11.5 Graphs of Polar Equations

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "11.5 Graphs of Polar Equations"

Transcription

1 9 Applications of Tigonomet.5 Gaphs of Pola Equations In this section, we discuss how to gaph equations in pola coodinates on the ectangula coodinate plane. Since an given point in the plane has infinitel man diffeent epesentations in pola coodinates, ou Fundamental Gaphing Pinciple in this section is not as clean as it was fo gaphs of ectangula equations on page. We state it below fo completeness. The Fundamental Gaphing Pinciple fo Pola Equations The gaph of an equation in pola coodinates is the set of points which satisf the equation. That is, a point P, is on the gaph of an equation if and onl if thee is a epesentation of P, sa,, such that and satisf the equation. Ou fist eample focuses on the some of the moe stuctuall simple pola equations. Eample.5.. Gaph the following pola equations.. =. =. = 5. = Solution. In each of these equations, onl one of the vaiables and is pesent making the othe vaiable fee. This makes these gaphs easie to visualize than othes.. In the equation =, is fee. The gaph of this equation is, theefoe, all points which have a pola coodinate epesentation,, fo an choice of. Gaphicall this tanslates into tacing out all of the points units awa fom the oigin. This is eactl the definition of cicle, centeed at the oigin, with a adius of. > 0 < 0 In =, is fee The gaph of =. Once again we have being fee in the equation =. Plotting all of the points of the fom, gives us a cicle of adius centeed at the oigin. See the discussion in Eample.. numbe a.

2 .5 Gaphs of Pola Equations 97 < 0 > 0 In =, is fee The gaph of =. In the equation = 5, is fee, so we plot all of the points with pola epesentation, 5. What we find is that we ae tacing out the line which contains the teminal side of = 5 when plotted in standad position. < 0 = 5 = 0 > 0 In = 5, is fee The gaph of = 5. As in the pevious eample, the vaiable is fee in the equation =. Plotting, fo vaious values of shows us that we ae tacing out the -ais.

3 98 Applications of Tigonomet > 0 = 0 = < 0 In =, is fee The gaph of = Hopefull, ou epeience in Eample.5. makes the following esult clea. Theoem.8. Gaphs of Constant and : Suppose a and α ae constants, a 0. The gaph of the pola equation = a on the Catesian plane is a cicle centeed at the oigin of adius a. The gaph of the pola equation = α on the Catesian plane is the line containing the teminal side of α when plotted in standad position. Suppose we wish to gaph = cos. A easonable wa to stat is to teat as the independent vaiable, as the dependent vaiable, evaluate = f at some fiendl values of and plot the esulting points. We geneate the table below. = cos, 0, 0, 0 0,,, 5, 5 0 0, 7, 7, Fo a eview of these concepts and this pocess, see Sections. and..

4 .5 Gaphs of Pola Equations 99 Despite having nine odeed pais, we get onl fou distinct points on the gaph. Fo this eason, we emplo a slightl diffeent stateg. We gaph one ccle of = cos on the -plane and use it to help gaph the equation on the -plane. We see that as anges fom 0 to, anges fom to 0. In the -plane, this means that the cuve stats units fom the oigin on the positive -ais = 0 and gaduall etuns to the oigin b the time the cuve eaches the -ais =. The aows dawn in the figue below ae meant to help ou visualize this pocess. In the -plane, the aows ae dawn fom the -ais to the cuve = cos. In the -plane, each of these aows stats at the oigin and is otated though the coesponding angle, in accodance with how we plot pola coodinates. It is a less-pecise wa to geneate the gaph than computing the actual function values, but it is makedl faste. uns fom 0 to Net, we epeat the pocess as anges fom to. Hee, the values ae all negative. This means that in the -plane, instead of gaphing in Quadant II, we gaph in Quadant IV, with all of the angle otations stating fom the negative -ais. uns fom to < 0 so we plot hee As anges fom to, the values ae still negative, which means the gaph is taced out in Quadant I instead of Quadant III. Since the fo these values of match the values fo in The gaph looks eactl like = cos in the -plane, and fo good eason. At this stage, we ae just gaphing the elationship between and befoe we intepet them as pola coodinates, on the -plane.

5 90 Applications of Tigonomet [ ] 0,, we have that the cuve begins to etace itself at this point. Poceeding futhe, we find that when, we etace the potion of the cuve in Quadant IV that we fist taced out as. The eade is invited to veif that plotting an ange of outside the inteval [0, ] esults in etacting some potion of the cuve. We pesent the final gaph below. = cos in the -plane = cos in the -plane Eample.5.. Gaph the following pola equations.. = sin. = + cos. = 5 sin. = cos Solution.. We fist plot the fundamental ccle of = sin on the -aes. To help us visualize what is going on gaphicall, we divide up [0, ] into the usual fou subintevals [ 0, ] [, [ ] [, ],, and, ], and poceed as we did above. As anges fom 0 to, deceases fom to. This means that the cuve in the -plane stats units fom the oigin on the positive -ais and gaduall pulls in towads the oigin as it moves towads the positive -ais. uns fom 0 to The gaph of = cos looks suspiciousl like a cicle, fo good eason. See numbe a in Eample...

6 .5 Gaphs of Pola Equations 9 Net, as uns fom to, we see that inceases fom to. Picking up whee we left off, we gaduall pull the gaph awa fom the oigin until we each the negative -ais. uns fom to Ove the inteval [, ], we see that inceases fom to. On the -plane, the cuve sweeps out awa fom the oigin as it tavels fom the negative -ais to the negative -ais. uns fom to Finall, as takes on values fom to, deceases fom back to. The gaph on the -plane pulls in fom the negative -ais to finish whee we stated. uns fom to We leave it to the eade to veif that plotting points coesponding to values of outside the inteval [0, ] esults in etacing potions of the cuve, so we ae finished.

7 9 Applications of Tigonomet = sin in the -plane = sin in the -plane.. The fist thing to note when gaphing = + cos on the -plane ove the inteval [0, ] is that the gaph cosses though the -ais. This coesponds to the gaph of the cuve passing though the oigin in the -plane, and ou fist task is to detemine when this happens. Setting = 0 we get + cos = 0, o cos =. Solving fo in [0, ] gives = and =. Since these values of ae impotant geometicall, we beak the inteval [0, ] into si subintevals: [ 0, ] [,, ] [,, ], [, ] [,, ] [ and, ]. As anges fom 0 to, deceases fom to. Plotting this on the -plane, we stat units out fom the oigin on the positive -ais and slowl pull in towads the positive -ais. uns fom 0 to On the inteval [, ], deceases fom to 0, which means the gaph is heading into and will eventuall coss though the oigin. Not onl do we each the oigin when =, a theoem fom Calculus 5 states that the cuve hugs the line = as it appoaches the oigin. 5 The tangents at the pole theoem fom second semeste Calculus.

