Review. Scan Conversion. Rasterizing Polygons. Rasterizing Polygons. Triangularization. Convex Shapes. Utah School of Computing Spring 2013

Size: px
Start display at page:

Download "Review. Scan Conversion. Rasterizing Polygons. Rasterizing Polygons. Triangularization. Convex Shapes. Utah School of Computing Spring 2013"

Transcription

1 Uth Shool of Computing Spring 2013 Review Leture Set 4 Sn Conversion CS5600 Computer Grphis Spring 2013 Line rsteriztion Bsi Inrementl Algorithm Digitl Differentil Anlzer Rther thn solve line eqution t eh piel, use evlution of line from previous piel nd slope to pproimte line eqution Bresenhm Use integer rithmeti nd midpoint disrimintor to test etween two possile piels (over vs. over-nd-up) Rsterizing Polgons In intertive grphis, polgons rule the world Two min resons: Lowest ommon denomintor for surfes Cn represent n surfe with ritrr ur Splines, mthemtil funtions, volumetri isosurfes Mthemtil simpliit lends itself to simple, regulr rendering lgorithms Like those we re out to disuss Suh lgorithms emed well in hrdwre Rsterizing Polgons Tringle is the miniml unit of polgon All polgons n e roken up into tringles Conve, onve, omple Tringles re gurnteed to e: Plnr Conve Wht etl does it men to e onve? Conve Shpes A two-dimensionl shpe is onve if nd onl if ever line segment onneting two points on the oundr is entirel ontined. Conve polgons esil tringulted Tringulriztion Conve polgons present hllenge Computer Grphis CS5600

2 Uth Shool of Computing Spring 2013 Rsterizing Tringles Sn Conversion Intertive grphis hrdwre sometimes uses edge wlking or edge eqution tehniques for rsterizing tringles Intertive grphis hrdwre more ommonl uses rentri oordintes for rsterizing tringles In snline rendering surfes re projeted on the sreen nd spe filling rsterizing lgorithms re used to fill in the olor. Color vlues from light re pproimted. Tringle Rsteriztion Issues Etl whih piels should e lit? A: Those piels inside the tringle edges Wht out piels etl on the edge? Drw them: order of tringles mtters (it shouldn t) Don t drw them: gps possile etween tringles We need onsistent (if ritrr) rule Emple: drw piels on left nd ottom edge, ut not on right or top edge Tringle Rsteriztion Issues Sliver Tringle Rsteriztion Issues Moving Slivers Tringle Rsteriztion Issues Shred Edge Ordering Computer Grphis CS5600

3 Uth Shool of Computing Spring 2013 Computer Grphis CS5600

4 Uth Shool of Computing Spring 2013 Edge Equtions How do we know if it s inside? An edge eqution is simpl the eqution of the line defining tht edge Q: Wht is the impliit eqution of line? A: A + B + C = 0 Q: Given point (,), wht does plugging & into this eqution tell us? A: Whether the point is: On the line: A + B + C = 0 Aove the line: A + B + C > 0 Below the line: A + B + C < 0 Edge Equtions Edge equtions thus define two hlf-spes: Edge Equtions And tringle n e defined s the intersetion of three positive hlf-spes: A 3 + B 3 + C 3 < 0 A 3 + B 3 + C 3 > 0 A 2 + B 2 + C 2 < 0 A 2 + B 2 + C 2 > 0 A 1 + B 1 + C 1 > 0 A 1 + B 1 + C 1 < 0 Edge Equtions So simpl turn on those piels for whih ll edge equtions evlute to > 0: Computer Grphis CS5600

