to trget cn I get this helicopter to lnd, on verge? This vrition is just like experiment 2, bove, except tht the response vrible mesured is the distnce tht the helicopter lnds from desired point trget, rther thn the time it took to fll. NCSSM pge 2

3 Pired differences t: 4. Does this long-rotor helicopter tke different length of time to fll, on verge, thn this short-rotor helicopter? Mke one long-rotor nd one short-rotor helicopter. All drops of both helicopters will be performed by the sme person nd from the sme height. Flip coin to see which one will be dropped first; fter it is dropped, then drop the other one. Repet this process 3 times, lwys dropping one nd then the other. The pired t test of significnce is pproprite for this experimentl design becuse the helicopter drops re pired by order. Once the dt re collected, use t test on the pired differences to see whether there is significnt evidence ginst H : µ long short = vs. H long short. Alterntively, crete confidence intervl estimte of µ long short. In either cse, the scope of inference is ll the possible drops of these two prticulr helicopters with this prticulr dropper. (It is not possible, sttisticlly, to generlize to the long- nd short-rotor designs.) 5. Do helicopters tke different length of time, on verge, to fll indoors versus outdoors? Mke 25 long-rotor helicopters. Ech helicopter is dropped once indoors nd once outdoors from the sme fixed height. As in experiment 4, do the drops in pirs, using coin flip for ech tril to determine whether the helicopter is dropped indoors first nd then outdoors, or vice-vers. Mesure the times of descent, nd test H in out = vs. H in out. Extrpolte to the popultion of ll long-rotor helicopters dropped by this dropper. A vrition would be to hve different dropper for ech tril. Then the results could be extrpolted to drops by ny dropper. 6. Are Glori nd Floyd eqully good, on verge, t mking their helicopters lnd ner specified trget? This experiment involves two droppers nd 2 longrotor helicopters, ll dropped from the sme fixed height. Choose helicopter t rndom nd flip coin to see who drops it first. Both droppers drop the sme helicopter, then move on to nother rndomly selected helicopter, gin choosing who drops first by rndom coin flip. The response vrible is the distnce from desired point trget tht the helicopter lnded. The pired t will test the significnce of evidence for H : µ Glori Floyd = vs. H Glori Floyd. Piring controls for wer nd ter on helicopter, incresed bility over time, environmentl fctors, etc. Any significnt difference my be extrpolted to Floyd s nd Glori s bility with ll long-rotor helicopters. NCSSM pge 3

4 Two smple t: 7. Do short-rotor nd long-rotor helicopters tke different lengths of time, on verge, to fll? Construct 25 short-rotor nd 25 long-rotor helicopters. One dropper (or ll different droppers, for lrger scope of inference) drops ll 5 helicopters from the sme fixed height nd their times of descent re mesured nd recorded. The order of the 5 drops is completely rndom determined, sy, by shuffling 25 blck crds nd 25 red crds nd then drwing them one t time, dropping short-rotor for the blck crds nd long-rotor for the red crds. This completely rndom order (s opposed to dropping the helicopters in pirs s in experiment 4) is wht mkes this design require two-smple t inference procedures rther thn one-smple t on pired differences. Test H : µ short = µ long vs. H short µ long or crete confidence intervl estimte of µ short µ long. 8. Do long-rotor helicopters tke different length of time, on verge, to fll indoors compred to outdoors? 4 different long-rotor helicopters re constructed nd re rndomly llocted to two groups: indoor nd outdoor. Ech is dropped once in its pproprite loction from the sme fixed height (obviously this is crucil!) with the order of the 4 drops being completely rndom, s in experiment 7. The time of descent is mesured nd recorded for ech drop. Use 2-smple t procedures to test whether the men flling time is different for indoors nd outdoors. One smple proportion: 9. How good is Brnby t hitting trget with long-rotor helicopters? 3 longrotor helicopters re constructed. A trget is drwn on the ground below dropping pltform. (The trget should be lrge enough tht it will get hit t lest 1 times nd will lso be missed t lest 1 times in order to hve np ˆ > 1 nd ( pˆ ) n 1 > 1.) One person drops ll 3 helicopters in rndom order nd records how mny of the drops hit the trget. A confidence intervl estimte of the proportion of hits is then constructed. NCSSM pge 4