8 .5 Gaphs of Pola Equations 9 = On the inteval [, ], anges fom 0 to. Since 0, the cuve passes though the oigin in the -plane, following the line = and continues upwads though Quadant IV towads the positive -ais. Since is inceasing fom 0 to, the cuve pulls awa fom the oigin to finish at a point on the positive -ais. = Net, as pogesses fom to, anges fom to 0. Since 0, we continue ou gaph in the fist quadant, heading into the oigin along the line =. Recall that one wa to visualize plotting pola coodinates, with < 0 is to stat the otation fom the left side of the pole - in this case, the negative -ais. Rotating between and adians fom the negative -ais in this case detemines the egion between the line = and the -ais in Quadant IV.

9 9 Applications of Tigonomet = On the inteval [, ], etuns to positive values and inceases fom 0 to. We hug the line = as we move though the oigin and head towads the negative -ais. = As we ound out the inteval, we find that as uns though to, inceases fom out to, and we end up back whee we stated, units fom the oigin on the positive -ais. uns fom to

10 .5 Gaphs of Pola Equations 95 Again, we invite the eade to show that plotting the cuve fo values of outside [0, ] esults in etacing a potion of the cuve alead taced. Ou final gaph is below. = = = + cos in the -plane = + cos in the -plane. As usual, we stat b gaphing a fundamental ccle of = 5 sin in the -plane, which in this case, occus as anges fom 0 to. We patition ou inteval into subintevals to help us with the gaphing, namel [ 0, ] [,, ] [,, ] [ and, ]. As anges fom 0 to, inceases fom 0 to 5. This means that the gaph of = 5 sin in the -plane stats at the oigin and gaduall sweeps out so it is 5 units awa fom the oigin on the line =. 5 5 Net, we see that deceases fom 5 to 0 as uns though [, ], and futhemoe, is heading negative as cosses. Hence, we daw the cuve hugging the line = the -ais as the cuve heads to the oigin.

11 9 Applications of Tigonomet 5 5 As uns fom to, becomes negative and anges fom 0 to 5. Since 0, the cuve pulls awa fom the negative -ais into Quadant IV. 5 5 Fo, inceases fom 5 to 0, so the cuve pulls back to the oigin. 5 5

12 .5 Gaphs of Pola Equations 97 Even though we have finished with one complete ccle of = 5 sin, if we continue plotting beond =, we find that the cuve continues into the thid quadant! Below we pesent a gaph of a second ccle of = 5 sin which continues on fom the fist. The boed labels on the -ais coespond to the potions with matching labels on the cuve in the -plane We have the final gaph below = 5 sin in the -plane = 5 sin in the -plane. Gaphing = cos is complicated b the, so we solve to get = ± cos = ± cos. How do we sketch such a cuve? Fist off, we sketch a fundamental peiod of = cos which we have dotted in the figue below. When cos < 0, cos is undefined, so we don t have an values on the inteval,. On the intevals which emain, cos anges fom 0 to, inclusive. Hence, cos anges fom 0 to as well. 7 Fom this, we know = ± cos anges continuousl fom 0 to ±, espectivel. Below we gaph both = cos and = cos on the plane and use them to sketch the coesponding pieces of the cuve = cos in the -plane. As we have seen in ealie 7 Owing to the elationship between = and = ove [0, ], we also know p cos cos wheeve the fome is defined.

13 98 Applications of Tigonomet eamples, the lines = and =, which ae the zeos of the functions = ± cos, seve as guides fo us to daw the cuve as is passes though the oigin. = = = cos and = cos As we plot points coesponding to values of outside of the inteval [0, ], we find ouselves etacing pats of the cuve, 8 so ou final answe is below. = = = ± cos in the -plane = cos in the -plane A few emaks ae in ode. Fist, thee is no elation, in geneal, between the peiod of the function f and the length of the inteval equied to sketch the complete gaph of = f in the plane. As we saw on page 99, despite the fact that the peiod of f = cos is, we sketched the complete gaph of = cos in the -plane just using the values of as anged fom 0 to. In Eample.5., numbe, the peiod of f = 5 sin is, but in ode to obtain the complete gaph of = 5 sin, we needed to un fom 0 to. While man of the common pola gaphs can be gouped into families, 9 the authos tul feel that taking the time to wok though each gaph in the manne pesented hee is the best wa to not onl undestand the pola 8 In this case, we could have geneated the entie gaph b using just the plot = p cos, but gaphed ove the inteval [0, ] in the -plane. We leave the details to the eade. 9 Numbes and in Eample.5. ae eamples of limaçons, numbe is an eample of a pola ose, and numbe is the famous Lemniscate of Benoulli.

14 .5 Gaphs of Pola Equations 99 coodinate sstem, but also pepae ou fo what is needed in Calculus. Second, the smmet seen in the eamples is also a common occuence when gaphing pola equations. In addition to the usual kinds of smmet discussed up to this point in the tet smmet about each ais and the oigin, it is possible to talk about otational smmet. We leave the discussion of smmet to the Eecises. In ou net eample, we ae given the task of finding the intesection points of pola cuves. Accoding to the Fundamental Gaphing Pinciple fo Pola Equations on page 9, in ode fo a point P to be on the gaph of a pola equation, it must have a epesentation P, which satisfies the equation. What complicates mattes in pola coodinates is that an given point has infinitel man epesentations. As a esult, if a point P is on the gaph of two diffeent pola equations, it is entiel possible that the epesentation P, which satisfies one of the equations does not satisf the othe equation. Hee, moe than eve, we need to el on the Geomet as much as the Algeba to find ou solutions. Eample.5.. Find the points of intesection of the gaphs of the following pola equations.. = sin and = sin. = and = cos. = and = cos. = sin and = cos Solution.. Following the pocedue in Eample.5., we gaph = sin and find it to be a cicle centeed at the point with ectangula coodinates 0, with a adius of. The gaph of = sin is a special kind of limaçon called a cadioid. 0 = sin and = sin It appeas as if thee ae thee intesection points: one in the fist quadant, one in the second quadant, and the oigin. Ou net task is to find pola epesentations of these points. In 0 Pesumabl, the name is deived fom its esemblance to a stlized human heat.