5 Uth Shool of Computing Spring 2013 Sweep-line Sweep-line: Notes Bsi ide: Drw edges vertill Interpolte olors up/down edges Fill in horizontl spns for eh snline At eh snline, interpolte edge olors ross spn Order three tringle verties in nd Find middle point in dimension nd ompute if it is to the left or right of polgon. Also ould e flt top or flt ottom tringle We know where left nd right edges re. Proeed from top snline downwrds (nd other w too) Fill eh spn Until ottom/top verte is rehed Advntge: n e mde ver fst Disdvntges: Lots of finik speil ses Sweep line: Disdvntges Frtionl offsets: Be reful when interpolting olor vlues! Bewre of gps etween djent edges Bewre of dupliting shred edges Computer Grphis CS5600

6 Uth Shool of Computing Spring 2013 Polgon Sn Conversion Intersetion Points Other points in the spn Computer Grphis CS5600

7 Uth Shool of Computing Spring 2013 Determining Inside vs. Outside Verties nd Prit Use the odd-prit rule Set prit even initill Invert prit t eh intersetion point Drw piels when prit is odd, do not drw when it is even How do we ount verties, i.e., do we invert prit when verte flls etl on sn line? Sn line????? How do we ount the interseting verte in the prit omputtion? Verties nd Prit We need to either ount it 0 times, or 2 times to keep prit orret. Wht out: Sn line We need to ount this verte one????? Verties nd Prit If we ount verte s one intersetion, the seond polgon gets drwn orretl, ut the first does not. If we ount verte s zero or two intersetions, the first polgon gets drwn orretl, ut the seond does not. How do we hndle this? Count onl verties tht re the min verte for tht line Verties nd Prit How do we del with horizontl edges???? Horizontl Edges Both ses now work orretl Don t ount their verties in the prit lultion! Computer Grphis CS5600

8 Uth Shool of Computing Spring 2013 Effet of onl ounting min : Top spns of polgons re not drwn Top Spns of Polgons Shred Polgon Edges Drw Lst polgon wins Wht if two polgons shre n edge? If two polgons shre this edge, it is not prolem. Wht out if this is the onl polgon with tht edge? Ornge lst Solution: Spn is losed on left nd open on right ( min < m ) Sn lines losed on ottom nd open on top ( min < m ) Blue lst Generl Piel Ownership Rule Hlf-plne rule: A oundr piel (whose enter flls etl on n edge) is not onsidered prt of primitive if the hlf plne formed the edge nd ontining the primitive lies to the left or elow the edge. Applies to ritrr polgons s well s to retngles... Shred edge Consequenes: Spns re missing the right-most piel Eh polgon is missing its top-most spn Generl Polgon Rsteriztion Consider the following polgon: D B A F C How do we know whether given piel on the snline is inside or outside the polgon? E Inside-Outside Points Polgon Rsteriztion Polgon Rsteriztion Inside-Outside Points Computer Grphis CS5600

9 Uth Shool of Computing Spring 2013 Generl Polgon Rsteriztion Bsi ide: use prit test for eh snline edgecnt = 0; for eh piel on snline (l to r) if (oldpiel->newpiel rosses edge) edgecnt ++; // drw the piel if edgecnt odd if (edgecnt % 2) setpiel(piel); Generl Polgon Rsteriztion Count our verties refull G F I H E C J D A B Fster Polgon Rsteriztion How n we optimize the ode? for eh snline edgecnt = 0; for eh piel on snline (l to r) if (oldpiel->newpiel rosses edge) edgecnt ++; // drw the piel if edgecnt odd if (edgecnt % 2) setpiel(piel); Big ost: testing piels ginst eh edge Solution: tive edge tle (AET) Ative Edge Tle Ative Edge Tle Ide: Edges interseting given snline re likel to interset the net snline The order of edge intersetions doesn t hnge muh from snline to snline Preproess: Sort on Y Edge Tle Y m, t min,slope Computer Grphis CS5600

