SE1CY15 Differentiation and Integration Part B

Transcription

1 SECY5 Diffrniion nd Ingrion Pr B Diffrniion nd Ingrion 6 Prof Richrd Michll Tody w will sr o look mor ypicl signls including ponnils, logrihms nd hyprbolics Som of his cn b found in h rcommndd books Crof 67-9; Jms -, 5-57 Sroud 86-9,,95-57; Singh -5,66,6; Don forg o nd h uorils o g prcic Also, r suppor is vilbl from hp:// cnr nd hp:// p RJM /9/5 Prof Richrd Michll 5 Eponnil, Logrihm, Hyprbolic W m ponnils in lcur, in h RC circui hough hy cn b sn in mny sysms. Th currn dcyd smoohly ponnilly. In his lcur w will look h ponnil funcion, is diffrnil nd is ingrl. Logrihm is invrs ponnil w will considr i nd som pplicions Hyprbolic funcions r combinions of ponnils, so hs oo will b considrd. p RJM /9/5 Prof Richrd Michll 5 On Eponnils In lcur w rgud h vriion of I nd V Eponnils Formlly Funcion: p() or whr ~.788 is ; ~.78; - ~.7; ; - / I K.7K V E.6E TRC T W quod h quion for I E I RC R E V + I R, so V E I R V E - E RC L s look formlly E E Hnc RC, I.7 ; V E - E.6E R R As -, Ep() so dos slop Ep() so is slop As +, Ep() so dos slop d(p()) Suggss p() p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Mor on Eponnils Ep(- - ) ; Slop - Ep() ; Slop is - Ep(- ) ; Slop d(p(-)) So -p(-) Ep(- ) ; Slop Ep(*) ; Slop is Ep( ) ; Slop d(p()) So p() So for Eponnils d n From rsuls, gnrlis: n n n Thrfor n + c n Rmmbr for h RC Circui E I RC V E - E dv RC Also I C R RC dv de de RC E - - E RC RC RC dv E RC E C C RC I RC R p5 RJM /9/5 Prof Richrd Michll 5 p6 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5

2 SECY5 Diffrniion nd Ingrion Pr B In Clss Ercis For h RC circui, E 5V, R Ω nd C.F 5 Thus I..5 V I Find n prssion for V, givn h V whn s V * c A, -5 + c; so c 5 So V c Logrihm Invrs of Eponnil If y hn log y or ln(y) is log o bs Log o bs log usd in informion hory, for insnc Log ofn usd for sound : dcibls * log (volg) If log (y) hn y Ruls log( * b) log () + log (b) log ( n ) n * log () So log (/b) log ( * b - ) log() log(b) log () s for insnc log b(x) To chng bs : logx log b () p7 RJM /9/5 Prof Richrd Michll 5 p8 RJM /9/5 Prof Richrd Michll 5 Diffrnil of Logrihm Diffrnil of is sy, bu wh of is invrs ln? W find i using n rick, nmly ln(y) So y p() d(ln(y)) dy d(p()) p() y dy dy No, iniilly h rsul is funcion of, bu w cn hn chng i o on of y. Considr lso: Diffrnil of sin - Th diffrnil for sin - illusrs his furhr sin - () so sin() d(sin - ()) d d(sin()) cos() This is funcion of, should b funcion of. Bu s cos () + sin (), cos() ( - sin () ) d(sin - ()) So d - sin () - Similr mhods cn b usd for cos - nd n -. p9 RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 f - () y - f(y) dy If y sin - (), sin(y) Ingrion of sin - So sin - () y - sin(y) dy y + cos(y) y f(y) dy y f - () - f () Logrihmic Grphs A norml linrly scld grph hs s lik h following Th icks on h s r linr, hnc linr grph Answr should b funcion of : Shows grph of : y - y sin - (); cos(y) - sin (y) - So sin - () sin - () c Somims br for on/boh s o b logrihmic. Usd, for insnc, whn vribls ovr wid rng p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5

3 SECY5 Diffrniion nd Ingrion Pr B Applicion Frquncy Rspons If inpu o RC circui is K sin(), oupu is K sin(-). K Gin nd Phs -n - ( CR) K + ( CR) Frquncy Rspons : how Gin nd Phs vry wih ω Plo Grphs : Gin vs ω nd Phs vrsus ω Rlvn vriion of Gin nd Phs rquirs h ω is from ~ /RC o /RC if RC. from. o rd/s Th following log scl is ppropri : ch powr of qul spc Rngs: RC., ω. o rd/s Gin from o + ( CR).9 9 Phs -n - ( CR): -n O o -n O So plo log(gin) vs log(ω) nd Phs vs log(ω) p RJM /9/5 Prof Richrd Michll 5 - p RJM /9/ Prof Richrd Michll 5 Approiming Gin Grph Gin + ( CR) If CR, Gin ; If CR, Gin ; + ( CR) CR For formr, grph is horizonl lin For lr, Plo log vs log( ); CR i Plo log( CR) - -log( ) log(cr) vs log( ) i. Plo srigh lin : grdin is -, offs log(cr) - Gin grph wih sympos Th srigh lins, sympos, m ω /RC - Gin hn is.77 - Gin grph srs on firs sympo nd movs wy vnully ouching scond sympo Approiming Frq Rspons lik his is powrful ool p5 RJM /9/5 Prof Richrd Michll 5 p6 RJM /9/5 Prof Richrd Michll 5 Th hyprbolic sin, cosin nd ngn funcions r: sinh cosh sinh nh cosh p7 RJM /9/5 Hyprbolic Funcions y sinh y cosh Prof Richrd Michll y nh p8 RJM /9/5 Applicions of Hyprbolics In som nurl nworks, h oupu of nuron is summion pssd o diffrnibl civion funcion If h oupu is o b.., f(z) + -z sigmoid(z) z - -z If h oupu is o b -.., f(z) nh(z) z + -z A cnry is chin hnging undr is own wigh. Lr w driv n prssion for i, in rms of uni wigh w nd nsion T T w y cosh w T Prof Richrd Michll 5 Prof Richrd Michll, 5