15 950 Applications of Tigonomet ode fo a point P to be on the gaph of = sin, it must have a epesentation P, which satisfies = sin. If P is also on the gaph of = sin, then P has a possibl diffeent epesentation P, which satisfies = sin. We fist t to see if we can find an points which have a single epesentation P, that satisfies both = sin and = sin. Assuming such a pai, eists, then equating the epessions fo gives sin = sin o sin =. Fom this, we get = + k o = 5 + k fo integes k. Plugging = into = sin, we get = sin = =, which is also the value we obtain when we substitute it into = sin. Hence,, is one epesentation fo the point of intesection in the fist quadant. Fo the point of intesection in the second quadant, we t = 5. Both equations give us the point, 5, so this is ou answe hee. What about the oigin? We know fom Section. that the pole ma be epesented as 0, fo an angle. On the gaph of = sin, we stat at the oigin when = 0 and etun to it at =, and as the eade can veif, we ae at the oigin eactl when = k fo integes k. On the cuve = sin, howeve, we each the oigin when =, and moe geneall, when = + k fo integes k. Thee is no intege value of k fo which k = + k which means while the oigin is on both gaphs, the point is neve eached simultaneousl. In an case, we have detemined the thee points of intesection to be,,, 5 and the oigin.. As befoe, we make a quick sketch of = and = cos to get feel fo the numbe and location of the intesection points. The gaph of = is a cicle, centeed at the oigin, with a adius of. The gaph of = cos is also a cicle - but this one is centeed at the point with ectangula coodinates, 0 and has a adius of. = and = cos We have two intesection points to find, one in Quadant I and one in Quadant IV. Poceeding as above, we fist detemine if an of the intesection points P have a epesentation, which satisfies both = and = cos. Equating these two epessions fo, we get cos =. To solve this equation, we need the accosine function. We get We ae eall using the technique of substitution to solve the sstem of equations j = sin = sin

16 .5 Gaphs of Pola Equations 95 = accos + k o = accos + k fo integes k. Fom these solutions, we get, accos as one epesentation fo ou answe in Quadant I, and, accos as one epesentation fo ou answe in Quadant IV. The eade is encouaged to check these esults algebaicall and geometicall.. Poceeding as above, we fist gaph = and = cos to get an idea of how man intesection points to epect and whee the lie. The gaph of = is a cicle centeed at the oigin with a adius of and the gaph of = cos is anothe fou-leafed ose. = and = cos It appeas as if thee ae eight points of intesection - two in each quadant. We fist look to see if thee an points P, with a epesentation that satisfies both = and = cos. Fo these points, cos = o cos =. Solving, we get = + k o = 5 + k fo integes k. Out of all of these solutions, we obtain just fou distinct points epesented b,,, 5,, 7 and,. To detemine the coodinates of the emaining fou points, we have to conside how the epesentations of the points of intesection can diffe. We know fom Section. that if, and, epesent the same point and 0, then eithe = o =. If =, then = +k, so one possibilit is that an intesection point P has a epesentation, which satisfies = and anothe epesentation, +k fo some intege, k which satisfies = cos. At this point, f we eplace eve occuence of in the equation = cos with +k and then see if, b equating the esulting epessions fo, we get an moe solutions fo. Since cos + k = cos + k = cos fo eve intege k, howeve, the equation = cos + k educes to the same equation we had befoe, = cos, which means we get no additional solutions. Moving on to the case whee =, we have that = + k + fo integes k. We look to see if we can find points P which have a epesentation, that satisfies = and anothe, See Eample.5. numbe. The authos have chosen to eplace with +k in the equation = cos fo illustation puposes onl. We could have just as easil chosen to do this substitution in the equation =. Since thee is no in =, howeve, this case would educe to the pevious case instantl. The eade is encouaged to follow this latte pocedue in the inteests of efficienc.

17 95 Applications of Tigonomet, + k +, that satisfies = cos. To do this, we substitute fo and + k + fo in the equation = cos and get = cos + k +. Since cos + k + = cos + k + = cos fo all integes k, the equation = cos + k + educes to = cos, o = cos. Coupling this equation with = gives cos = o cos =. We get = +k o = +k. Fom these solutions, we obtain 5 the emaining fou intesection points with epesentations,,,,, and, 5, which we can eadil check gaphicall.. As usual, we begin b gaphing = sin and = cos. Using the techniques pesented in Eample.5., we find that we need to plot both functions as anges fom 0 to to obtain the complete gaph. To ou supise and/o delight, it appeas as if these two equations descibe the same cuve! = sin and = cos appea to detemine the same cuve in the -plane To veif this incedible claim, we need to show that, in fact, the gaphs of these two equations intesect at all points on the plane. Suppose P has a epesentation, which satisfies both = sin and = cos. Equating these two epessions fo gives the equation sin = cos. While nomall we discouage dividing b a vaiable epession in case it could be 0, we note hee that if cos = 0, then fo ou equation to hold, sin = 0 as well. Since no angles have both cosine and sine equal to zeo, we ae safe to divide both sides of the equation sin = cos b cos to get tan = which gives = + k fo integes k. Fom these solutions, howeve, we Again, we could have easil chosen to substitute these into = which would give =, o =. 5 We obtain these epesentations b substituting the values fo into = cos, once again, fo illustation puposes. Again, in the inteests of efficienc, we could plug these values fo into = whee thee is no and get the list of points: `,, `,, `, as `,, we still get the same set of solutions. A quick sketch of = sin ` these ae two diffeent animals. and `, 5. While it is not tue that `, epesents the same point and = cos ` in the -plane will convince ou that, viewed as functions of,

18 .5 Gaphs of Pola Equations 95 get onl one intesection point which can be epesented b,. We now investigate othe epesentations fo the intesection points. Suppose P is an intesection point with a epesentation, which satisfies = sin and the same point P has a diffeent epesentation, + k fo some intege k which satisfies = cos. Substituting into the latte, we get = cos [ + k] = cos + k. Using the sum fomula fo cosine, we epand cos + k = cos cosk sin sin k = ± cos, since sink = 0 fo all integes k, and cos k = ± fo all integes k. If k is an even intege, we get the same equation = cos as befoe. If k is odd, we get = cos. This latte epession fo leads to the equation sin = cos, o tan =. Solving, we get = + k fo integes k, which gives the intesection point,. Net, we assume P has a epesentation, which satisfies = sin and a epesentation, + k + which satisfies = cos fo some intege k. Substituting fo and + k + in fo into = cos gives = cos [ + k + ]. Once again, we use the sum fomula fo cosine to get cos [ + k + ] = cos + k+ = cos cos k+ = ± sin whee the last equalit is tue since cos k+ = 0 and sin Hence, = cos [ + k + ] can be ewitten as = ± sin then sin k+ sin sin k+ k+ = ± fo integes k.. If we choose k = 0, = sin =, and the equation = cos [ + k + ] in this case educes to = sin, o = sin which is the othe equation unde consideation! What this means is that if a pola epesentation, fo the point P satisfies = sin, then the epesentation, + fo P automaticall satisfies = cos. Hence the equations = sin and = cos detemine the same set of points in the plane. Ou wok in Eample.5. justifies the following. Guidelines fo Finding Points of Intesection of Gaphs of Pola Equations To find the points of intesection of the gaphs of two pola equations E and E : Sketch the gaphs of E and E. Check to see if the cuves intesect at the oigin pole. Solve fo pais, which satisf both E and E. Substitute + k fo in eithe one of E o E but not both and solve fo pais, which satisf both equations. Keep in mind that k is an intege. Substitute fo and + k + fo in eithe one of E o E but not both and solve fo pais, which satisf both equations. Keep in mind that k is an intege.