10 Uth Shool of Computing Spring 2013 Ative Edge Tle Ative Edge Tle Preproess: Sort on Y Edge Tle Preproess: Sort on Y Edge Tle AB: Y m, t min,slope Y m, t min,slope AB: 3 7-5/2 CB: Ative Edge Tle Ative Edge Tle Preproess: Sort on Y Edge Tle Y m, t min,slope AB: 3 7-5/2 CB: 5 7 6/4 CD: Preproess: Sort on Y Edge Tle Y m, t min,slope AB: 3 7-5/2 CB: 5 7 6/4 CD: DE: Ative Edge Tle Ative Edge Tle Preproess: Sort on Y Edge Tle Y m, t min,slope AB: 3 7-5/2 CB: 5 7 6/4 CD: DE: /4 EF: Preproess: Sort on Y Edge Tle Y m, t min,slope AB: 3 7-5/2 CB: 5 7 6/4 CD: DE: /4 EF: 9 7-5/2 FA: Computer Grphis CS5600

11 Uth Shool of Computing Spring 2013 Ative Edge Tle Ative Edge Tle Preproess: Sort on Y Edge Tle Y m, t min,slope AB: 3 7-5/2 CB: 5 7 6/4 CD: DE: /4 EF: 9 7-5/2 FA: Preproess: Sort on Y Edge Tle Y m, t min,slope AB: 3 7-5/2 CB: 5 7 6/4 CD: DE: /4 EF: 9 7-5/2 FA: Wht out Y min? Ative Edge Tle Ative Edge Tle Preproess: Sort on Y Edge Tle Y m, min,slope Algorithm: snline from ottom to top Sort ll edges their minimum oord (lst slide) Strting t smllest Y oord with in entr in edge tle For eh snline: Add edges with Y min = Y (move edges in edge tle to AET) Retire edges with Y m < Y (ompleted edges) Sort edges in AET intersetion Wlk from left to right, setting piels prit rule Inrement snline Relulte edge intersetions (how?) Stop when Y > Y m for edge tle nd AET is empt Ative Edge Tle Ative Edge Tle Emple Algorithm: snline from ottom to top Sort ll edges their minimum oord (lst slide) Strting t smllest Y oord with in entr in edge tle For eh snline: 1. Add edges with Y min = Y (move edges in edge tle to AET) 2. Retire edges with Y m < Y (ompleted edges) 3. Sort edges in AET intersetion 4. Wlk from left to right, setting piels prit rule 5. Inrement snline 6. Relulte edge intersetions (how?) For ever non-vertil edge in the AET updte for the new (lulte the net intersetion of the edge with the sn line). Stop when Y > Y m for edge tle nd AET is empt Emple of n AET ontining edges {FA, EF, DE, CD} on sn line 8: 1. : ( = 8) Get edges from ET uket (none in this se, = 8 hs no entr) 2. : Remove from the AET n entries where m = (none here) 3. : sort X 4. : Drw sn line. To hndle multiple edges, group in pirs: {FA,EF}, {DE,CD} 5. : = +1 ( = 8+1 = 9) 6. : Updte for non-vertil edges, s in simple line drwing. Snline Y vl Current X Slope (FvDFH pges 92, 99) Computer Grphis CS5600

12 Uth Shool of Computing Spring 2013 Ative Edge Tle Emple (ont.) Tringles (ont.) 1. : ( = 9) Get edges from ET uket (none in this se, = 9 hs no entr in ET) Sn line 9 shown in fig 3.28 elow 2. : Remove from the AET n entries with m = (remove FA, EF) 3. : Sort X 4. : Drw sn line etween {DE, CD} 5. : = +1 = : Updte in {DE, CD} 7. : ( = 10) (Sn line 10 shown in fig 3.28 elow) 8. And so on Rsteriztion lgorithms n tke dvntge of tringle properties Grphis hrdwre is optimized for tringles Beuse tringle drwing is so fst, mn sstems will sudivide polgons into tringles prior to sn onversion Y vl Current X Slope (FvDFH pges 92, 99) Wh re Brentri oordintes useful? Brentri Coordintes For n point, if the rentri representtion of tht point:,, < 1 Also,, n e used s mss funtion ross the surfe of tringle to e used for interpoltion. This is used to interpolte normls ross the surfe of tringle to mke polgon surfes look rounder. Consider tringle defined three points,, nd. Define new oordinte sstem in whih is the origin, nd define the oordinte sstem sis vetors Note tht the oordinte sstem will e nonorthogonl. Brentri Coordintes Brentri Coordintes - - With this new oordinte sstem, n point n e written s: rerrnging terms, we get: let then p ( ) ( ) p (1 ) ( 1 ) p Computer Grphis CS5600

