10.2 Parametric Calculus

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Claude Griffin
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1 10. Paametic Calculus Let s now tun ou attention to figuing out how to do all that good calculus stuff with a paametically defined function. As a woking eample, let s conside the cuve taced out by a point on the edge of a olling tie. We call this cuve a cycloid and, as you can imagine, it cannot be descibed nicely in the fom y f (). 1

2 To descibe the cycloid, let s conside a point on a olling cicle of adius initially at the oigin and label its and position in tems of the angle olled. (t) y This gives ise to the paametic equations t sin t and y(t) 1 cos t Ou goal now will be to compute thee things: 1) The slope of a tangent,. ) The aea unde the cuve, A. t 3) The ac length of the cuve, L.

3 Tangents: (t) y(t) y f () We need to compute but we don t have. We only have and. That s okay, we can use the chain ule. fo 0 Note, we can also use this tick to compute highe ode deivatives. Fo eample, at second ode we have d y d 3

4 Eample: Find all points on the cycloid with hoizontal tangent lines. (t) t sin t 1 cos t y(t) So we find 1 cos t sin t sin t sin t 1 cos t 1 cos t 4

5 Since sin t 1 cos t the cycloid will have a hoizontal tangent when sin t 0 and 1cos t 0 t (n 1) fo n This coesponds to points (, y) ((n 1),) y

6 Aea: Now let s find the aea unde one ach of a cycloid. Nomally we would compute the aea unde a cuve by integating y 0 A 1 f () How can we eintepet this as an integal in tems of the paamete? t y f () is just the height of the function so we can eplace it with y y(t) The diffeential detemines which vaiable we integate ove. This can be eplaced using '(t) 6

7 We must also change ou limits of integation to eflect ou shift fom integating ove to integating ove t. The esult is A t t 1 y(t)'(t) Eample: Find the aea unde one ach of a cycloid. Fom the tangent poblem we have y(t)'(t) 1cos t y What ae the limits of integation though? 0 7

8 We find the new limits the same way we would fo any substitution: y If we wee integating ove, the limits would be 0 and 0 We then use (t) t sin t to wok out the value of the paamete t coesponding to these values. 0 t sin t 0 t 0 t sin t t So the aea is A 0 1 cos t 3 Eecise: Veify this esult. 8

9 Ac Length Finally, let s look at how to compute ac length given a paameteized cuve. Rathe than manipulating ou eisting epession fo ac length (which we could do), let s go back to fist pinciples. Recall, we compute ac length by diving a cuve into dl infinitesimal segments of length y dl and then integating. When the cuve is defined paametically, we ewite the diffeentials and in tems of, '(t) and y'(t) 9

10 Then, dl ' y' ' y' Use and '(t) y'(t) Facto out Theefoe, we aive at the ac length fomula L t t 1 Eecise: Compute the ac length of one ach of the cycloid (t) t sin t y(t) 1cos t defined by and. 10

11 Eample: Conside a piece of a paabola paameteized by (t) t y(t) 4t t and fo t [0,4] a) Sketch this cuve. b) Find the location whee the tangent line is hoizontal using the paametic equations fo the cuve. c) Find the aea unde the cuve using the paametic equations. 11

Compute the distance will you cove elative to the gound afte one evolution

12 Eample: If you walk adially outwads fom the cente of mey-go-ound spinning, you will tace out a spial path elative to a stationay obseve. Compute the distance will you cove elative to the gound afte one evolution stating at t 0 given the paameteization (t) t cos(t) and y(t) tsin(t) Achimedean spial 1

13 Eample: A binay black hole system loses enegy via gavitational waves causing the black holes to spial inwads. The path of one of the black holes is given by (t) What distance does this black hole cove between t 0 and t? e (time in units of seconds, distance in units of 1,000,000 km) t cos(t) and y(t) e t sin(t) 13

14 13. Calculus with Vecto Functions Often 3D paametically defined functions o vecto functions ae used to descibe tajectoies. In this case, we would like to be able to compute deivatives (e.g., to find velocity fom displacement) o integate (e.g., to do the evese). So how do we diffeentiate o integate something like (t) (t)î y(t)ĵ z(t)kˆ It tuns out we can simply diffeentiate o integate component-by-component: E.g., d dz (t) î ĵ kˆ 14

Eample: An electon tavels though a magnetic field in a helical tajectoy with velocity R v(t) R sin( t),r cos( t), v 0 whee is the adius of the heli, is the angula

15 Eample: An electon tavels though a magnetic field in a helical tajectoy with velocity R v(t) R sin( t),r cos( t), v 0 whee is the adius of the heli, is the angula fequency, and is the speed in the z-diection. v 0 a a) Find the acceleation of the electon at t /. b) Find the displacement between t 0 and t 3 / (i.e., afte 1.5 evolutions). 15

16 Miem Pep: Review Sessions Satuday, Feb. 0, 6 8 PM in DC 1351 (OneMatch club) Sunday, Feb. 1, 4 PM in MC 4059 (with me) Stu Tips Wok though quizzes and woksheets ty to do them again without looking at solutions fist. Wok though old miem (time youself and lean whee you weaknesses may be). Solutions ae posted. Wok though eamples fom class. Do poblems fom the tetbook. (Also, take time off to ela mental health is impotant). Office Hous duing Reading Week to be announced and by appointment (but afte Wednesday). 16

17 Miem Mateial: Integation by Pats: u dv uv vdu Tig Integals: sin m ()cos n () and tan m ()sec n () Tig Substitutions: a a Patial Factions: Integate and a A a b asin a tan asec A B a b c Impope Integals: Infinite limits, discontinuous integands, convegence/divegence, compaison test 17

18 Miem Mateial: Volumes of Revolution: Disks, washes, & cylindical shells. Ac Length: min 0 L Suface Aea by Revolution: b a b ma V 1 f '() a ma min S f () 1 (f ') S g(y) d yc h() 1 (g') 18

19 Miem Mateial: Fomulating Diffeential Equations: Given a statement, wite down a diffeential equation that models it. Diection Fields: Daw; autonomous; locate solutions Sepaable DEs: g()h(y) 1 h(y) g() Linea DEs: Rewite in fom Find integating facto Solve y() y' P()y Q() I() e 1 I() P() I()Q() C 19

ME 210 Applied Mathematics for Mechanical Engineers

ME 210 Applied Mathematics for Mechanical Engineers Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the