An Insight into Differentiation and Integration

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Madlyn Butler
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1 Differetiatio A Isigt ito Differetiatio a Itegratio Differetiatio is basically a task to fi out ow oe variable is cagig i relatio to aoter variable, te latter is usually take as a cause of te cage. For istace, y We migt cosier takig values:,,,., + successively by icremet, te y is icreasig by a icremet ( + ) + at Te average icremet over te iterval (, + ) of is ( + ) + To be more geerally, ( + ) te average icremet over te iterval (, + ) of is + By ecreasig te amout of icremet of to be, te average icremet over te iterval (, + ) of is ( + ) + ( ) Geeralizig, te average icremet over te iterval (, + ) of is ( + ) + ( ) Goig o tis way, we te come to te iea tat te istataeous average icremet at is, we te iterval becomes witless. Tis is efie as te rate of cage of wit respective to at ay value of. To epress te process of work, wic we efie as ifferetiatio, a te result, we write ( + ) lim We ave tree fuametal results of ifferetiatio: for positive iteger si cos (Base o te fuametal limit result is i raias e e siδ lim ), we

2 From tese, wit te elps of a umber of rules suc as Te Prouct Rules, Te Quotiet Rule, Te Cai Rule, Implicit Fuctio Rule we euce may staar results icluig for ay ratioal umber (Fially, for ay real umber ) Of course, we soul be able to establis all oter results from first priciples, wic meas a work relyig oly o te efiitio f( + ) f( ) f( ) lim Itegratio Differetiatio is importat ot oly tat it is a tool to track te cage of oe variable wit respect to aoter, also, kowig of te erivative of a fuctio leas us to te ietificatio of te fuctio itself. For eample, if we fi tat f ( ) g( ) u( ), te f ( ) a g( ) ca oly iffer by a costat So, give f ( ) u ( ), to fi f ( ), we first look for g( ) wic as te kow erivative u ( ) Te job is u ( ), wic gives a result peig a costat We write f ( ) g( ) + c If we kow tat f( a ), te c g( a), f ( ) g( ) g( a). Tis is a efiite result, wic gives value of f ( ) at ay value, for eample, at b, b f () b g() b g() a, for wic we write u ( ) a Te value is efiite a te work is calle a efiite itegratio Cases of usig erivative to fi te origial fuctios A well-kow case is fiig istace travele wit kowlege of velocity v. If s is te istace travele i time t, we ave s v t, base o te geuie efiitio of v For some oter cases, te erivative of a variable must be reasoe out, toug tey are usually take ituitively. Take te case we we come to suc work like fiig area uer a curve.

3 Y y y+ δ y A() δ A O a X If we efie A( ) to be te area uer te curve from A Ituitively, we take y To be a bit more rigorous We first claim tat δ A lies betwee yδ a ( y+ δy) δ, δ A So, lies betwee y a ( y + δ y) A δ A lim y a to, so A y, Area A( ) ca te be fou Take aoter case, for fiig te volume V() of a rigt circular coe of eigt Deotig V( ) to be te volume of te circular coe, of te same sape, wit a eigt, cosiere as a variable, S to be te surface area of te circular cross-sectio at eigt,

4 A S r B δ V() O We claim tat δ V lies betwee S δ a (S + δs) δ δ V lies betwee S a (S + δ S) δ V δ V so lim S. δ δ By similarity, r k, for a costat k k k c ( ) π k + c π r+ c π r if we take V for V S π( ) π + Eercise (witout solutio attace). Differetiate ta wit respect to from first priciples. siδ (You may use te result lim ). Base o te result for positive iteger, a te rules of ifferetiatio, prove tat Prove also tat A particle is movig vertically upwars wit velocity v 5t 5t Sow tat it is movig owwars after t Fi te istace of te particle from its positio at t 4

5 (i) We t 8 (ii) We t s (5t 5 t ) t 5 [5 t t ] 7 4. Give tat y sec, fi y Give a reaso wy i Calculus, raia measure is preferre rater ta egree measure. 5. A kite is at a orizotal istace l from te flyer a a vertical istace above te same. We it is risig up at a velocity v, te flyer let off more strig to keep te orizotal istace costat. Assumig te strig to be straigt, a te agle of elevatio of te kite to be θ raias, fi te rate of cage of θ wit respective to time t * 6. Let ( e ) ( + ), were is a large umber * Sow tat ( e ) * ( e ) 5

6 Eercise (wit solutio attace). Differetiate ta wit respect to from first priciples. siδ (You may use te result lim ) Solutio: ta( + ) ta si( + )cos cos( + )si δ[cos( + δ)cos ] si( + ) si δ[cos( + δ) cos ] δ cos( + δ) cos ta( + ) ta siδ So, lim lim[ ] δ cos( + δ)cos sec. Base o te result for positive iteger, a te rules of ifferetiatio, prove tat Prove also tat Solutio: Let y y y, te y, y, y For provig of, let z. A particle is movig vertically upwars wit velocity v 5t 5t Sow tat it is movig owwars after t Fi te istace of te particle from its positio at t (iii) We t 8 (iv) We t Solutio: v 5t 5 t 5 t( t) < we t > So, te particle is movig owwars after t (i) If s is te istace move upwars after t s v 5t 5 t, otig tat tis is true for all t, t tat is weter t < or t s 8 (5t 5 t ) t 5 8 [5 t t ] 746 (ii) s v 5t 5 t, eve toug t t s (5t 5 t ) t 5 [5 t t ] 7 6

7 4. Give tat y sec, fi y Give a reaso wy i Calculus, raia measure is preferre rater ta egree measure. Solutio Let y sec secu, were u raias egrees. Te 8u π y y u π π (secuta u) (sec ta ) u 8 8 π As see from te above, sec (sec ta ) 8 r Wereas secu secuta u u 5. A kite is at a orizotal istace l from te flyer a a vertical istace above te same. We it is risig up at a velocity v, te flyer let off more strig to keep te orizotal istace costat. Assumig te strig to be straigt, a te agle of elevatio of te kite to be θ raias, fi te rate of cage of θ wit respective to time t Solutio We ave ta θ, were is a variable a l is a costat l Toug a epressio for θ is wat we are lookig for, we ee ot t writeθ ta ( ), a o θ.we ca coose to o iirectly l t θ Differetiatig bot sies of tis equatio, we get sec θ v t l t l θ So (cos θ ) v, toug te result is ot totally i terms of te variable. t l * 6. Let ( e ) ( + ), were is a large umber * Sow tat ( e ) * ( e ) Solutio * ( e ) ( ) ( ) ( ) ( ) ( + ) e, as ( + ) * ( ) ( + ) ( + ) 7

LIMITS AND DERIVATIVES

Capter LIMITS AND DERIVATIVES. Overview.. Limits of a fuctio Let f be a fuctio defied i a domai wic we take to be a iterval, say, I. We sall study te cocept of it of f at a poit a i I. We say f ( ) is