13 Uth Shool of Computing Spring 2013 = 2 = 1 =0 = -1 Brentri Coordintes - =-1 - =0 =1 =2 Brentri Coordintes Now n point in the plne n e represented using its rentri oordintes If p then the point lies somewhere in the tringle Brentri Coordintes Computing Brentri Coordintes If one of the oordintes is zero nd the other two re etween 0 nd 1, the point is on n edge If two oordintes re zero nd the other is one, the point is t verte. The rentri oordinte is the signed sled distne from the point to the line pssing through the other two tringle points Impliit form etween two points (,) nd (,) f (, ) ( ) ( ) =-1 =0 =1 f (, ) ( ) ( ) - d=1 = 2 Brentri Coordintes =-1 =0 =1 =2 PDF Slides = 1 - =0 - = -1 Computer Grphis CS5600

14 Uth Shool of Computing Spring 2013 Computer Grphis CS5600 Computing Brentri Coordintes To ompute the rentri oordintes of point: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 Brentri Coordinte Applet Rsterize This! (Rsteriztion intuition) When we render tringle we wnt to determine if piel is within tringle. (rentri oords) Clulte the olor of the piel (use rentri oors). Drw the piel. Repet until the tringle is ppropritel filled. Rsteriztion Pseudo Code Rsteriztion Rsteriztion

15 Uth Shool of Computing Spring 2013 Rsteriztion Bounding Bo Ym, Xm Ym, Xm Y???, Xmin Y???, Xmin Ymin, X??? Ymin, X??? weighted omintion of verties P P 1 0,, 1 onve omintion of points Brentri Coordintes 1 P2 P3 P 3 P 1 P (1,0,0) 0 (0,0,1) 0. 5 P 2 1 (0,1,0) Brentri Coordintes for I nterpoltion how to ompute,,? use iliner interpoltion or plne equtions interpolte,, z d... one omputed, use to interpolte n # of prmeters from their verte vlues r r1 r2 r3 g g g et. 1 2 g3 Interpolttion: Gourud Shding Gourud Shding Snline Alg need liner funtion over tringle tht ields originl verte olors t verties use rentri oordintes for this ever piel in interior gets olors resulting from miing olors of verties with weights orresponding to rentri oordintes olor t piels is ffine omintion of olors t verties lgorithm modif snline lgorithm for polgon sn-onversion : linerl interpolte olors long edges of tringle to otin olors for endpoints of spn of piels linerl interpolte olors from these endpoints within the snline Color( 1 Color( ) Color( 1 ) : ) Color( ) 3 X min X X X X X * C m ur m ur Cmin 1 * m X min X m X min X ur m X m Computer Grphis CS5600

16 Uth Shool of Computing Spring 2013 Filling Tehniques Another pproh to polgon fill is using filling tehnique, rther thn sn onversion Pik point inside the polgon, then fill neighoring piels until the polgon oundr is rehed Boundr Fill Approh: Drw polgon oundr in the frme uffer Determine n interior point Strting t the given point, do If the point is not the oundr olor or the fill olor Set this piel to the fill olor Propgte to the piel s neighors nd ontinue Filling Tehniques Flood Fill Approh: Set ll interior piels to ertin olor The oundr n e n other olor Pik n interior point nd set it to the polgon olor Propgte to neighors, s long s the neighor is the interior olor This is used for regions with multi-olored oundries Propgting to Neighors Most frequentl used pprohes: 4-onneted re 8-onneted re Region to e filled 4-onneted 8-onneted Fill lgorithms hve potentil prolems E.g., 4-onneted re fill: Fill Prolems Fill Prolems Similrl, 8-onneted n lek over to nother polgon Strting point Fill omplete Strting point Fill omplete Another prolem: the lgorithm is highl reursive Cn use stk of spns to redue mount of reursion Computer Grphis CS5600

