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# Elementary Functions MTH 112

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This 7 page Class Notes was uploaded by August Feeney on Monday October 19, 2015. The Class Notes belongs to MTH 112 at Portland Community College taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/224645/mth-112-portland-community-college in Math at Portland Community College.

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Date Created: 10/19/15

Haberman MTH 112 Section IV Parametric and Implicit Equations 3990 f 394 s Ar Module 1 Parametric Equations EXAMPLE 1 Consider a figure skater carving a figureeight on the ice Figure 1 below shows the skater s path we ve embedded a coordinate plane on the icerink 39y Figure 1 The skater s figureeight Clearly the yvalues are not a function of the xvalues since this graph fails the vertical line testquot But it should also be clear that at each moment in time the skater is in exactly one location In other words the skater s location is a function of time We can use a system of parametric equations to define the skater s location as a function of time A system of parametric equations consists of a pair of functions one that describes the x coordinate and the other that describes the ycoordinate typically the input variable for these functions represents time It turns out that the system of parametric equations given below describes the skater s motion that we ve graphed in Figure 1 above xt 4sint yt 8005 t Let s sketch a graph of this system of parametric equations for 0 S t S 8 Let s assume that t is measured in seconds so we re graphing the figure skater s movement during her first 8 seconds of skating Note that we should expect to obtain the same graph as given in Figure 1 but we need to study how to obtain such a graph ourselves is CLICK HERE to see the system of parametric equations that describe the figure skater s path graphed by hand Note that if you want to graph this function on your calculator you need to change the graph mode of your calculator to parametric EXAMPLE 2 Suppose that a robot is moving around a coordinate plane and that x ft represents that xcoordinate and y gt represents the ycoordinate of the robots location on the plane as functions of time t in seconds The graphs of x ft and y gt during the first four minutes of the robot s travels are given in Figure 2 sketch a graph of the robot s movement 274 J E 23 1 L i 39 I 39 39 r I 2 4 l 2 3 4 77 1u 2 r 7 2 39 The graph of xft The graph of ygt Figure2 SOLUTION To graph the movement of the robot we need to find and plot ordered pairs x y ft gt To find these ordered pairs we need to choose values for t and find the corresponding values of x ft and y gt t0 0 0 x f Sotherobotisat 0 1whent0 yg01 t1 xf12 So the robot is at 2 1 when t 1 y g1 1 So the robot is at 1 1 when t 2 y g2 1 t3 xf31 yg32 t2 xf21 So the robot is at 1 2 when t 3 t4 4 0 x f Sothe robotis at 0 0 whent4 yg40 Now we can plot these five points and connect them in order starting with the point representing t 0 and ending with the point representing I 4 See Figure 3 below We use arrows to keep track of the direction of travel Q EXAMPLE 3 Suppuse that the x7 and yrcuurdmates enne muvement en a rubut are gwe by the ewean funmuns uf Ume z m secunds WE H assume that these Equatmns appmerau z gt x s e 22 y 72 4 sketch a graph the rubut s muvement SOLUT ON Tu graph the muvement enne mm we need me nd and mm urdered pans x y Tu elf x and y usmgthe p rametrwcequatwuns gwen abuve z 0 x s 7 20 7 s 39 Sothevobot sat 672 Men 1 0 y 72 40 72 11 x621 e4 Sothevobohs at 4 2 Men ye241 2 z 2 x s 7 22 2 Somewme 2 6 Men 1 2 y 72 42 s Sothevobohsat 010 when 24z 10 New We earr etet these pumts arre eerrrreet them m urder stamrrg thh the pumt representmg z 0 e gure4 e use arruvvstu keep track er the etreetterr ettraveh arre te seggestthe etreetterr thatthe reeetvhu travet m the future C Ea y the reeet S m vmg hhearty We eah eliminate the parameter 1 te nd a sthgte Equatmn that desmbes the rubut s muvement and we shumd Expect eer Equatmn te ee hhear z m x672t y7241 furt arre therr subs mute the reset mm uur ether Equatmn Let s setve the Equatmn that represents the xrcuurdmate furt arre subs utute mm the Equatmn that represents the y e e dm te x672t 2 2267 1 sex y724t 32449054 226x 1072x Thus path that the robot follows is represented by the equation y 10 2x Notice that this equation does in fact represent the line we graphed above Although this single equation is arguably more efficient than the system of parametric equations the advantage of the system of parametric equations is that in addition to describing the path that the robot travels it tells us WHEN the robot is in at each point along the path A system of parametric equations involves three variables so it conveys three pieces of information rather than just the two pieces of information conveyed by a single equation in twovariables Typically the parameter t represents time so that in addition to describing the path that an object travels a system of parametric equations also describes the time that it takes the object to travel along the path EXAMPLE 4 The x and ycoordinates of the movement of a particle are given by the following functions of time t where 0 S t S 2n x cost y sint I The graph of this system is given in Figure 5 You should graph it on your graphing calculator and make sure that you are able to obtain the same graph x cost Figure 5 The graph of y Slnt Let s eliminate the parameter r from this system in order to find a single equation involving x and y that represents the path ie the unit circle that the particle travels along In this case rather than using a substitution technique like we used in the previous example we ll utilize a trigonometric identity We ll use an identity since identities are ALWAYS true no matter what the values of the variables In this case we need an identity that involves cost and sint since these expressions are involved in our system of parametric equation Let s use the Pythagorean Identity sinzt 00521 1 and substitute x and y for cost and sint respectively sin2t coszt 1 3 y2 x2 1 Thus the equation x2 y2 1 describes the path that the particle travels along which is what we should have expected since the path is a unit circle Note that the drawback to this equation is that unlike the system of parametric equations it doesn t tell us WHEN the particle is at each point EXAMPLE 5 The x and y coordinates of the movement of a particle are given by the following functions of time t where 0 S t S 2n x cos7rt 1 y 25in7rt 2 The graph of this system is given in Figure 6 You should graph it on your graphing calculator and make sure that you are able to obtain the same graph x 2 1 1 2 1 Figure6 The graph of he particle s movement Let s eliminate the parameter r from this system in order to find a single equation involving x and y that represents the path that the particle travels along As we did in the previous example we can use the Pythagorean Identity sin2t9 00526 1 But first we need to solve the equations in our system for cos7rt and sin7rt so that we can then substitute these expressions for 0056 and sint9 in the Pythagorean Identity x 0050 1 y 25in7rt 2 3 COS7TI x 1 and 3 Sin t Thus sin2t9 00526 1 2 2 2 x1 y 2 1 2 x12 y z2 1 So the equation x 12 y 22 1 describes the path that the particle travels along In the next two modules we ll study implicit equations like this one This equation describes an ellipse ie an oval EXAMPLES Recall that the equation rt9 in polar coordinates represents the Archimedean spiral Express this spiral in parametric equations SOLUTION To parameterize the polar equation r 9 we need too find two functions one that represents the xcoordinate and one that represents the ycoordinate In Section III Module 1 we found that the polar coordinates r 0 can be translated into rectangular coordinates x y by using the following formulas x rcost9 and y rsint9 We can use these formulas to parameterize the equation r 9 The equation tells us that we can substitute 0 for r since they are equal x rcost9 and y rsint9 3 x 60056 and y 65int9 Finally by substituting t for 0 since I is the variable we usually use for our parameter we obtain the following parameterization of the polar equation r 9 x tcost y tsint Be sure to graph this system of parametric equations on your graphing calculator to verify that these equations give us the Archimedean spiral

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