19 95 Applications of Tigonomet Ou last eample ties togethe gaphing and points of intesection to descibe egions in the plane. Eample.5.. Sketch the egion in the -plane descibed b the following sets.. {, 0 5 sin, 0 }. {, cos, 0 }. {, + cos 0,. {, 0 sin, 0 } {, 0 sin, } } Solution. Ou fist step in these poblems is to sketch the gaphs of the pola equations involved to get a sense of the geometic situation. Since all of the equations in this eample ae found in eithe Eample.5. o Eample.5., most of the wok is done fo us.. We know fom Eample.5. numbe that the gaph of = 5 sin is a ose. Moeove, we know fom ou wok thee that as 0, we ae tacing out the leaf of the ose which lies in the fist quadant. The inequalit 0 5 sin means we want to captue all [ of ] the points between the oigin = 0 and the cuve = 5 sin as uns though 0,. Hence, the egion we seek is the leaf itself. 5 5 { }, 0 5 sin, 0. We know fom Eample.5. numbe that = and = cos intesect at =, so the egion that is being descibed hee is the set of points whose diected distance fom the oigin is at least but no moe than cos as uns fom 0 to. In othe wods, we ae looking at the points outside o on the cicle since but inside o on the ose since cos. We shade the egion below. = = and = cos { }, cos, 0

20 .5 Gaphs of Pola Equations 955. Fom Eample.5. numbe, we know that the gaph of = + cos is a limaçon whose inne loop is taced out as uns though the given values to. Since the values takes on in this inteval ae non-positive, the inequalit + cos 0 makes sense, and we ae looking fo all of the points between the pole = 0 and the limaçon as anges ove the inteval [ ]. In othe wods, we shade in the inne loop of the limaçon., = = {, + cos 0, }. We have two egions descibed hee connected with the union smbol. We shade each in tun and find ou final answe b combining the two. In Eample.5., numbe, we found that the cuves = sin and = sin intesect when =. Hence, fo the fist egion, {, 0 sin, 0 }, we ae shading the egion between the oigin = 0 out to the cicle = sin as anges fom 0 to, which is the angle of intesection of the two cuves. Fo the second egion, {, 0 sin, }, picks up whee it left off at and continues to. In this case, howeve, we ae shading fom the oigin = 0 out to the cadioid = sin which pulls into the oigin at =. Putting these two egions togethe gives us ou final answe. = = sin and = sin { }, 0 sin, 0 {, 0 sin, }

21 95 Applications of Tigonomet.5. Eecises In Eecises - 0, plot the gaph of the pola equation b hand. Caefull label ou gaphs.. Cicle: = sin. Cicle: = cos. Rose: = sin. Rose: = cos 5. Rose: = 5 sin. Rose: = cos5 7. Rose: = sin 8. Rose: = cos 9. Cadioid: = cos 0. Cadioid: = sin. Cadioid: = + cos. Cadioid: = sin. Limaçon: = cos. Limaçon: = sin 5. Limaçon: = + cos. Limaçon: = 5 cos 7. Limaçon: = 5 sin 8. Limaçon: = + 7 sin 9. Lemniscate: = sin 0. Lemniscate: = cos In Eecises - 0, find the eact pola coodinates of the points of intesection of gaphs of the pola equations. Remembe to check fo intesection at the pole oigin.. = cos and = + cos. = + sin and = cos. = sin and =. = cos and = 5. = cos and = sin. = cos and = sin 7. = cos and = 8. = sin and = 9. = cos and = 0. = sin and = In Eecises - 0, sketch the egion in the -plane descibed b the given set.. {, 0, 0 }. {, 0 sin, 0 }. {, 0 cos, }. {, 0 sin, 0 }

22 .5 Gaphs of Pola Equations {, 0 cos, }. {, cos, } 7. {, + cos cos, } { 8., } sin, 7 9. {, 0 sin, 0 } {, 0 cos, } 0. {, 0 sin, 0 } {, 0, } In Eecises - 50, use set-builde notation to descibe the pola egion. Assume that the egion contains its bounding cuves.. The egion inside the cicle = 5.. The egion inside the cicle = 5 which lies in Quadant III.. The egion inside the left half of the cicle = sin.. The egion inside the cicle = cos which lies in Quadant IV. 5. The egion inside the top half of the cadioid = cos. The egion inside the cadioid = sin which lies in Quadants I and IV. 7. The inside of the petal of the ose = cos which lies on the positive -ais 8. The egion inside the cicle = 5 but outside the cicle =. 9. The egion which lies inside of the cicle = cos but outside of the cicle = sin 50. The egion in Quadant I which lies inside both the cicle = as well as the ose = sin While the authos tul believe that gaphing pola cuves b hand is fundamental to ou undestanding of the pola coodinate sstem, we would be deelict in ou duties if we totall ignoed the gaphing calculato. Indeed, thee ae some impotant pola cuves which ae simpl too difficult to gaph b hand and that makes the calculato an impotant tool fo ou futhe studies in Mathematics, Science and Engineeing. We now give a bief demonstation of how to use the gaphing calculato to plot pola cuves. The fist thing ou must do is switch the MODE of ou calculato to POL, which stands fo pola.

23 958 Applications of Tigonomet This changes the Y= menu as seen above in the middle. Let s plot the pola ose given b = cos fom Eecise 8 above. We tpe the function into the = menu as seen above on the ight. We need to set the viewing window so that the cuve displas popel, but when we look at the WINDOW menu, we find thee eta lines. In ode fo the calculato to be able to plot = cos in the -plane, we need to tell it not onl the dimensions which and will assume, but we also what values of to use. Fom ou pevious wok, we know that we need 0, so we ente the data ou see above. I ll sa moe about the -step in just a moment. Hitting GRAPH ields the cuve below on the left which doesn t look quite ight. The issue hee is that the calculato sceen is 9 piels wide but onl piels tall. To get a tue geometic pespective, we need to hit ZOOM SQUARE seen below in the middle to poduce a moe accuate gaph which we pesent below on the ight. In function mode, the calculato automaticall divided the inteval [Xmin, Xma] into 9 equal subintevals. In pola mode, howeve, we must specif how to split up the inteval [min, ma] using the step. Fo most gaphs, a step of 0. is fine. If ou make it too small then the calculato takes a long time to gaph. It ou make it too big, ou get chunk gabage like this. You will need to epeiment with the settings in ode to get a nice gaph. Eecises 5-0 give ou some cuves to gaph using ou calculato. Notice that some of them have eplicit bounds on and othes do not.

