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Mm 390.1(fo in 'HM. [4+ Ocul'vuku Fan/«Auk A curve C is defined by the parametric equations Example 1 .
m = t2 y = t3 — 3t 1. Show that C has two tangents at the point (3,0) and find
their equations. Wkln Aug C an" 'HAL Point (3,0) 2 ,1" +=E 4M 'i'i'J-g A curve C is defined by the parametric equations Example 1 m=t2 y=t3—3t 2. Find the .where the tangent is horizontal or vertical. Plow] HM. 44/le M10 by": pu'mjmz eiwiwi
.J - A curve C is defined by the parametric equations Example 1 . m=t2 y=t3—3t 3. Determine Where the curve is concave upward or downward.
/\_, A curve C is defined by the parametric equations Example 1 . m = t2 y = t3 — 3t
4. Sketch the curve. Consider the cycloid x : TH? — 31116), 3; : 'r(1 — COS :9). Example 2 . 1. Find the tangent to the cycloid at the point where 9 = arr/3 "cm 9 (8 'HM Pdvm-lev MA V' R a cwf'm'. Consider the cycloid a: : TU? — Sinfi'), y : 1"(1 — cos 6). Example 2 . 2. At what points is the tangent horizontal? Vertical? O:r(l-(o'é) A Otrn'na JV 1 .
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"or [ZOMLI Taullw'" /\/ Consider the eyeloid a: = TU? — Sinfl), y : 7"[1 — (303 6'). Example 2 . 2. At what points is the tangent horizontal? Vertical? AflDHM" Wm} +0 440w 'HNUL t4 k Vul'l'la' 'EKIWA Q {owl 'L'AX
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The area under the curve traced out once by the parametric equations
93=f(t} y=9(t) (1935 is given by 4 Han Tan,
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A., L 5mm" A? human (.4 = gm) 316W) Find the area under one arch of the eyeloid: Example 3. $=T(9—sin9) y=r(1—Cosi9)
hr Am: g W (M A0 21' = garb-(«9) '(l'éogé) 0W 1 | .
t—Lme + 2* aéxwy J9 '-'— 1'19 'lsme at £6 + 479'». 20) : r1(211+1r)
mlgga'V Jimlfl o >?'nv1 Arc Length Length of a Parametric Curve We Tcyfil; '. If a curve C is described by the parametric equations
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Fm) y=g(t) {15:33 . s: "A .3 +0 54
then the length of C is ("4". "a." 'k [M £1
l o C a} 'invuu'l {YALH7 (MN. as 'i' W'Kes 'Cyw Find the length of the parametric curve (OVAL 9C rum/LS I) Example 4.]. $=cost y=si11t 0£t£2fi ("4&qu Adria-hug)
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DC A ((4.1 wI'Hn VAAI'MS:| Find the length of the parametric curve Example 4.2 $=5in2t y=cos2t Ugthfl' This Cum k J40 A dull. 0C no't'M :'
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To LowpUSkl-Q Cr 'ngl we "luau ou(7 "1"th '(I'aw 0 'ILo'lTV. Find the length of one arch of the cycloid Example 5 . 53:?"(6—si119) y=r[1—cosl9) 04,9é2rr - 3E1 .
9 'l/(I'Qfa) J0 : lime (r(h(o(9))1 * ("in 6)1 '16 r1(|-cos9)l + v1<in19 A9
0 \ r 7:" 75W ("flu-I V 3' = g; V1(i'lc0494'(0516 +£11.19 JO 1 V gom A0 =1 Find the length of one arch of the cycleid Example 5 . 53:1"(6—51119) y:r(1—cesl9) Surface Area Surface Area of a Surface of Revoluticn We. aka f¢}uiV{: If a curve is described by the parametric equations
$=ffiJ y=9(t) agtsfl and is rotated about the m—axis then the area of the resulting L] W
W '9 d3: 2 dy 2
S=/u 213; (a) +(E) (it . Q, am} 13' am (owiiuuouf surface is Show that the surface area of a sphere of radius 'r is 4Tf'f'2. Example 6. TAM: Rotate a (awhciull
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g: g 7.113 g (£31 + (gay 0" '= (limiu'tmr 0"
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7' ngfl'rsiA [Tlciulbr (03+) oH' "- lTlrI g do!" J+ c l'frr1(rcog+) W '\ 0 =2:nr"(~(4)+ I) =qfi'zd/ 'fl'