17 Uth Shool of Computing Spring 2013 Pttern Filling Often we wnt to fill region with pttern, not just olor Define n n m pimp (or itmp) tht we wish to replite ross the region 54 pimp How do ou determine the nhor point A point on the polgon Left-most point? The pttern will move with the polgon Diffiult to deide the right nhor point Sreen (or window) origin Esier to determine nhor point The pttern does not move with the ojet Pttern Filling Ojet to e ptterned Finl ptterned ojet Pttern Filling How do we determine whih olor to olor point in the ojet? Use the MOD funtion to tile the pttern ross the polgon For point (, ) Use the pttern olor loted t ( MOD m, MOD n) Pttern Emple For the pttern shown, wht olor does the piel t lotion (235, 168) get olored, ssuming the pttern is nhored t the lower left orner of the ojet? Pttern ??? Pttern Emple Pttern Emple 168??? The pttern piel (2, 0) should mp to sreen lotion (235, 168) Pttern The pttern is Need to find the reltive distne to the point to drw: X = ( ) = 10 Y = ( ) = 5 (2, 0) Let s mp the pttern onto the polgon nd see ??? Net figure out whih pttern piel orresponds to this sreen piel: X pttern = 10 MOD 4 = 2 Y pttern = 5 MOD 5 = 0 (0, 0) pttern lotion Computer Grphis CS5600

18 Uth Shool of Computing Spring 2013 The End Leture Set 4 Sn Conversion 103 Computer Grphis CS5600

excenters and excircles

excenters and excircles 21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling

More information

Lesson 2.1 Inductive Reasoning

Lesson 2.1 Inductive Reasoning Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,

More information

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127 ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not

More information

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed

More information

How to Graphically Interpret the Complex Roots of a Quadratic Equation

How to Graphically Interpret the Complex Roots of a Quadratic Equation Universit of Nersk - Linoln DigitlCommons@Universit of Nersk - Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7-007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen

More information

SECTION 7-2 Law of Cosines

SECTION 7-2 Law of Cosines 516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished

More information

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn

Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?

More information

Ratio and Proportion

Ratio and Proportion Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty

More information

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5. . Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry

More information

Lesson 18.2: Right Triangle Trigonometry

Lesson 18.2: Right Triangle Trigonometry Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to non-right tringles (hpter

More information

1. Area under a curve region bounded by the given function, vertical lines and the x axis.

1. Area under a curve region bounded by the given function, vertical lines and the x axis. Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given

More information

Quick Guide to Lisp Implementation

Quick Guide to Lisp Implementation isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled S-epressions. The representtion o n S-epression n e roken into two piees, the

More information

Thank you for participating in Teach It First!

Thank you for participating in Teach It First! Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you

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

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

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

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

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

Maximum area of polygon

Maximum area of polygon Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is

More information

Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.

Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb. - Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help

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

8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary 8-1 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted

More information

Chapter. Contents: A Constructing decimal numbers

Chapter. Contents: A Constructing decimal numbers Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

More information

Words Symbols Diagram. abcde. a + b + c + d + e

Words Symbols Diagram. abcde. a + b + c + d + e Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

More information

Right-angled triangles

Right-angled triangles 13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Right-ngled tringles Syllus referene Mesurement 4 Right-ngled tringles

More information

Intersection Problems

Intersection Problems Intersetion Prolems Determine pirs of interseting ojets? C A B E D Complex shpes forme y oolen opertions: interset, union, iff. Collision etetion in rootis n motion plnning. Visiility, olusion, renering

