9.1. Click here for answers. Click here for solutions. PARAMETRIC CURVES

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1 SECTION 9. PARAMETRIC CURVES 9. PARAMETRIC CURVES A Click here for answers. S Click here for solutions. 5 (a) Sketch the curve b using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. Eliminate the parameter to find a Cartesian equation of the curve.. t 4,. t, t,. t, t 4, 4. t, 5. t, 6. t, t, 7. t, 5t, 8. t, 9. cos, sin,. cos,. e t, st,. e t,. cos t, e t 4. t, t t 6 t t t sin cos 4 t 5. t, t, t t t t t 4 t t t t 6 9 (a) Eliminate the parameter to find a Cartesian equation of the curve. Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. 6. cos, sin, 7. cos, 8. tan sec, tan sec, 9. cos t, Describe the motion of a particle with position, as t varies in the given interval.. 4 4t, t 5,. tan t, cot t,. 8t, t,. sin t, csc t, ; 4 6 Graph and as functions of t and observe how and increase or decrease as t increases. Use these observations to make a rough sketch b hand of the parametric curve. Then use a graphing device to check our sketch. 4. t, 5. cos t, 6. t 4, sin cos t t t tan t t t 6 t t 6 t Copright, Cengage Learning. All rights reserved.

2 SECTION 9. PARAMETRIC CURVES 9. ANSWERS E Click here for eercises. S Click here for solutions.. (a). (a). (a). (a) =. (a) = 4. (a). (a) = e, Or: = ln, e (, ) 4. (a) =, > t= t= t t= _ =, + =, = 4 ( ) +4 = ( 6) 9 5. (a) (, ) = +, 5. (a) 6. (a) Or: = ( ), + (, ) =5 7. (a) + =, 8. (a) (a) + (/) = = 5 ( ), += ( +) 4 7. (a) = 4, 9. (a). (a) 8. (a) = /, > 9. (a) +=, Copright, Cengage Learning. All rights reserved = + =,

3 SECTION 9. PARAMETRIC CURVES. Moves along = +7from (4, 5) to ( 4, 9). Moves along the first quadrant branch of =/ from (, ) (, ) to. Moves along +8 =from (, ) to (5, ). Moves down the first quadrant branch of =from (, ) to (sin, csc ) Copright, Cengage Learning. All rights reserved.

4 4 SECTION 9. PARAMETRIC CURVES 9. SOLUTIONS E Click here for eercises.. (a) =t +4, = t t (a) = t, =6 t t =t +4, = t =( +)+4= +6or =. (a) = t, =t, t 4 t =6 t t =6 t = 6 ( ) 6 = t = = ( 6) 9 5. (a) = t t = =t =( ) =. (a) = t, = t +4, t = t, =+t =+( ) =5,so + =5 6. (a) t t =t, = t, t =( ) =, so + =,with (a) Copright, Cengage Learning. All rights reserved. = t t = t = ( ) = t +4= +4= 4 ( ) +4or = =t, =+5t, t ( ) = = 5 5 ( ),

5 SECTION 9. PARAMETRIC CURVES 5 8. (a) 4. (a) t= =t, = t ( ) + =,so += ( +) 4 9. (a) 5. (a) t=_ t= = t +t, = t +t + =, (, ) =cosθ, =sinθ, θ π ( ) ( ) + =cos θ + sin θ =,or. (a). (a) = =cos θ, =sinθ + =cos θ +sin θ =, (, ) = t +t, = t, t = +, ( ) Or: =, + 6. (a) =cosθ, = sin θ, θ π. ( ) ( ) =cos θ + sin θ = +,so / + (/) =. = e t, = t = e, Or: = ln, e. (a) 7. (a) =cosθ, =sin θ. ( ) =cos θ + sin θ = +, so = 4,. The curve is at (, ) whenever θ =πn. = e t, = e t =, >. (a) (, ) 8. (a) =tanθ + sec θ, =tanθ sec θ, π <θ< π. =tan θ sec θ = = /, >. Copright, Cengage Learning. All rights reserved. =cos t, =cos 4 t =,

6 6 SECTION 9. PARAMETRIC CURVES 9. (a) =cost, =cost. =cost =cos t =,so +=,. t, π as oscillates.. =4 4t, =t +5, t. =4 (t) =4 ( 5) = +4, so the particle moves along the line = +7from (4, 5) to ( 4, 9).. =tant, =cott, π t π. =/ for 6. The particle moves along the first quadrant ( branch of the hperbola =/ from, ) to (, ).. =8t, = t, t =8( ) = 8, so the particle moves along the line +8 =from (, ) to (5, ).. = csc t = / sin t = /. The particle slides down the first quadrant branch of the hperbola =from (, ) to (sin, csc ) (.8447,.884) as t goes from π to From the graphs, it seems that as t, and. So the point ( (t),(t)) will move from far out in the fourth quadrant as t increases. At t =,both and are, so the graph passes through the origin. After that the graph passes through the second quadrant ( is negative, is positive), then intersects the -ais at = 9 when t =. After this, the graph passes through the third quadrant, going through the origin again at t =, and then as t, and. Note that for ever point ( (t),(t)) = ( ( t ),t t ), we can substitute t to get the corresponding point ( ( t),( t)) = ( [ ( t) ], ( t) ( t) ) = ( (t), (t)) andsothegraphissmmetricaboutthe-ais. The first figure was obtained using = t, = ( t ) ; = t, = t t;and π t π. 6. As t, and. The graph passes through the origin at t =, and then goes through the second quadrant ( negative, positive), passing through the point (, ) at t =.Ast increases, the graph passes through the point (, ) at t =, and then as t,both and approach. The first figure was obtained using = t, = t 4 ; = t, = t +;and π t π. Copright, Cengage Learning. All rights reserved. 5. As t, π and oscillates between and. Then, as t increases through, increases while continues to oscillate, and the graph passes through the origin. Then, as