5 Two smple proportions: 1. Do Zck nd Xen both hve the sme probbility of hitting trget with this prticulr helicopter? One long-rotor helicopter is constructed. Ech of two people will drop it 25 times towrds lrge trget from fixed height, with the 5 drops being performed in completely rndom order (s in experiment 7). The response vrible recorded is whether or not the helicopter hit the trget. One my then test the hypothesis H : pzck = pxen vs. H A: pzck pxenor crete confidence intervl estimte of pzck pxen. Vritions on this experiment: Do long- nd short-rotor helicopters hve n equl chnce of hitting trget? Is it esier for person to hit trget when dropping from lower height thn higher height? (This ltter vrition would be good exmple of one-sided test.) Chi-squre goodness-of-fit test: 11. Does this helicopter, under these environmentl conditions, hve directionl bis in where it lnds? Drw verticl nd horizontl line on the ground with their intersection directly beneth the dropping point. One person drops single long-rotor helicopter 4 times from fixed height. The qudrnt tht the helicopter lnds in is the response vrible. Under the hypothesis of no directionl bis, one would expect: H : pq1 = pq2 = pq3 = pq4 =.25. A chisqure goodness-of-fit test will test tht hypothesis. Evidence ginst it would suggest either directionlly-bised helicopter or, more likely, consistent wind direction. Chi-squre test of independence: 12. Is closeness to trget (concentric circles drwn round trget on the ground like drtbord) independent of rotor length? Construct 3 long-rotor nd 3 short-rotor helicopters. One person drops them in completely rndom order towrds drtbord trget drwn on the ground: two concentric circles seprting the ground into the regions close, medium, nd fr. Count how mny of ech type lnd in ech region nd test the independence of rotor length nd closeness to trget by using chi-squre sttistic on 2 3 tble. (Note: Evidence of difference in the two does not necessrily indicte tht either one is significntly better i.e., lnds significntly closer on verge thn the other. It only indictes difference in the distribution of where they lnd.) NCSSM pge 5

6 13. Is the probbility of hitting trget relted to drop height? This experiment is similr to number 12. Use low, medium, nd high for drop heights nd hit, not hit for the response vrible, with counts recorded in 3 2 tble. Use single helicopter nd dropper, or else single dropper with multiple helicopters of the sme type for lrger scope of inference. A chi-squre test of independence will indicte whether there is significnt evidence of ssocition between drop height nd successful trget-hitting. Vrition on this experiment: Is the probbility of hitting trget relted to the weight of the helicopter? Use pper clips, smll binder clips, nd lrger binder clips to crete different weights. Slope of Regression Line: Note: both of the experiments below re best done where helicopters cn be dropped from gret height (t lest 5 meters). 14. Does dding weight to helicopter speed up its descent? Construct 3 helicopters nd rndomly llocte them to 5 groups of 6, ech group of which will correspond to prticulr dded weight. In completely rndom order, drop ech helicopter from the sme height, recording the time of descent. Construct lestsqures line using the weight s the explntory vrible nd the time of descent s the response vrible. One my either estimte β with confidence intervl or test H : β = vs. H : β <. A 15. Do helicopters dropped from higher height tke longer on verge to fll thn helicopters dropped from lower height? Or: How fst do these helicopters fll once they rech their terminl velocity? Construct 5 long-rotor helicopters nd rndomly llocte them to 5 different fixed drop heights. Then mke ll 5 drops in completely rndom order, timing the descent of ech helicopter. Construct regression line using height s the explntory vrible nd time of descent s the response vrible. A test of H : β = vs. H A : β > should produce significnt evidence for the ltter. One my lso construct confidence intervl estimte of β, which is the reciprocl of the verge flling speed of the helicopters fter reching terminl velocity. (The intercept of the regression line results from the initil period of fluttering before the helicopter settles into its terminl velocity. It is likely to be negtive, since the helicopters tend to fll fster before they rech their terminl velocity, shortening the times of their flls.) NCSSM pge 6

Tests for One Poisson Mean

Tests for One Poisson Mean Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution

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

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

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

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

Name: Lab Partner: Section:

Name: Lab Partner: Section: Chpter 4 Newton s 2 nd Lw Nme: Lb Prtner: Section: 4.1 Purpose In this experiment, Newton s 2 nd lw will be investigted. 4.2 Introduction How does n object chnge its motion when force is pplied? A force

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

N Mean SD Mean SD Shelf # Shelf # Shelf #

N Mean SD Mean SD Shelf # Shelf # Shelf # NOV xercises smple of 0 different types of cerels ws tken from ech of three grocery store shelves (1,, nd, counting from the floor). summry of the sugr content (grms per serving) nd dietry fiber (grms

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

STA 2023 Test #3 Practice Multiple Choice

STA 2023 Test #3 Practice Multiple Choice STA 223 Test #3 Prctice Multiple Choice 1. A newspper conducted sttewide survey concerning the 1998 rce for stte sentor. The newspper took rndom smple (ssume it is n SRS) of 12 registered voters nd found

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

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

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

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding

On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl

More information

Chapter 6 Solving equations

Chapter 6 Solving equations Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign

More information

Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.

Homework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule. Text questions, Chpter 5, problems 1-5: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed

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

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:

Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period: Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A

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

Answer, Key Homework 8 David McIntyre 1

Answer, Key Homework 8 David McIntyre 1 Answer, Key Homework 8 Dvid McIntyre 1 This print-out should hve 17 questions, check tht it is complete. Multiple-choice questions my continue on the net column or pge: find ll choices before mking your

More information

Theory of Forces. Forces and Motion

Theory of Forces. Forces and Motion his eek extbook -- Red Chpter 4, 5 Competent roblem Solver - Chpter 4 re-lb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics - Everything

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

Biostatistics 103: Qualitative Data Tests of Independence

Biostatistics 103: Qualitative Data Tests of Independence Singpore Med J 2003 Vol 44(10) : 498-503 B s i c S t t i s t i c s F o r D o c t o r s Biosttistics 103: Qulittive Dt Tests of Independence Y H Chn Prmetric & non-prmetric tests (1) re used when the outcome

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables

Section 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

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

Arc Length. P i 1 P i (1) L = lim. i=1

Arc Length. P i 1 P i (1) L = lim. i=1 Arc Length Suppose tht curve C is defined by the eqution y = f(x), where f is continuous nd x b. We obtin polygonl pproximtion to C by dividing the intervl [, b] into n subintervls with endpoints x, x,...,x

More information

MATLAB: M-files; Numerical Integration Last revised : March, 2003

MATLAB: M-files; Numerical Integration Last revised : March, 2003 MATLAB: M-files; Numericl Integrtion Lst revised : Mrch, 00 Introduction to M-files In this tutoril we lern the bsics of working with M-files in MATLAB, so clled becuse they must use.m for their filenme

More information

Net Change and Displacement

Net Change and Displacement mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

More information

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd processes. Introducing technology

More information

Double Integrals over General Regions

Double Integrals over General Regions Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing

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

Small Business Networking

Small Business Networking Why network is n essentil productivity tool for ny smll business Effective technology is essentil for smll businesses looking to increse the productivity of their people nd business. Introducing technology

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

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

5.2 The Definite Integral

5.2 The Definite Integral 5.2 THE DEFINITE INTEGRAL 5.2 The Definite Integrl In the previous section, we sw how to pproximte totl chnge given the rte of chnge. In this section we see how to mke the pproximtion more ccurte. Suppose

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

Lecture 3 Basic Probability and Statistics

Lecture 3 Basic Probability and Statistics Lecture 3 Bsic Probbility nd Sttistics The im of this lecture is to provide n extremely speedy introduction to the probbility nd sttistics which will be needed for the rest of this lecture course. The

More information

AAPT UNITED STATES PHYSICS TEAM AIP 2010

AAPT UNITED STATES PHYSICS TEAM AIP 2010 2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS - 75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD

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

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

274 Chapter 13. Chapter 13

274 Chapter 13. Chapter 13 74 hpter 3 hpter 3 3. () ounts will be obtined from the smples so th problem bout compring proportions. (b) h n observtionl study compring rndom smples selected from two independent popultions. 3. () cores

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

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

Homework #6: Answers. a. If both goods are produced, what must be their prices?