4 SECY5 Diffrniion nd Ingrion Pr B Diffrnils / Ingrls sinh cosh dsinh cosh( ) dcosh Cn show h sinh( ) Thus cosh sinh( ) c Thus sinh cosh( ) c Hyprbolic Idniis cosh sinh cosh sinh - - cosh sinh cosh() cosh p9 RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Summry Tody w hv lookd ponnils, logrihms nd hyprbolics funcions ofn usd in sysms W hv rgud grphiclly hir diffrnil Tuoril Wk 6 Qs, nd 6. Find h following diffrnils, d5-5 p(-) d.5*ln() d cosh(5) ) b) c) Th of p() is Th of ln() is Th of cosh() is p() / sinh() 6. Find h following ingrls, ) p(-.) b) -5p(-) c) cosh() Agin h invrs mns p(-) is - ½ p(-) + c N wk : chin rul nd ingrion by subsiuion. 6. ) Find h quion of h ngn o h grph givn by y - 5 p(-). b) Find h mn of f() from - o ; p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Tuoril Wk 6 Qs nd 5 6. Th volg V in RC circui is givn by h following whn h inpu volg is. ; V dv ) Find dv b) Show h V is soluion of h quion. - V 6.5 Th phs shif of n RC circui is P -n -.5. dp Find. d Tuoril Wk 6 Q For h mss spring sysm blow, h cclrion d of h mss,, is givn by A, is vlociy is m/s nd is posiion is m. ) Find n prssion for posiion. d b) Show h 5 Oupu posiion, Spring, k (vlociy, v) Dshpo, F Objc, mss m p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5

5 SECY5 Diffrniion nd Ingrion Pr B Tuoril -Wk 6 -Hins 6. Apply sndrd ruls 6. Dio 6. ) m dy/ ; c y() m b) Us sndrd ingrl for mns 6. ) srighforwrd b) show boh sids of quion qul. 6.5 Us invrs funcion no diff of n() is sc (). 6.6 ) Ingr, find consn, ingr, find consn b) vlu lf hnd sid of quion show is 5. Diffrniion nd Ingrion 7 Prof Richrd Michll Tody : Chin Rul nd Ingrion by subsiuion Som of his cn b found in h rcommndd books Crof 75-7,8-86; Jms 97,5-57,55-55 Sroud 8-87, 87-89; Singh 69-8, Don forg o nd h uorils o g prcic Also, r suppor is vilbl from hp:// cnr nd hp:// p5 RJM /9/5 Prof Richrd Michll 5 p6 RJM /9/5 Prof Richrd Michll 5 Chin Rul Why Nd From ls wo wks w sblishd grphiclly h d - n d - d, - nd gnrlisd: n n n Thrfor n + c n dsin() dsin() cos() nd cos() d d dsin( ) In gnrl sin( ) d W will n considr h chin rul which cn formlly confirm hs, nd hn h quivln for ingrion. Th Chin Rul for sin(ω) sin(ω) simpl g of funcion dfind by wo funcions: f(z) z g() For sin (ω), z ω, nd sin (z). For diffrnil of wih rspc o us chin rul: So, for sin(ω), dz dz d sin(z) cos(z) dz dz dsin( ) Hnc cos(z) cos( ) dz d( ) p7 RJM /9/5 Prof Richrd Michll 5 p8 RJM /9/5 Prof Richrd Michll 5 p9 RJM /9/5 Empl for RC Circui V 5 5 p(-/): find I dv/ To diffrni p(-/), l z -/ nd p(z) dz dp(z) d(-/) dz dz dv d5 Thus 5 * -.*p( - ) p( - ) If you fl confidn (coms wih prcic), jus wri p(z) * -/ - -.*p( ) d 5p - dv d5 - -*- p - p - Prof Richrd Michll 5 p RJM /9/5 Sigmoid in Nurl Nworks Cybrniciss nd Compur Sciniss us nurl ns. In som nworks nuron s oupu is givn by sigmoid O -z {z is wighd sum of nuron inpus} For lrning, w nd o know h diffrnil of O wr z -z L +, so - -z dz - do - O, so -z + do do - -z -z So - dz dz + -z -z + Prof Richrd Michll 5 Prof Richrd Michll, 5 5