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

MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

More information

Calculating Principal Strains using a Rectangular Strain Gage Rosette

Calculating Principal Strains using a Rectangular Strain Gage Rosette Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used

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

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

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m

Vectors Summary. Projection vector AC = ( Shortest distance from B to line A C D [OR = where m1. and m . Slr prout (ot prout): = osθ Vetors Summry Lws of ot prout: (i) = (ii) ( ) = = (iii) = (ngle etween two ientil vetors is egrees) (iv) = n re perpeniulr Applitions: (i) Projetion vetor: B Length of projetion

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

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

The area of the larger square is: IF it s a right triangle, THEN + =

The area of the larger square is: IF it s a right triangle, THEN + = 8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht

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

Heron s Formula for Triangular Area

Heron s Formula for Triangular Area Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,

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

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001

CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001 CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic

More information

50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS

50 MATHCOUNTS LECTURES (10) RATIOS, RATES, AND PROPORTIONS 0 MATHCOUNTS LECTURES (0) RATIOS, RATES, AND PROPORTIONS BASIC KNOWLEDGE () RATIOS: Rtios re use to ompre two or more numers For n two numers n ( 0), the rtio is written s : = / Emple : If 4 stuents in

More information

GRADE 4. Fractions WORKSHEETS

GRADE 4. Fractions WORKSHEETS GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh

More information

SOLVING EQUATIONS BY FACTORING

SOLVING EQUATIONS BY FACTORING 316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting

More information

Interior and exterior angles add up to 180. Level 5 exterior angle

Interior and exterior angles add up to 180. Level 5 exterior angle 22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, efinitions

More information

Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.

Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names. 1- The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor Self-Chek Quiz Three squres with

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

Section 5-5 Solving Right Triangles*

Section 5-5 Solving Right Triangles* 5-5 Solving Right Tringles 379 79. Geometry. The re of retngulr n-sided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml

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

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

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.

Exponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep. Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:

More information

SOLVING QUADRATIC EQUATIONS BY FACTORING

SOLVING QUADRATIC EQUATIONS BY FACTORING 6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving

More information

Variable Dry Run (for Python)

Variable Dry Run (for Python) Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 20-50 minutes

More information

PLWAP Sequential Mining: Open Source Code

PLWAP Sequential Mining: Open Source Code PL Sequentil Mining: Open Soure Code C.I. Ezeife Shool of Computer Siene University of Windsor Windsor, Ontrio N9B 3P4 ezeife@uwindsor. Yi Lu Deprtment of Computer Siene Wyne Stte University Detroit, Mihign

More information

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.

Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right. Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction

More information

8. Hyperbolic triangles

8. Hyperbolic triangles 8. Hyperoli tringles Note: This yer, I m not doing this mteril, prt from Pythgors theorem, in the letures (nd, s suh, the reminder isn t exminle). I ve left the mteril s Leture 8 so tht (i) nyody interested

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

Volumes by Cylindrical Shells: the Shell Method

Volumes by Cylindrical Shells: the Shell Method olumes Clinril Shells: the Shell Metho Another metho of fin the volumes of solis of revolution is the shell metho. It n usull fin volumes tht re otherwise iffiult to evlute using the Dis / Wsher metho.

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

Assembly & Installation Instructions: Impulse 2-Leg Workcenter IM3E48-78-30D-X

Assembly & Installation Instructions: Impulse 2-Leg Workcenter IM3E48-78-30D-X Assemly & Instlltion Instrutions: Impulse -Leg Workenter IM3E48-78-30D-X Prts Inluded, Frme Set A Impulse Bse Assemly Qty: D M5 0 mm Phillips Wood Srews Qty: 8 M0 35 mm Soket ed Srews Qty: 8 Box E Swith