24 .5 Gaphs of Pola Equations =, 0 5. = ln, 5. = e., 0 5. =, = sin5 cos 5. = sin + cos 57. = actan, 58. = 59. = cos 0. = cos cos. How man petals does the pola ose = sin have? What about = sin, = sin and = sin5? With the help of ou classmates, make a conjectue as to how man petals the pola ose = sinn has fo an natual numbe n. Replace sine with cosine and epeat the investigation. How man petals does = cosn have fo each natual numbe n? Looking back though the gaphs in the section, it s clea that man pola cuves enjo vaious foms of smmet. Howeve, classifing smmet fo pola cuves is not as staight-fowad as it was fo equations back on page. In Eecises -, we have ou and ou classmates eploe some of the moe basic foms of smmet seen in common pola cuves.. Show that if f is even 7 then the gaph of = f is smmetic about the -ais. a Show that f = + cos is even and veif that the gaph of = + cos is indeed smmetic about the -ais. See Eample.5. numbe. b Show that f = sin is not even, et the gaph of = sin is smmetic about the -ais. See Eample.5. numbe.. Show that if f is odd 8 then the gaph of = f is smmetic about the oigin. a Show that f = 5 sin is odd and veif that the gaph of = 5 sin is indeed smmetic about the oigin. See Eample.5. numbe. b Show that f = cos is not odd, et the gaph of = cos is smmetic about the oigin. See Eample.5. numbe.. Show that if f = f fo all in the domain of f then the gaph of = f is smmetic about the -ais. a Fo f = sin, show that f = f and the gaph of = sin is smmetic about the -ais, as equied. See Eample.5. numbe. 7 Recall that this means f = f fo in the domain of f. 8 Recall that this means f = f fo in the domain of f.

25 90 Applications of Tigonomet b Fo f = 5 sin, show that f f, et the gaph of = 5 sin is smmetic about the -ais. See Eample.5. numbe. In Section.7, we discussed tansfomations of gaphs. classmates eploe tansfomations of pola gaphs. In Eecise 5 we have ou and ou 5. Fo Eecises 5a and 5b below, let f = cos and g = sin. a Using ou gaphing calculato, compae the gaph of = f to each of the gaphs of = f +, = f +, = f and = f. Repeat this pocess fo g. In geneal, how do ou think the gaph of = f + α compaes with the gaph of = f? b Using ou gaphing calculato, compae the gaph of = f to each of the gaphs of = f, = f, = f and = f. Repeat this pocess fo g. In geneal, how do ou think the gaph of = k f compaes with the gaph of = f? Does it matte if k > 0 o k < 0?. In light of Eecises -, how would the gaph of = f compae with the gaph of = f fo a geneic function f? What about the gaphs of = f and = f? What about = f and = f? Test out ou conjectues using a vaiet of pola functions found in this section with the help of a gaphing utilit. 7. With the help of ou classmates, eseach cadioid micophones. 8. Back in Section., in the paagaph befoe Eecise 5, we gave ou this link to a fascinating list of cuves. Some of these cuves have pola epesentations which we invite ou and ou classmates to eseach.

26 .5 Gaphs of Pola Equations 9.5. Answes. Cicle: = sin. Cicle: = cos. Rose: = sin. Rose: = cos = = 5. Rose: = 5 sin. Rose: = cos5 = 5 = = 7 0 = 0 = 9 0 =

27 9 Applications of Tigonomet 7. Rose: = sin = = 8. Rose: = cos = 5 = 8 8 = 7 8 = 8 9. Cadioid: = cos 0. Cadioid: = sin Cadioid: = + cos. Cadioid: = sin

28 .5 Gaphs of Pola Equations 9. Limaçon: = cos =. Limaçon: = sin = 5 = = 5 5. Limaçon: = + cos + = 5 + = 7. Limaçon: = 5 cos = accos 5 = accos 5 7. Limaçon: = 5 sin = acsin 5 8 = acsin 5 8. Limaçon: = + 7 sin = + acsin 7 = acsin 7 8 9

29 9 Applications of Tigonomet 9. Lemniscate: = sin 0. Lemniscate: = cos = =. = cos and = + cos,,, 5, pole. = + sin and = cos +,,, 7, pole

30 .5 Gaphs of Pola Equations 95. = sin and =, 7,,. = cos and =,,,,, 0 5. = cos and = sin,, pole

31 9 Applications of Tigonomet. = cos and = sin 0 0, actan, pole 7. = cos and =,,, 5,, 7,, 8. = sin and =,,, 5,,,, 7

32 .5 Gaphs of Pola Equations = cos and =,,, 5,,,, 7,,,,,,, 5, 0. = sin and =,,, 5, 7,, 9, 7,,,,,,,,

33 98 Applications of Tigonomet. {, 0, 0 }. {, 0 sin, 0 }. {, 0 cos, }. {, 0 sin, 0 } 5. {, 0 cos, }. {, cos, }

34 .5 Gaphs of Pola Equations {, + cos cos, } 8. {, } sin, 7 9. {, 0 sin, 0 } {, 0 cos, }

35 970 Applications of Tigonomet 0. {, 0 sin, 0 } {, 0, }. {, 0 5, 0 }. {, 0 5, }. {, 0 sin, }. {, cos 0, } 5. {, 0 cos, 0 }. {, 0 sin, 0 } {, 0 sin, } o {, 0 sin, 5 } 7. {, 0 cos, 0 } { 8, 0 cos, 5 8 } o {, 0 cos, 8 } 8 8. {, 5, 0 } 9. {, 0 cos, 0} {, sin cos, 0 actan} 50. {, 0 sin, 0 } {, 0, 5 } {, 0 sin, 5 }

UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

More information

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities.