6 SECY5 Diffrniion nd Ingrion Pr B Anohr Empl Th gin of sysm, G, s funcion of ngulr frquncy ω, is G ( -. ω ).ω dg Find h hr vlus of such h dω L G so z ( -. ω ). ω z Simplify : z -.ω.ω. ω -.6ω.ω dz -. ω. ω d z -.6ω.ω dg G z so - z dz Coninud dz -. ω. ω d dg dg dz d dz d dg -.ω.ω - z -.ω. ω d z Wn ω whr his is zro nd numror only i.. -.ω. ω or ω(-.. ω ) or ω or.. ω i.. ω 8 So rquird rsul is ω or ω 8 rd/s p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 A Rld Tchniqu Find grdin poin,y on circl dfind by y r - Wn dy/; could k squr roo + crry on s norml. Esir wy : diffrni boh sids wih rspc o. dr- RHS: - - dy dy dy dy LHS: y dy dy So y r No nd o hv rsul wih no y Ercis You r o find how h volum of sphr (V / π r ) chngs wih im if r is 5 5 -/. dv dr dv Find, nd hnc dr Answr dv r r dr dr 5 -/ -/ dv dv dr r 5 5 dr -/ -/ -/ p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Ingrion Using Subsiuion This is rld o h chin rul usd in diffrniion. L's pply o sin( ) Simplify by subsiuing ω, nd hn ingr sin( ) sin() Bu mus ingr f() wr As,, so Thus sin( ) sin() -cos() * c Bu w inrocd o hlp us, nswr should hv sin( ) cos c Applicion - Roo Mn Squr Mn of on cycl of sin( ) sin( ) - cos( ) - - No usful msur, so ofn us roo mn squr or rms vlu of signl. rms sin( ) sin ( ) rms Looks wkwrd : us rig idniy sin () ½ ( cos()) b f() b- p5 RJM /9/5 Prof Richrd Michll 5 p6 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5 6

7 SECY5 Diffrniion nd Ingrion Pr B RMS of Sinusoid Som mor mpls of subsiuion rms sin ( ) sin () ½ ( cos()) sin(ω + ) : L ω + ; / ω rms -cos() sin( ) sin() -cos() -cos( ) + c + c - sin( ) - sin( ) sin() ; - ; -; - so ln() c ln( ) + c p7 RJM /9/5 Prof Richrd Michll 5 p8 RJM /9/5 Prof Richrd Michll 5 Dfini Ingrls nd Subsiuion Bwr of limis : rmmbr 5 f() mns ingr f() nd vlu o 5 Cn us subsiuion bu mus chng limis ccordingly w us 5+7, so 5 or (5 7) 5 limis 5*+77 nd 5* (5 7) Ingrion by Pril Frcions A chniqu o simplify ingrls o ons w cn solv. Th vlociy v of flling objc im found using. Find s funcion of v, hn v(). - v v fcoriss s ( v) nd ( + v) W solv h bov by rorgnising s follows A B + A nd B consns - v + v Hnc -Aln( - v) + Bln( + v) + c Wh r A nd B? p9 RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Finding A nd B nd hn A B + - v - v + v A( + v) B( - v) A( + v) + B( - v) + - v ( - v)( + v) ( - v)( + v) - v Numrors mus b qul A( + v) + B( - v) (A + B) + v(a - B) Equing cofficins: (A + B) (A - B) so A B nd A ln( - v) +.5 ln( + v) + c - v + v Compling Problm + v -.5 ln( - v) +.5 ln( + v) + c.5 ln + c - v Ok, bu wn v() no (v) + v + v - c.5 ln or ( - c) ln - v - v Tk p of boh sids ( - c) + v - v Ghr v rms ( - c) ( - v) + v ( - c) - v( + ( - c) ) ( - c) - v + ( - c) p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5 7