More information

Chapter. Fractions. Contents: A Representing fractions

Chapter. Fractions. Contents: A Representing fractions Chpter Frtions Contents: A Representing rtions B Frtions o regulr shpes C Equl rtions D Simpliying rtions E Frtions o quntities F Compring rtion sizes G Improper rtions nd mixed numers 08 FRACTIONS (Chpter

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

BUSINESS PROCESS MODEL TRANSFORMATION ISSUES The top 7 adversaries encountered at defining model transformations

BUSINESS PROCESS MODEL TRANSFORMATION ISSUES The top 7 adversaries encountered at defining model transformations USINESS PROCESS MODEL TRANSFORMATION ISSUES The top 7 dversries enountered t defining model trnsformtions Mrion Murzek Women s Postgrdute College for Internet Tehnologies (WIT), Institute of Softwre Tehnology

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting

More information

CS 316: Gates and Logic

CS 316: Gates and Logic CS 36: Gtes nd Logi Kvit Bl Fll 27 Computer Siene Cornell University Announements Clss newsgroup reted Posted on we-pge Use it for prtner finding First ssignment is to find prtners P nd N Trnsistors PNP

More information

Lecture 3: orientation. Computer Animation

Lecture 3: orientation. Computer Animation Leture 3: orienttion Computer Animtion Mop tutoril sessions Next Thursdy (Feb ) Tem distribution: : - :3 - Tems 7, 8, 9 :3 - : - Tems nd : - :3 - Tems 5 nd 6 :3 - : - Tems 3 nd 4 Pper ssignments Pper ssignment

More information

Chapter 10 Geometry: Angles, Triangles and Distance

Chapter 10 Geometry: Angles, Triangles and Distance hpter 10 Geometry: ngles, Tringles nd Distne In setion 1 we egin y gthering together fts out ngles nd tringles tht hve lredy een disussed in previous grdes. This time the ide is to se student understnding

More information

The art of Paperarchitecture (PA). MANUAL

The art of Paperarchitecture (PA). MANUAL The rt of Pperrhiteture (PA). MANUAL Introution Pperrhiteture (PA) is the rt of reting three-imensionl (3D) ojets out of plin piee of pper or ror. At first, esign is rwn (mnully or printe (using grphil

More information

Equivalence Checking. Sean Weaver

Equivalence Checking. Sean Weaver Equivlene Cheking Sen Wever Equivlene Cheking Given two Boolen funtions, prove whether or not two they re funtionlly equivlent This tlk fouses speifilly on the mehnis of heking the equivlene of pirs of

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

UNCORRECTED SAMPLE PAGES

UNCORRECTED SAMPLE PAGES 6 Chpter Length, re, surfe re n volume Wht you will lern 6A Length n perimeter 6B Cirumferene of irles n perimeter of setors 6C Are of qurilterls n tringles 6D Are of irles 6E Perimeter n re of omposite

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

Student Access to Virtual Desktops from personally owned Windows computers

Student Access to Virtual Desktops from personally owned Windows computers Student Aess to Virtul Desktops from personlly owned Windows omputers Mdison College is plesed to nnoune the ility for students to ess nd use virtul desktops, vi Mdison College wireless, from personlly

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

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

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent! MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more

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

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;

Assuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C; B-26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndom-numer genertor supplied s stndrd with ll computer systems Stn Kelly-Bootle,

More information

Homework 3 Solutions

Homework 3 Solutions CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

More information

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2

. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2 7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6

More information

Enterprise Digital Signage Create a New Sign

Enterprise Digital Signage Create a New Sign Enterprise Digitl Signge Crete New Sign Intended Audiene: Content dministrtors of Enterprise Digitl Signge inluding stff with remote ess to sign.pitt.edu nd the Content Mnger softwre pplition for their

More information

Inter-domain Routing

Inter-domain Routing COMP 631: COMPUTER NETWORKS Inter-domin Routing Jsleen Kur Fll 2014 1 Internet-sle Routing: Approhes DV nd link-stte protools do not sle to glol Internet How to mke routing slle? Exploit the notion of

More information

1 GSW IPv4 Addressing

1 GSW IPv4 Addressing 1 For s long s I ve een working with the Internet protools, people hve een sying tht IPv6 will e repling IPv4 in ouple of yers time. While this remins true, it s worth knowing out IPv4 ddresses. Even when

More information

The Pythagorean Theorem

The Pythagorean Theorem The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully

More information

Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002

Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002 dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl

More information

If two triangles are perspective from a point, then they are also perspective from a line.