Homework #6: Answers. a. If both goods are produced, what must be their prices? Text questions, hpter 7, problems 1-2. Homework #6: Answers 1. Suppose there is only one technique tht cn be used in clothing production. To produce one unit of clothing requires four lbor-hours nd one

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

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

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

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

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes

14.2. The Mean Value and the Root-Mean-Square Value. Introduction. Prerequisites. Learning Outcomes he Men Vlue nd the Root-Men-Squre Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men

More information

1 Numerical Solution to Quadratic Equations

1 Numerical Solution to Quadratic Equations cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll

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

Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and

Rational Functions. Rational functions are the ratio of two polynomial functions. Qx bx b x bx b. x x x. ( x) ( ) ( ) ( ) and Rtionl Functions Rtionl unctions re the rtio o two polynomil unctions. They cn be written in expnded orm s ( ( P x x + x + + x+ Qx bx b x bx b n n 1 n n 1 1 0 m m 1 m + m 1 + + m + 0 Exmples o rtionl unctions

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

Biostatistics 102: Quantitative Data Parametric & Non-parametric Tests

Biostatistics 102: Quantitative Data Parametric & Non-parametric Tests Singpore Med J 2003 Vol 44(8) : 391-396 B s i c S t t i s t i c s F o r D o c t o r s Biosttistics 102: Quntittive Dt Prmetric & Non-prmetric Tests Y H Chn In this rticle, we re going to discuss on the

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

Small Business Networking

Small Business Networking Why Network is n Essentil Productivity Tool for Any Smll Business TechAdvisory.org SME Reports sponsored by Effective technology is essentil for smll businesses looking to increse their productivity. Computer

More information

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS

PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,

More information

Lecture 15 - Curve Fitting Techniques

Lecture 15 - Curve Fitting Techniques Lecture 15 - Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting - motivtion For root finding, we used given function to identify where it crossed zero where does fx

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

Scalar Line Integrals

Scalar Line Integrals Mth 3B Discussion Session Week 5 Notes April 6 nd 8, 06 This week we re going to define new type of integrl. For the first time, we ll be integrting long something other thn Eucliden spce R n, nd we ll

More information

r 2 F ds W = r 1 qe ds = q

r 2 F ds W = r 1 qe ds = q Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study

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

1 of 7 9/14/15, 10:27 AM

1 of 7 9/14/15, 10:27 AM Stuent: Dte: Instructor: Doug Ensle Course: MAT117 1 Applie Sttistics - Ensle Assignment: Online 6 - Section 3.2 1 of 7 9/14/15, 1:27 AM 1. 2 of 7 9/14/15, 1:27 AM The t on the right shows the percent

More information

Sect 8.3 Triangles and Hexagons

Sect 8.3 Triangles and Hexagons 13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed two-dimensionl geometric figure consisting of t lest three line segments for its

More information

Geometry Notes SIMILAR TRIANGLES

Geometry Notes SIMILAR TRIANGLES Similr Tringles Pge 1 of 6 SIMILAR TRIANGLES Objectives: After completing this section, you shoul be ble to o the following: Clculte the lengths of sies of similr tringles. Solve wor problems involving

More information

Pythagoras theorem and trigonometry (2)

Pythagoras theorem and trigonometry (2) HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

More information

Lesson 10. Parametric Curves

Lesson 10. Parametric Curves Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find

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

Physics 101: Lecture 15 Rolling Objects Today s lecture will cover Textbook Chapter

Physics 101: Lecture 15 Rolling Objects Today s lecture will cover Textbook Chapter Exm II Physics 101: Lecture 15 olling Objects Tody s lecture will cover Textbook Chpter 8.5-8.7 Physics 101: Lecture 15, Pg 1 Overview eview K rottion = ½ I w Torque = Force tht cuses rottion t = F r sin

More information

1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE

1 PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE PRECALCULUS READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the smples, work the problems, then check your nswers t the end of ech topic. If you don t get the nswer given, check your work nd look

More information

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING

TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING TITLE THE PRINCIPLES OF COIN-TAP METHOD OF NON-DESTRUCTIVE TESTING Sung Joon Kim*, Dong-Chul Che Kore Aerospce Reserch Institute, 45 Eoeun-Dong, Youseong-Gu, Dejeon, 35-333, Kore Phone : 82-42-86-231 FAX

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

For the Final Exam, you will need to be able to:

For the Final Exam, you will need to be able to: Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny

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

Why is the NSW prison population falling?