8 SECY5 Diffrniion nd Ingrion Pr B p RJM /9/5 Summry Tody w inrocd h Chin rul for diffrniion nd som simpl Ingrion by subsiuion. Th Chin Rul llows us o show dsin(5-7) Subsiuion llows us o show h 9 cos(5-7) + c 5 ln(9-) + c Thr r mny mor wys of subsiuing som rlvn o Enginring Mhs r givn n wk Prof Richrd Michll 5 Tuoril Wk 7 Q, nd 7. Th currn i hrough diod in rms of volg V cross i is i.5( V ) Find h.c. rsisnc r which is dfind s r. di dv 7. In rlibiliy nginring, h disribuion funcion, F, for s of componns is givn by F df Find h dnsiy funcion. 7.. Th gin of mss-spring, s funcion of ω, is dg G. Find vlus of ω whr. ω -9ω +5 dω Clcul G ll hs vlus o find whr G m. p RJM /9/5 Prof Richrd Michll 5 Tuoril Wk 7 Q, 5 nd 6 7. Th posiion, s, of n objc moving in srigh lin is s 8. + Find s givn h s whn. 7.5 Th sinusoidl oupu of sysm is givn by O 5 sin ( -.) Find is mn vlu bwn nd Find is rms vlu bwn nd.5. Tuoril -Wk 7 -Hins 7. Us Chin rul o find di/dv 7. Agin us h chin rul 7. Agin h chin rul look for whr numror is rmmbr cn hv ngiv frquncis. 7. Us subsiuion 7.5 nd 6 Us sm subsiuion for boh mn nd rms rmmbr o chng limis nd o us rdins. p5 RJM /9/5 Prof Richrd Michll 5 p6 RJM /9/5 Prof Richrd Michll 5 Diffrniion nd Ingrion 8 Prof Richrd Michll Ingrion Mor Subsiuion W hv lookd using simpl subsiuion for indfini nd dfini ingrls Tody : Volums/Surfcs; Mor Subsiuion. Som of his cn b found in h rcommndd books Crof 87-8, 85-89; Jms Sroud 89-8,86-87,9-99; Singh 7-8,- Don forg o nd h uorils o g prcic Also, r suppor is vilbl from hp:// cnr nd hp:// p7 RJM /9/5 Prof Richrd Michll 5 Rvision Now w shll look mor chniqus nd pplicions rld o using subsiuion in ingrion Firs w will s som gomric pplicions, which will provid som usful mpls lr Sring wih volums of rvoluion which r nsions of rs undr curvs p8 RJM /9/5 cos sin Prof Richrd Michll 5 Prof Richrd Michll, 5 8

9 SECY5 Diffrniion nd Ingrion Pr B p9 RJM /9/5 Volums of Solids of Rvoluion y f(): f() r undr curv: sum rs of srips: b f() * b As, bcoms f() If f rod round is hv solid of rvoluion: volum found by summing volums of cylindrs, rdius y & wih δ b V y As : V y f() b Prof Richrd Michll 5 b Lnghs of Curvs nd Surfc Ars Lngh of pr, ssum srigh lin, so s +y s y ds dy Div by : s b dy Hnc lngh from o b is: s Surfc r, whn curv rod, found similrly s p5 RJM /9/5 Prof Richrd Michll 5 b dy S y Empl Lin Rod o Con y f() Ingrion of funcion * driviv Find surfc r whn y rod bou is...5 dy ;.5 S 9 ; V Hr h 'min' funcion is 9, funcion of, muliplid by k* bing mulipl of h driviv of dy ; 9 S 6 s Th chniqu is o mk h subsiuion So or so S 9 No rms (h min funcion s driviv) cncl p5 RJM /9/5 Prof Richrd Michll 5 p5 RJM /9/5 Prof Richrd Michll 5 Compling h Empl Ingrls of f () / f().5 Wn S 9 ; L, ; For o.5, o S NB could hv chngd o f() & usd limis: bov sir No, h sm subsiuion lso works whn ingring h diffrnil of funcion DIVIDED by h funcion sin( ) g n( ) cos( ) dz L z cos( ); sin( ) sin( ) n( ) dz dz sin( )z z ln(z) c - ln cos( ) c p5 RJM /9/5 Prof Richrd Michll 5 p5 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5 9

10 SECY5 Diffrniion nd Ingrion Pr B Ercis Find h cnr of mss, C, of h smi-circulr lmin rdius : C my - y dy o m is mss pr uni r Us - y y or dy Whn y, ; whn y, dy -y y C m y -m y -y -m - -m m y Subsiuion Using Trigonomry Mos Mhs books covr ingrls lik Which dos no hv Enginring pplicion ( ) Hr h is wkwrd nd w subsiu o rmov i. cos - sin. L sin(), so cos() cos() cos() sin () cos() - + c sin + c Som Usful Subsiuions (ohrs my work) If n ingrl hs ( - ) + ( - ) ( + ) Try using sin() n () sc() sinh() p55 RJM /9/5 Prof Richrd Michll 5 p56 RJM /9/5 Prof Richrd Michll 5 Empl Forc F pplid o br, nsion l givn by: L F F l + L L n() so sc () d L F L F l sc () d sc () d + n () sc () L L F F - F - L l n n Ingrion solving Diff Eqns This is on pplicion of ingrion ss up n mpl g objc, posiion, cclring from iniil vlociy 5 + As RHS is funcion of, cn ingr boh sids wr 5 + Hnc c Somims w nd o invr h diff quion p57 RJM /9/5 Prof Richrd Michll 5 p58 RJM /9/5 Prof Richrd Michll 5 Applicion solving Diff Eqn dv E - V Th diffrnil quion for h RC circui: RC Bu RHS is funcion of V no : RC So w invr h quion nd g dv E - V RC Ingring boh sids wr V dv dv dv E - V RC Hnc dv E - V Rclling -ln ( - ) So -RC ln (E-V) + c - Coninud : Bu wn V s funcion of, so rrrng - c ln (E - V) -RC - c so -RC ln (E - V) E - V - c c - - Bu -RC RC * RC Ris boh sids o powr of - - so k RC E - V or V E - k RC - If V, E - k ; k E. Thus V E - E RC So -RC ln (E-V) + c c RC is consn, cll i k p59 RJM /9/5 Prof Richrd Michll 5 p6 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5