If two triangles are perspective from a point, then they are also perspective from a line. Mth 487 hter 4 Prtie Prolem Solutions 1. Give the definition of eh of the following terms: () omlete qudrngle omlete qudrngle is set of four oints, no three of whih re olliner, nd the six lines inident

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

Strong acids and bases

Strong acids and bases Monoprotic Acid-Bse Equiliri (CH ) ϒ Chpter monoprotic cids A monoprotic cid cn donte one proton. This chpter includes uffers; wy to fi the ph. ϒ Chpter 11 polyprotic cids A polyprotic cid cn donte multiple

More information

Brillouin Zones. Physics 3P41 Chris Wiebe

Brillouin Zones. Physics 3P41 Chris Wiebe Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction

More information

One Minute To Learn Programming: Finite Automata

One Minute To Learn Programming: Finite Automata Gret Theoreticl Ides In Computer Science Steven Rudich CS 15-251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge

More information

Released Assessment Questions, 2015 QUESTIONS

Released Assessment Questions, 2015 QUESTIONS Relesed Assessmet Questios, 15 QUESTIONS Grde 9 Assessmet of Mthemtis Ademi Red the istrutios elow. Alog with this ooklet, mke sure you hve the Aswer Booklet d the Formul Sheet. You my use y spe i this

More information

EXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE Real-World Connection EXAMPLE

EXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE Real-World Connection EXAMPLE - Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse hek Skills You

More information

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00

c b 5.00 10 5 N/m 2 (0.120 m 3 0.200 m 3 ), = 4.00 10 4 J. W total = W a b + W b c 2.00 Chter 19, exmle rolems: (19.06) A gs undergoes two roesses. First: onstnt volume @ 0.200 m 3, isohori. Pressure inreses from 2.00 10 5 P to 5.00 10 5 P. Seond: Constnt ressure @ 5.00 10 5 P, isori. olume

More information

Scan Tool Software Applications Installation and Updates

Scan Tool Software Applications Installation and Updates Sn Tool Softwre Applitions Instlltion nd Updtes Use this doument to: Unlok softwre pplitions on Sn Tool Instll new softwre pplitions on Sn Tool Instll the NGIS Softwre Suite pplitions on Personl Computer

More information

End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.

End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below. End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.

More information

Angles and Triangles

Angles and Triangles nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir

More information

CHAPTER 31 CAPACITOR

CHAPTER 31 CAPACITOR . Given tht Numer of eletron HPTER PITOR Net hrge Q.6 9.6 7 The net potentil ifferene L..6 pitne v 7.6 8 F.. r 5 m. m 8.854 5.4 6.95 5 F... Let the rius of the is R re R D mm m 8.85 r r 8.85 4. 5 m.5 m

More information

Active Directory Service

Active Directory Service In order to lern whih questions hve een nswered orretly: 1. Print these pges. 2. Answer the questions. 3. Send this ssessment with the nswers vi:. FAX to (212) 967-3498. Or. Mil the nswers to the following

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

Square Roots Teacher Notes

Square Roots Teacher Notes Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

More information

1 Fractions from an advanced point of view

1 Fractions from an advanced point of view 1 Frtions from n vne point of view We re going to stuy frtions from the viewpoint of moern lger, or strt lger. Our gol is to evelop eeper unerstning of wht n men. One onsequene of our eeper unerstning

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