Graphs of Equations. A coordinate system is a way to graphically show the relationship between 2 quantities. Gaphs of Equations CHAT Pe-Calculus A coodinate sstem is a wa to gaphicall show the elationship between quantities. Definition: A solution of an equation in two vaiables and is an odeed pai (a, b) such

More information

Trigonometric Functions of Any Angle

Trigonometric Functions of Any Angle Tigonomet Module T2 Tigonometic Functions of An Angle Copight This publication The Nothen Albeta Institute of Technolog 2002. All Rights Reseved. LAST REVISED Decembe, 2008 Tigonometic Functions of An

More information

CHAT Pre-Calculus Section 10.7. Polar Coordinates

CHAT Pre-Calculus Section 10.7. Polar Coordinates CHAT Pe-Calculus Pola Coodinates Familia: Repesenting gaphs of equations as collections of points (, ) on the ectangula coodinate sstem, whee and epesent the diected distances fom the coodinate aes to

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

Skills Needed for Success in Calculus 1

Skills Needed for Success in Calculus 1 Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

More information

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90

2.2. Trigonometric Ratios of Any Angle. Investigate Trigonometric Ratios for Angles Greater Than 90 . Tigonometic Ratios of An Angle Focus on... detemining the distance fom the oigin to a point (, ) on the teminal am of an angle detemining the value of sin, cos, o tan given an point (, ) on the teminal

More information

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES . TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

More information

Trigonometry in the Cartesian Plane

Trigonometry in the Cartesian Plane Tigonomet in the Catesian Plane CHAT Algeba sec. 0. to 0.5 *Tigonomet comes fom the Geek wod meaning measuement of tiangles. It pimail dealt with angles and tiangles as it petained to navigation astonom

More information

Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of

Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of Oiginall TRIGONOMETRY was that banch of mathematics concened with solving tiangles using tigonometic atios which wee seen as popeties of tiangles athe than of angles. The wod Tigonomet comes fom the Geek

More information

Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algebra and Trig. I. A point is a location or position that has no size or dimension. Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

More information

Chapter 3: Vectors and Coordinate Systems

Chapter 3: Vectors and Coordinate Systems Coodinate Systems Chapte 3: Vectos and Coodinate Systems Used to descibe the position of a point in space Coodinate system consists of a fied efeence point called the oigin specific aes with scales and

More information

LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

More information

Modern Linear Algebra

Modern Linear Algebra Hochschule fü Witschaft und Recht Belin Belin School of Economics and Law Wintesemeste 04/05 D. Hon Mathematics fo Business and Economics LV-N. 0069.0 Moden Linea Algeba (A Geometic Algeba cash couse,

More information

Section 5-3 Angles and Their Measure

Section 5-3 Angles and Their Measure 5 5 TRIGONOMETRIC FUNCTIONS Section 5- Angles and Thei Measue Angles Degees and Radian Measue Fom Degees to Radians and Vice Vesa In this section, we intoduce the idea of angle and two measues of angles,

More information

Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction.

Review of Vectors. Appendix A A.1 DESCRIBING THE 3D WORLD: VECTORS. 3D Coordinates. Basic Properties of Vectors: Magnitude and Direction. Appendi A Review of Vectos This appendi is a summa of the mathematical aspects of vectos used in electicit and magnetism. Fo a moe detailed intoduction to vectos, see Chapte 1. A.1 DESCRIBING THE 3D WORLD:

More information

Write and Graph Equations of Circles

Write and Graph Equations of Circles 0.7 Wite and Gaph Equations of icles Befoe You wote equations of lines in the coodinate plane. Now You will wite equations of cicles in the coodinate plane. Wh? So ou can detemine zones of a commute sstem,

More information

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

More information

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables.

On Correlation Coefficient. The correlation coefficient indicates the degree of linear dependence of two random variables. C.Candan EE3/53-METU On Coelation Coefficient The coelation coefficient indicates the degee of linea dependence of two andom vaiables. It is defined as ( )( )} σ σ Popeties: 1. 1. (See appendi fo the poof

More information

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

More information

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature.

In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Radians At school we usually lean to measue an angle in degees. Howeve, thee ae othe ways of measuing an angle. One that we ae going to have a look at hee is measuing angles in units called adians. In

More information

Transformations in Homogeneous Coordinates

Transformations in Homogeneous Coordinates Tansfomations in Homogeneous Coodinates (Com S 4/ Notes) Yan-Bin Jia Aug, 6 Homogeneous Tansfomations A pojective tansfomation of the pojective plane is a mapping L : P P defined as u a b c u au + bv +

More information

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E

Samples of conceptual and analytical/numerical questions from chap 21, C&J, 7E CHAPTER 1 Magnetism CONCEPTUAL QUESTIONS Cutnell & Johnson 7E 3. ssm A chaged paticle, passing though a cetain egion of space, has a velocity whose magnitude and diection emain constant, (a) If it is known

More information

Mechanics 1: Motion in a Central Force Field

Mechanics 1: Motion in a Central Force Field Mechanics : Motion in a Cental Foce Field We now stud the popeties of a paticle of (constant) ass oving in a paticula tpe of foce field, a cental foce field. Cental foces ae ve ipotant in phsics and engineeing.

More information

Fluids Lecture 15 Notes

Fluids Lecture 15 Notes Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

More information

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360! 1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

More information

Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapter 3 Savings, Present Value and Ricardian Equivalence Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

More information

TRIGONOMETRY REVIEW. The Cosines and Sines of the Standard Angles

TRIGONOMETRY REVIEW. The Cosines and Sines of the Standard Angles TRIGONOMETRY REVIEW The Cosines and Sines of the Standad Angles P θ = ( cos θ, sin θ ) . ANGLES AND THEIR MEASURE In ode to define the tigonometic functions so that they can be used not only fo tiangula

More information

Revision Guide for Chapter 11

Revision Guide for Chapter 11 Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams

More information

Financing Terms in the EOQ Model

Financing Terms in the EOQ Model Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

More information

mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2 " 2 = GM . Combining the results we get !

mv2. Equating the two gives 4! 2. The angular velocity is the angle swept per GM (2! )2 4! 2  2 = GM . Combining the results we get ! Chapte. he net foce on the satellite is F = G Mm and this plays the ole of the centipetal foce on the satellite i.e. mv mv. Equating the two gives = G Mm i.e. v = G M. Fo cicula motion we have that v =!

More information

Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0.