11 SECY5 Diffrniion nd Ingrion Pr B Cnry: Cbl Hnging Own Wigh You r no pcd o rmmbr his, jus si bck nd pprci h pplicion of vrious chniqus. W will us gomric chniqus nd ohrs o dc diffrnil quions for his & hn solv by ingrion. A P, Horizonl posiion, Vricl posiion y; whn y A P nsion in cbl is T p, ngl θ p6 RJM /9/5 Horizonl nsion T Wigh of cbl pr uni lngh w Considr wh hppns P Prof Richrd Michll 5 Cnry Coninud If cbl no moving his forc quls vricl pr of T p So w * s T p sin(θ) A P lso, horiz componn is T p cos(θ) which mus qul T Tsin( p ) w*s Thus n( ) Tcos( p ) T Lngh of cbl, o P : dy s + A P forc down w * s dy Bu P is slop n( ) dy w*s w dy d y w dy * + T T or T p6 RJM /9/5 Prof Richrd Michll 5 Cnry Coninud d y w dy dy dz w uggh! Bu l z so z T T T T W cn solv by or dz dz w z w z dz T L z sinh() ; cosh(), so cosh() w + sinh () T T T cosh() + c sinh (z) + c w cosh() w w dy T dy Bu, z, so c, hnc sinh ( ) w Cnry Concludd T dy dy w Now sinh ( ), so sinh w T w T w y sinh cosh + c T w T Bu y whn T w* T T cosh + c + c; c - w T w w T w y cosh - w T N wk: diffrniion nd ingrion of proc p6 RJM /9/5 Prof Richrd Michll 5 p6 RJM /9/5 Prof Richrd Michll 5 Tuoril Wk 8 Q, nd 8. ) Find h volum of rvoluion of h solid formd whn y is rod bou h -is from o. b) A prbolic rflcor is formd by roing y bou h -is, from o. Find is surfc r. 8. An rofoil is dfind by y for - Find h r blow his funcion nd bov h -is. 8. Th vlociy v of flling objc is givn by g -.5v Show h v g(- -.5 ) if v Tuoril Wk 8 Q nd 5 8. Th vlociy v of flling objc is givn by v g(- -.5 ). Find h disnc droppd im. 8.5 Th figur shows wr in prismic chnnl, for which h wr high y is dfind in rms of horizonl posiion by h quion dy 6 Find n prssion for y if y whn.5: p65 RJM /9/5 Prof Richrd Michll 5 p66 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5

12 SECY5 Diffrniion nd Ingrion Pr B Tuoril Wk 8 Q6 Er dp 8.6 Th logisic quion for populion P is P( - P).. ) Show h + dp P -P A b) Find P() if P im in h form P -B -c Tuoril -Wk 8 -Hins 8. Us rlvn formul nd ingr. 8. Is ingrl of squrroo us rig subsiuion. 8. Invr qn: find (v) -> v(). 8. Ingr. 8.5 Us rlvn hyprbolic subsiuion. 8.6 Invr qn, us Pril Frcions nd procd Diffrniion nd Ingrion 9 Prof Richrd Michll Tody : Diff nd In of Procs. Som of his cn b found in h rcommndd books Crof 79-7, 85-8 ; Jms 96-5, Sroud 79-8, 8-86; Singh 8-85, 88-96; Don forg o nd h uorils o g prcic Also, r suppor is vilbl from hp:// cnr nd hp:// p67 RJM /9/5 Prof Richrd Michll 5 p68 RJM /9/5 Prof Richrd Michll 5 Diffrniion of Proc Diffrniing Dmpd Sinusoid W cn diffrni h sum of wo (or mor) rms; nd w know h chin rul for mor complicd rms. N w will considr h proc of wo funcions. If h wo funcions r u nd v, hn w us h rul If drop mss, i oscills bu oscillions di ou - sin(b) d(uv) u + v Mss How dos chng?.g. Find vlociy of n objc wih posiion For : u so ; v - - so Thus vlociy is - ( * - + *) (6 - ) - u ; so - v sin(b); so b cos(b) d( - sin (b) - b cos(b) + sin(b) (- )) - b cos(b) - sin(b) p69 RJM /9/5 Prof Richrd Michll 5 p7 RJM /9/5 Prof Richrd Michll 5 Ercis If hr is considrbl fricion, h mss dos no oscill Thn is posiion vris by -. Find Answr - u ; so - v ; so d( - (- - )) ( - ) Diffrniion of Quoin Considr mchin wih grs, usd o driv is oupu, dfind in rms of n, h rio of on s o nohr. Hr h mchin cclrion is n n Suppos w wn o find how chngs wih n: d/dn. Hr w us h quoin rul, gin in rms of u nd v d u v v - u v (This cn b drivd from h of u * v - ) p7 RJM /9/5 Prof Richrd Michll 5 p7 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5