Why is the NSW prison population falling? NSW Bureu of Crime Sttistics nd Reserch Bureu Brief Issue pper no. 80 September 2012 Why is the NSW prison popultion flling? Jcqueline Fitzgerld & Simon Corben 1 Aim: After stedily incresing for more thn

More information

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT

COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE. Skandza, Stockholm ABSTRACT COMPARISON OF SOME METHODS TO FIT A MULTIPLICATIVE TARIFF STRUCTURE TO OBSERVED RISK DATA BY B. AJNE Skndz, Stockholm ABSTRACT Three methods for fitting multiplictive models to observed, cross-clssified

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

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

Chapter 8 - Practice Problems 1

Chapter 8 - Practice Problems 1 Chpter 8 - Prctice Problems 1 MULTIPLE CHOICE. Choose the one lterntive tht best completes the sttement or nswers the question. A hypothesis test is to be performed. Determine the null nd lterntive hypotheses.

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

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

Plotting and Graphing

Plotting and Graphing Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured

More information

Solutions to Section 1

Solutions to Section 1 Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this

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

Introduction to Mathematical Reasoning, Saylor 111

Introduction to Mathematical Reasoning, Saylor 111 Frction versus rtionl number. Wht s the difference? It s not n esy question. In fct, the difference is somewht like the difference between set of words on one hnd nd sentence on the other. A symbol is

More information

AP STATISTICS SUMMER MATH PACKET

AP STATISTICS SUMMER MATH PACKET AP STATISTICS SUMMER MATH PACKET This pcket is review of Algebr I, Algebr II, nd bsic probbility/counting. The problems re designed to help you review topics tht re importnt to your success in the clss.

More information

The IS-LM Model. Underlying model due to John Maynard Keynes Model representation due to John Hicks Used to make predictions about

The IS-LM Model. Underlying model due to John Maynard Keynes Model representation due to John Hicks Used to make predictions about The S-LM Model Underlying model due to John Mynrd Keynes Model representtion due to John Hicks Used to mke predictions bout nterest rtes Aggregte spending ( ggregte output) mportnt ssumption: price is

More information

Diffraction and Interference of Light

Diffraction and Interference of Light rev 12/2016 Diffrction nd Interference of Light Equipment Qty Items Prt Number 1 Light Sensor CI-6504 1 Rotry Motion Sensor CI-6538 1 Single Slit Set OS-8523 1 Multiple Slit Set OS-8523 1 Liner Trnsltor

More information

Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt

Newton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt Newton s Three Lws For couple centuries before Einstein, Newton s Lws were the bsic principles of Physics. These lws re still vlid nd they re the bsis for much engineering nlysis tody. Forml sttements

More information

The Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof.

The Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof. The Prllelogrm Lw Objective: To tke students through the process of discovery, mking conjecture, further explortion, nd finlly proof. I. Introduction: Use one of the following Geometer s Sketchpd demonstrtion

More information

Lecture 1: Introduction to Economics

Lecture 1: Introduction to Economics E111 Introduction to Economics ontct detils: E111 Introduction to Economics Ginluigi Vernsc Room: 5B.217 Office hours: Wednesdy 2pm to 4pm Emil: gvern@essex.c.uk Lecture 1: Introduction to Economics I

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

Uniform convergence and its consequences

Uniform convergence and its consequences Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,

More information

Complementary Coffee Cups

Complementary Coffee Cups Complementry Coffee Cups Thoms Bnchoff Tom Bnchoff (Thoms Bnchoff@brown.edu) received his B.A. from Notre Dme nd his Ph.D. from the University of Cliforni, Berkeley. He hs been teching t Brown University

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

Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers

Redistributing the Gains from Trade through Non-linear. Lump-sum Transfers Redistributing the Gins from Trde through Non-liner Lump-sum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lump-sum trnsfer rules to redistribute the gins from

More information