Unit Vectors. the unit vector rˆ. Thus, in the case at hand, 5.00 rˆ, means 5.00 m/s at 36.0. Unit Vectos What is pobabl the most common mistake involving unit vectos is simpl leaving thei hats off. While leaving the hat off a unit vecto is a nast communication eo in its own ight, it also leads

More information

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

92.131 Calculus 1 Optimization Problems

92.131 Calculus 1 Optimization Problems 9 Calculus Optimization Poblems ) A Noman window has the outline of a semicicle on top of a ectangle as shown in the figue Suppose thee is 8 + π feet of wood tim available fo all 4 sides of the ectangle

More information

Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

More information

Lab 5: Circular Motion

Lab 5: Circular Motion Lab 5: Cicula motion Physics 193 Fall 2006 Lab 5: Cicula Motion I. Intoduction The lab today involves the analysis of objects that ae moving in a cicle. Newton s second law as applied to cicula motion

More information

Physics: Electromagnetism Spring PROBLEM SET 6 Solutions

Physics: Electromagnetism Spring PROBLEM SET 6 Solutions Physics: Electomagnetism Sping 7 Physics: Electomagnetism Sping 7 PROBEM SET 6 Solutions Electostatic Enegy Basics: Wolfson and Pasachoff h 6 Poblem 7 p 679 Thee ae si diffeent pais of equal chages and

More information

Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Work, Power and Kinetic Energy Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

More information

Macroeconomics I. Antonio Zabalza. University of Valencia 1. Class 5. The IS-LM model and Aggregate Demand

Macroeconomics I. Antonio Zabalza. University of Valencia 1. Class 5. The IS-LM model and Aggregate Demand Macoeconomics I. Antonio Zabalza. Univesity of Valencia 1 Class 5. The IS-LM model and Aggegate Demand 1. Use the Keynesian coss to pedict the impact of: a) An incease in govenment puchases. b) An incease

More information

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1)

So we ll start with Angular Measure. Consider a particle moving in a circular path. (p. 220, Figure 7.1) Lectue 17 Cicula Motion (Chapte 7) Angula Measue Angula Speed and Velocity Angula Acceleation We ve aleady dealt with cicula motion somewhat. Recall we leaned about centipetal acceleation: when you swing

More information

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

More information

Model Question Paper Mathematics Class XII

Model Question Paper Mathematics Class XII Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

More information

CHAPTER 10 Aggregate Demand I

CHAPTER 10 Aggregate Demand I CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

More information

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M + + + + M M + O A Moment about an abita

More information

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

The Detection of Obstacles Using Features by the Horizon View Camera

The Detection of Obstacles Using Features by the Horizon View Camera The Detection of Obstacles Using Featues b the Hoizon View Camea Aami Iwata, Kunihito Kato, Kazuhiko Yamamoto Depatment of Infomation Science, Facult of Engineeing, Gifu Univesit aa@am.info.gifu-u.ac.jp

More information

4.1 - Trigonometric Functions of Acute Angles

4.1 - Trigonometric Functions of Acute Angles 4.1 - Tigonometic Functions of cute ngles a is a half-line that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one

More information

New proofs for the perimeter and area of a circle

New proofs for the perimeter and area of a circle New poofs fo the peimete and aea of a cicle K. Raghul Kuma Reseach Schola, Depatment of Physics, Nallamuthu Gounde Mahalingam College, Pollachi, Tamil Nadu 64001, India 1 aghul_physics@yahoo.com aghulkumak5@gmail.com

More information

Experiment 6: Centripetal Force

Experiment 6: Centripetal Force Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

More information

Review Module: Dot Product

Review Module: Dot Product MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics 801 Fall 2009 Review Module: Dot Poduct We shall intoduce a vecto opeation, called the dot poduct o scala poduct that takes any two vectos and

More information

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool

Universal Cycles. Yu She. Wirral Grammar School for Girls. Department of Mathematical Sciences. University of Liverpool Univesal Cycles 2011 Yu She Wial Gamma School fo Gils Depatment of Mathematical Sciences Univesity of Livepool Supeviso: Pofesso P. J. Giblin Contents 1 Intoduction 2 2 De Buijn sequences and Euleian Gaphs

More information

Th Po er of th Cir l. Lesson3. Unit UNIT 6 GEOMETRIC FORM AND ITS FUNCTION

Th Po er of th Cir l. Lesson3. Unit UNIT 6 GEOMETRIC FORM AND ITS FUNCTION Lesson3 Th Po e of th Ci l Quadilateals and tiangles ae used to make eveyday things wok. Right tiangles ae the basis fo tigonometic atios elating angle measues to atios of lengths of sides. Anothe family

More information

Some text, some maths and going loopy. This is a fun chapter as we get to start real programming!

Some text, some maths and going loopy. This is a fun chapter as we get to start real programming! Chapte Two Some text, some maths and going loopy In this Chapte you ae going to: Lean how to do some moe with text. Get Python to do some maths fo you. Lean about how loops wok. Lean lots of useful opeatos.

More information

Solutions to Homework Set #5 Phys2414 Fall 2005

Solutions to Homework Set #5 Phys2414 Fall 2005 Solution Set #5 1 Solutions to Homewok Set #5 Phys414 Fall 005 Note: The numbes in the boxes coespond to those that ae geneated by WebAssign. The numbes on you individual assignment will vay. Any calculated

More information

The LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero.

The LCOE is defined as the energy price ($ per unit of energy output) for which the Net Present Value of the investment is zero. Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

More information

arxiv: v2 [math.ho] 13 Jul 2016

arxiv: v2 [math.ho] 13 Jul 2016 axiv:1603.0854v [mat.ho] 13 Jul 016 IT IS NOT A COINCIDENCE! ON CURIOUS PATTERNS IN CALCULUS OPTIMIZATION PROBLEMS MARIA NOGIN Abstact. In te fist semeste calculus couse we lean ow to solve optimization

More information

1.4 Phase Line and Bifurcation Diag

1.4 Phase Line and Bifurcation Diag Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes

More information

Gauss Law. Physics 231 Lecture 2-1

Gauss Law. Physics 231 Lecture 2-1 Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

More information

The Binomial Distribution

The Binomial Distribution The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

More information

NURBS Drawing Week 5, Lecture 10

NURBS Drawing Week 5, Lecture 10 CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

More information

Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43 Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

More information

Chapter 6. Gradually-Varied Flow in Open Channels

Chapter 6. Gradually-Varied Flow in Open Channels Chapte 6 Gadually-Vaied Flow in Open Channels 6.. Intoduction A stea non-unifom flow in a pismatic channel with gadual changes in its watesuface elevation is named as gadually-vaied flow (GVF). The backwate

More information

PY1052 Problem Set 8 Autumn 2004 Solutions

PY1052 Problem Set 8 Autumn 2004 Solutions PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers

The Critical Angle and Percent Efficiency of Parabolic Solar Cookers The Citical Angle and Pecent Eiciency o Paabolic Sola Cookes Aiel Chen Abstact: The paabola is commonly used as the cuve o sola cookes because o its ability to elect incoming light with an incoming angle

More information

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years. 9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

More information

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS

BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR. John R. Graham Adapted from S. Viswanathan FUQUA SCHOOL OF BUSINESS BA 351 CORPORATE FINANCE LECTURE 4 TAXES AND THE MARGINAL INVESTOR John R. Gaham Adapted fom S. Viswanathan FUQUA SCHOOL OF BUSINESS DUKE UNIVERSITY 1 In this lectue we conside the effect of govenment