13 SECY5 Diffrniion nd Ingrion Pr B n n u n + ; so n dn Using h Quoin Rul u v v 8 - n; so - dn d n 8-n (8 - n)n - (n + )(-) Thus dn (8 - n) 6n - n + n + + 6n - n (8 - n) (8 - n) d v - u v Quoin Rul for n(ω) dsin( ) dcos( ) cos( ) W know cos( ) & so - sin( ); d n( ) sin( ) wh of? n( ) L u sin( ); v cos( ) d(n( )) cos( )*( cos( )) - sin( )*(-sin( )) So cos ( ) cos ( ) + sin ( ) sc ( ) cos ( ) NB sc()/cos() p7 RJM /9/5 Prof Richrd Michll 5 p7 RJM /9/5 Prof Richrd Michll 5 Applicion : Min/M - Opimision A bry, inrnl rsisnc r, wih vribl lod R. Find R o mimis powr. V E R P R r + R Find dp/dr nd hn R so dp/dr u E R so E ; v (r + R) so (r + R) dr dr dp (r + R) E - E R*(r + R) dr (r + R) E (r + R)(r + R - R) E (r - R) (r + R) (r + R) Clrly, his is zro whn R r. Ingrion By Prs This is mhod for ingring proc of wo rms. Is nsion of mhod for diffrniing procs d(uv) u + v Ingring boh sids givs d(uv) uv u + v Which rrrngd givs u uv - v p75 RJM /9/5 Prof Richrd Michll 5 p76 RJM /9/5 Prof Richrd Michll 5 Ky o Using Ingrion By Prs u uv - v For ingring proc of wo funcions, u nd /. Th ky is o slc which funcion is u nd which /. Th im is o choos hs suibly so h h rm v is sir o solv hn u Considr Showing Effc of Wrong Choic If l u cos() & cos() -sin() nd v u uv - v So ingrl is cos() -cos () + sin() Dfini vrsion of ingrl: p p p u uv q - v q q This hs md mrs hrdr. p77 RJM /9/5 Prof Richrd Michll 5 p78 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5

14 SECY5 Diffrniion nd Ingrion Pr B Bu wih righ choic Empl Mss Spring Sysm Agin Bu if u nd cos() cos() u uv - v Spring, k Dshpo, F Oupu posiion, (vlociy, v) Objc, mss m Typicl in conrol L m kg, F Nsm -, k Nm - nd v cos() sin() So cos() *sin() - *sin() sin() + cos() + c Shows why imporn o mk righ choic! Mss pulld, Spring ndd, forcd bck, opposd by fricion. Thn -* - * v Suppos v - -. Find if, im, m Scond rm sy, Us ingrion by prs for firs: p79 RJM /9/5 Prof Richrd Michll 5 p8 RJM /9/5 Prof Richrd Michll 5 Ingring - So o ingr - w choos u nd v rms snsibly L u nd - - Thn nd v - So u uv - v - - v - Compling Problm W now know Also - So v c - + c - - A, ; so - + c; c ; - Ension: vrify h nd v r soluions of - * - * v ( would nd o diffrni v o solv his problm ) p8 RJM /9/5 Prof Richrd Michll 5 p8 RJM /9/5 Prof Richrd Michll 5 Applicion Solving Diff Eqn dv Suppos w wn o solv + 5V 5 5 dv 5 5 If muliply by, g + 5V d(v 5 ) dv 5 5 Bu + V5 d(v 5 ) So or V 5 W us ingrion by prs o vlu 5 And hn divid by o find V: V 5 5 Compling Problm 5 For 5 5 : L u so ; so v. So c So V c -5 Thus V c Suppos V : A : -. + c; so c. Hnc V p8 RJM /9/5 Prof Richrd Michll 5 p8 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5

15 SECY5 Diffrniion nd Ingrion Pr B Mns Using Dfini Vrsion of Ingrl W quod rlir h sw() ~ sin() sin() sin() sin() T n n f() cos T T T T n b n f() sin T T T cos(n ) /n if n is odd - n -/n if n is vn should b L s do b n Finding b n for Swooh b n sin(n) { rmmbr n is consn } L u, nd sin(n) So ; nd v - cos(n) n So b n - cos(n) - - cos(n) π n n cos(n) + sin(n) n - n - p85 RJM /9/5 Prof Richrd Michll 5 p86 RJM /9/5 Prof Richrd Michll 5 Empl Coninud b n - cos(n) + sin(n) n - - n Bu, cos() cos(-); So cos(n) - is cos(n ) - - cos(-n ) cos(n ) Also sin() sin(-); sin(nπ) (n ingr) So sin(n) - is sin(n ) - sin(-n ) sin(n ) cos(n ) So b n * - * *cos(n ) - (s sd) n n You find n rm in uoril! Summry Hr w hv lookd h diffrnil nd ingrl of proc (or quoin rms) d(uv) u + v u uv - v d u v v - u v Th ky o ingrion is o choos u so scond ingrl is sir. For 5, wh is u nd wh is v? u 5; so v p87 RJM /9/5 Prof Richrd Michll 5 p88 RJM /9/5 Prof Richrd Michll 5 Tuoril Wk 9 Q Tuoril Wk 9 Q 9. In h mss-spring sysm blow, h posiion of h mss is givn by - ( sin() cos()) ) Show h im, h vlociy v is - m/s. d d b) Find nd hnc show h 5 Oupu posiion, Spring, k (vlociy, v) m kg; Objc, mss m F Ns/m; k 5 N/m Dshpo, F 9. For h mss-spring sysm, v, h mss' vlociy, is givn by v ) Find - b) Hnc find, h posiion of h mss, if, m. d d c) Find nd so show h 8 6 Spring, k Oupu posiion, (vlociy, v) m kg; Objc, mss m F 8 Ns/m; k 6 N/m Dshpo, F p89 RJM /9/5 Prof Richrd Michll 5 p9 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5 5

16 SECY5 Diffrniion nd Ingrion Pr B p9 RJM /9/5 Tuoril Wk 9 Q 9. Diffrni h following using h quoin rul sin( ) ) f() sinc() [usd in informion hory] b) Th displcmn of dmpd vricl pnlum cos(5) p() c) Th volg in n lcronic circui V p(.) Prof Richrd Michll 5 Tuoril Wk 9 Q,5 nd 6 9. Th gin of n lcronic circui, in rms of ngulr frquncy is givn by G Find such h G is mimisd. 9.5 Find h n rm for h Fourir Sris of swooh. i. find n cos(n) { rcll n is consn } Find r A undr of K for > : K nd consn i.. solv A K - ; Nb s, so Er! p9 RJM /9/5 Prof Richrd Michll 5 Tuoril Wk 9 Hins 9. ) Find /, nd pu. Us diff of proc b) Us diff of proc nd hn show LHS RHS 9. ) Choos u nd / s pr mpls b) Srighforwrd c) Diff of proc hn show LHS RHS 9. Srighforwrd. 9. Us diff of quoin o find dg/dω, find ω whr his is zro nd vlu G hs vlus. 9.5 Us Ingrion by Prs 9.6 Dio Diffrniion nd Ingrion Prof Richrd Michll Numricl Diffrniion nd Ingrion Som of his cn b found in h rcommndd books Crof 89-8; Jms 6-, 79-; Sroud 67-68; Singh 6-9; Don forg o nd h uorils o g prcic Also, r suppor is vilbl from hp:// cnr nd hp:// p9 RJM /9/5 Prof Richrd Michll 5 p9 RJM /9/5 Prof Richrd Michll 5 Numricl Diffrniion If hv funcion, cn lmos lwys diffrni i. Bu if signl is jus numbrs, cn only diffrni by siming: clld numricl diffrniion. In conrol sysms, common conrollr is PID, i ks signl & rurns P * + I * ingrl() + D * diff() Suppos d vlus ims h pr (g f(), f(+h)) Givn rul for diffrniion Migh pc sim of diff f(+ ) - f() f'() lim f(+h) - f() f'() h Howvr As wn diffrnil, don jus us grdin bfor, rhr us grdin bfor nd fr. Us f(+h), f(-h) f'() f(+h) - f(-h) *h.g. siming f () for f() : h.. f(.9).6 f(). f(.). Mhod : (.-.)/.. Mhod : (.-.6)/.. p95 RJM /9/5 Prof Richrd Michll 5 p96 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5 6

17 SECY5 Diffrniion nd Ingrion Pr B Why Avoid Numricl Diff Numricl Diffrniion should b voidd if possibl. This is cos i usully involvs dividing by smll numbr Th vlu bing dividd could hv rrors o poor msurmn or rounding rrors in rlir clculions. As dnominor <, hs rrors r mplifid. Suppos numror. bu should b : rror. Suppos lso, dnominor. Diffrnil simd s. /.. I should b /.. Thus rror of. mplifid so rror now.. Applicion : Smll Chngs A circulr pic of ml is o b copid, bu is rdius is mismsurd How do you find h rror in h r of h ml? For f(), if f is smll chng in f & smll chng in. df lim f For smll chngs, his cn b pproimd s df f So smll chng in f cn b simd by df f p97 RJM /9/5 Prof Richrd Michll 5 p98 RJM /9/5 Prof Richrd Michll 5 Applying o Circulr Disc Th r, A, is r, whr r is rdius. Th chng in r o smll chng in r is: da A r rr dr Suppos rdius ws msurd s., no, i.. r. Th rror in h r is: * *..68 Anohr mpl: h rror in volum of sphr whn rdius msurd s. whn i should hv bn. dv V r so r. Thus V *.. dr Why Us Numricl Ingrion Ofn hv funcion o b ingrd, for which n nlyicl funcion cn b found g cos() sin() + c For som funcions, subsiuion is ndd Howvr, for ohr funcions hr is no such soluion g p( ) Also, somims nginrs hv jus d vlus (from n primn sy), so no funcion o b ingrd Th soluion: pproim ingrion numriclly. Th concp cn lso b ndd for numricl soluion of diffrnil quions. p99 RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Bsic Concp If cn ingr f(), or don know f() nlyiclly, divid r ino srips, find r of ch nd sum. Rcngulr Ingrion Simpls (ls ccur) mhod; ssum ch srip rcngl Vrious wys of finding r of ch srip. Ech srip hs sm wih cll i h b Wn f() ( b 6 ) Ar of r'h rcngl wih * high h * f( r) 6 6 Ar of 7 srips: h *f( r) h * f( r) r r p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5 7