More information

Displacement, Velocity And Acceleration

Displacement, Velocity And Acceleration Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,

More information

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to . Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

More information

Uncertainties in Fault Tree Analysis

Uncertainties in Fault Tree Analysis ncetainties in Fault Tee nalysis Yue-Lung Cheng Depatment of Infomation Management Husan Chuang College 48 Husan-Chuang Rd. HsinChu Taiwan R.O.C bstact Fault tee analysis is one kind of the pobilistic

More information

Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M

Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuent-caying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to

More information

8-1 Newton s Law of Universal Gravitation

8-1 Newton s Law of Universal Gravitation 8-1 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,

More information

Road tunnel. Road tunnel information sheet. Think about. Using the information

Road tunnel. Road tunnel information sheet. Think about. Using the information Road tunnel This activity is about using a gaphical o algebaic method to solve poblems in eal contets that can be modelled using quadatic epessions. The fist poblem is about a oad tunnel. The infomation

More information

The Role of Gravity in Orbital Motion

The Role of Gravity in Orbital Motion ! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

More information

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

More information

PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013

PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013 PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80-kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,

More information

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

More information

TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS

TALLINN UNIVERSITY OF TECHNOLOGY, INSTITUTE OF PHYSICS 14. NEWTON'S RINGS 4. NEWTON'S RINGS. Obective Detemining adius of cuvatue of a long focal length plano-convex lens (lage adius of cuvatue).. Equipment needed Measuing micoscope, plano-convex long focal length lens, monochomatic

More information

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications

The Supply of Loanable Funds: A Comment on the Misconception and Its Implications JOURNL OF ECONOMICS ND FINNCE EDUCTION Volume 7 Numbe 2 Winte 2008 39 The Supply of Loanable Funds: Comment on the Misconception and Its Implications. Wahhab Khandke and mena Khandke* STRCT Recently Fields-Hat

More information

Valuation of Floating Rate Bonds 1

Valuation of Floating Rate Bonds 1 Valuation of Floating Rate onds 1 Joge uz Lopez us 316: Deivative Secuities his note explains how to value plain vanilla floating ate bonds. he pupose of this note is to link the concepts that you leaned

More information

9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics

9. Mathematics Practice Paper for Class XII (CBSE) Available Online Tutoring for students of classes 4 to 12 in Physics, Chemistry, Mathematics Available Online Tutoing fo students of classes 4 to 1 in Physics, 9. Mathematics Class 1 Pactice Pape 1 3 1. Wite the pincipal value of cos.. Wite the ange of the pincipal banch of sec 1 defined on the

More information

Continuous Compounding and Annualization

Continuous Compounding and Annualization Continuous Compounding and Annualization Philip A. Viton Januay 11, 2006 Contents 1 Intoduction 1 2 Continuous Compounding 2 3 Pesent Value with Continuous Compounding 4 4 Annualization 5 5 A Special Poblem

More information

An Introduction to Omega

An Introduction to Omega An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

More information

International Monetary Economics Note 1

International Monetary Economics Note 1 36-632 Intenational Monetay Economics Note Let me biefly ecap on the dynamics of cuent accounts in small open economies. Conside the poblem of a epesentative consume in a county that is pefectly integated

More information

Gravitation. AP Physics C

Gravitation. AP Physics C Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

More information

Converting knowledge Into Practice

Converting knowledge Into Practice Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

More information

Chapter 23: Gauss s Law

Chapter 23: Gauss s Law Chapte 3: Gauss s Law Homewok: Read Chapte 3 Questions, 5, 1 Poblems 1, 5, 3 Gauss s Law Gauss s Law is the fist of the fou Maxwell Equations which summaize all of electomagnetic theoy. Gauss s Law gives

More information

Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number.

Exam I. Spring 2004 Serway & Jewett, Chapters 1-5. Fill in the bubble for the correct answer on the answer sheet. next to the number. Agin/Meye PART I: QUALITATIVE Exam I Sping 2004 Seway & Jewett, Chaptes 1-5 Assigned Seat Numbe Fill in the bubble fo the coect answe on the answe sheet. next to the numbe. NO PARTIAL CREDIT: SUBMIT ONE

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

Lab #7: Energy Conservation

Lab #7: Energy Conservation Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

More information

PY1052 Problem Set 3 Autumn 2004 Solutions

PY1052 Problem Set 3 Autumn 2004 Solutions PY1052 Poblem Set 3 Autumn 2004 Solutions C F = 8 N F = 25 N 1 2 A A (1) A foce F 1 = 8 N is exeted hoizontally on block A, which has a mass of 4.5 kg. The coefficient of static fiction between A and the

More information

Semipartial (Part) and Partial Correlation

Semipartial (Part) and Partial Correlation Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

More information

Lesson 8 Ampère s Law and Differential Operators

Lesson 8 Ampère s Law and Differential Operators Lesson 8 Ampèe s Law and Diffeential Opeatos Lawence Rees 7 You ma make a single cop of this document fo pesonal use without witten pemission 8 Intoduction Thee ae significant diffeences between the electic

More information

Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria.

Infinite-dimensional Bäcklund transformations between isotropic and anisotropic plasma equilibria. Infinite-dimensional äcklund tansfomations between isotopic and anisotopic plasma equilibia. Infinite symmeties of anisotopic plasma equilibia. Alexei F. Cheviakov Queen s Univesity at Kingston, 00. Reseach

More information

General Physics (PHY 2130)

General Physics (PHY 2130) Geneal Physics (PHY 130) Lectue 11 Rotational kinematics and unifom cicula motion Angula displacement Angula speed and acceleation http://www.physics.wayne.edu/~apetov/phy130/ Lightning Review Last lectue:

More information

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it. Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

More information

Problem Set # 9 Solutions

Problem Set # 9 Solutions Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

More information

The Clustering Coefficient of Multiple Parallel Airlines AANET

The Clustering Coefficient of Multiple Parallel Airlines AANET Intenational Jounal of Futue Geneation Communication and Netwoking Vol. 9, No. 7 (016), pp. 15-144 http://dx.doi.og/10.1457/ijfgcn.016.9.7.1 The Clusteing Coefficient of Multiple Pael Ailines AANET Xue

More information

L19 Geomagnetic Field Part I

L19 Geomagnetic Field Part I Intoduction to Geophysics L19-1 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o

More information

Chris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment

Chris J. Skinner The probability of identification: applying ideas from forensic statistics to disclosure risk assessment Chis J. Skinne The pobability of identification: applying ideas fom foensic statistics to disclosue isk assessment Aticle (Accepted vesion) (Refeeed) Oiginal citation: Skinne, Chis J. (2007) The pobability

More information