18 SECY5 Diffrniion nd Ingrion Pr B Trpzoidl Ingrion Mor ccur ssum ch srip is rpzium in ffc pproim curv s srigh lins bwn poins Empl To s us funcion which hs is n nlyicl soluion so cn s how ccur h wo mhods r. ( b 6 ) Ar of r'h srip is h * f( r) + f( r+) Tol r is sum of hs If do his dircly coun f(n) wic, so br o do h * 5 6 f( ) + f( ) + f( r ) r 5 p Th corrc nswr is * p(.5) - p(.) 5.55 Firs, s h, so us vlus,,, nd 5. For h rcngulr mhod, h r is: * ( ) poor For h rpzoidl mhod, h r is: * (.5 * (. +.5 ) ) 5. ~ok p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Mking Srips Nrrowr You should pc br rsul if nrrowr srips usd Wih, h Rcngulr Trpzoidl Rc poor; Trp good wih h., or smllr Simpson s Rul Trpzium rul ssums srigh lin bwn djcn ps Simpson s ruls gos on sg furhr ; I uss n vn numbr of srips For firs hr djcn poins (nd hnc wo srips) find qudric funcion going hrough hm clcul r For poins,, 5 (i.. h following wo srips) find qudric funcion nd r c Sum ll hs rs..g p(/) for..5; us... nd...5 p5 RJM /9/5 Prof Richrd Michll 5 p6 RJM /9/5 Prof Richrd Michll 5 Finding Qudric Funcion Suppos hv hr vlus,, inrvls h, nd r rying o ingr f() knowing f( ), f( ) nd f( ). Us Lgrng Polynomils o find p() pssing hrough ps. Us shorhnd : f f( ), f f( ), f f( ). - - p() f f + f Bu - -h; - -h, c., so p() f - f + f h h h Simpson s Coninud For simpliciy, dfin r such h r * h whn, r So (r + ) * h whn, r - (r - ) * h whn, r + Thn Polynomil p() f - f + f h h h Bcoms p() (r-)*r*f - (r+)(r-)* f + (r+)*r*f W cn now ingr p() o g r. W no h h dr; p7 RJM /9/5 Prof Richrd Michll 5 p8 RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5 8

19 SECY5 Diffrniion nd Ingrion Pr B Simpson s Finlly Thn rquird r bcoms p() r- r+ *r*f - (r+)(r-)* f + *r*f h dr h r - r * f - r -r * f + r + r * f - h f( ) + f( ) + f( ) { rcll f shor for f( ) } For n wo srips, r is h f( ) + f( ) + f( ) h m/ (m-)/ Ar is * f( ) + f( m) + * f( r-) + * f( r) r r Simpson s On Ep(/) Applying o p(/), whr h * ( + ) + * Th corrc nswr is * p(.5) - p(.) Hr, vn wih wid srips, nswr vry good. If h.5, Simpson s rul givs If h., nswr is p9 RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Summry W hv discussd numricl diffrniion (o b voidd if poss) nd numricl ingrion. Ths concps will b ndd n rm o considr numricl soluion of diffrnil quions. This concluds his sris of lcurs on diffrniion nd ingrion N rm, h clculus hm coninus, wih rvision nd nsion of h opics covrd, nd will lso considr h drivion nd soluion of diffrnil quions. Prof Richrd Michll 5 Tuoril Wk Q nd. Smll Chngs - find chng in ) surfc r of sphr whn is rdius chngs from o. m. [Surfc r r ] b) powr in circui whn currn chngs from ma o.99 ma, whn pssing hrough k rsisor. [If I is currn going hrough rsisor R, Powr I R] c) gin of RC circui whn ngulr frquncy flls by % from. rd/s; if R*C hn Gin G. For f sim is diffrnil, for h. f( h) - f(-h) using h formul f'(). Commn. *h p RJM /9/5 Prof Richrd Michll 5 Tuoril Wk Q nd Q 5. Evlu sin. nlyiclly. Compr rsuls wih Rcngulr Trpzoidl nd Simpson mhods whr h is.. Rvision Th volg cross h cpcior in RLC sris circui is V - - ( sin () + cos () ) ) Find h currn in h circui, bing I.5 dv b) Find h volg cross h in cor, bing V di L c) Vrify h I + V L + V Tuoril Wk Q5.5 Rvision d(5v. ) ) Epnd dv b) An RC circui is dscribd by 5 + V Show h V. -.. c) Us ingrion by prs o find. d) Hnc find V givn h V im. Hins Q, nd us mhods in his wk s nos Q nd 5 look bck in prvious lcurs p RJM /9/5 Prof Richrd Michll 5 p RJM /9/5 Prof Richrd Michll 5 Prof Richrd Michll, 5 9