REVISION SHEET FP2 (Edx) CALCULUS. x x 0.5. x x 1.5. π π. Standard Calculus of Inverse Trig and Hyperbolic Trig Functions = + = + arcsin x = +

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1 the Further Mathematis netwk V 07 REVISION SHEET FP (Ed) CLCULUS The main ideas are: Calulus using inverse trig funtions & hperboli trig funtions and their inverses. Malaurin series Differentiating the Inverse Trig Funtions Befe the eam ou should know: That ou an differentiate the trig funtions, the hperboli trig funtions and their inverses. That ou an appl the standard rules f differentiation (produt rule, quotient rule and hain rule) to funtions whih involve the above. That ou an integrate, arsin(), aros(), artan(), arot(), arsinh(), arosh() et using integration b parts. Your trig identities and hperboli funtion identities and how to use them in integration problems. Partiularl get familiar with useful substitutions to make. How to onstrut and use redution fmulae How to alulate ar length and area of surfae of revolution..5 3 aros( ) d d artan( ) d d arsin( ) d d It is imptant to be aware of what the range is f eah of these, namel: π π π π arsin, 0 aros π, artan Standard Calulus of Inverse Trig and Hperboli Trig Funtions arsin( ) d d aros( ) d d artan( ) d d arsinh( ) d d + + ar osh( ) d d artan + + a a a a arsin + a a ar osh + a + a ar sinh + a Dislaimer: Ever efft has gone into ensuring the aura of this doument. However, the FM Netwk an aept no responsibilit f its ontent mathing eah speifiation eatl.

2 the Further Mathematis netwk V 07 Calulus using these funtions The eamples below are ver tpial and show most of the ommon triks. Note details of all substitutions have been omitted, make sure ou understand how to do them in this ase and also in the ase of a definite integral. + d arsinh d ( + ) d d d arsin arsin d d arosh ( ) 6 d arosh( ) d Some useful integration triks Splitting up an integration: e.g. (to see this use the hain rule, set z d d + d and then d d dz ). d dz d B inspetion: e.g. Sine ln( + ) gives + when differentiated, we have d ln( + ) + + sine ( + ) gives ( + ) when differentiated, we have d Using lever substitutions: e.g. the substitution u sinh( ) will help ou with + d. Redution Fmulae You should be able to derive and use redution fmulae f the evaluation of definite integrals in simple ases. i.e. to alulate ed.you should be able to find numerous eamples in our tet and ou should pratie 0 these to be omftable with the proedure involved. r Length and rea The length of an ar between points and B on a urve an be alulated b: B d + d d B d t B d d + d. In parametri fm this is: d t + dt dt The area of the surfae fmed when ar B is rotated ompletel about O is: B d π + d d B d π + d d (in parametri fm) π tb t dt d d + dt dt You should review eamples of how this tpe of question and how to solve them. This obviousl involves differentiation, algebrai manipulation and integration (often b substitution). dt Dislaimer: Ever efft has gone into ensuring the aura of this doument. However, the FM Netwk an aept no responsibilit f its ontent mathing eah speifiation eatl.

3 the Further Mathematis netwk V 07 REVISION SHEET FP (Ed) CO-ORDINTE SYSTEMS The main ideas are: Parametri and Cartesian Equations of the parabola, ellipse, hperbola and retangular hperbola Equations of tangents and nmals to the above Intrinsi odinates and radius of urvature Befe the eam ou should know: The parabola, ellipse and hperbola are eah loi of a point P whih moves so that its distane from the fied point (the fous) is in a onstant ration (e, the eentriit) to its distane from a fied line (the diretri). If the length of the ar P on a urve is s, and the tangent to the urve at P makes an angle of ψ with the positive -ais, then (s, ψ) are alled the intrinsi odinates of the point P. How to alulate the radius of urvature, ρ at a point P on a urve. The Parabola The parabola with equation a has fous at (a, 0) and diretri a. Parametriall the parabola with equation a is given b at, at. The tangent at (h, k) to the parabola has the equation k a( + h) and the tangent at (at, at) has the equation t + at. The responding nmal has the equation + t at + at 3. Eample Find the equation of the tangent to the parabola with equation a at the point T(at, at). If S is the fous find the equation of the hd QSR whih is parallel to the tangent at T. Prove that QR TS. Solution d d dt The gradient of the tangent at T is a. d dt d at t The tangent passes through T(at, at) and therefe has equation at ( at ) t + at. t The hd QSR is parallel to this tangent and so has the same gradient. Sine the hd passes through the fous (a, 0) the equation of hd QSR is 0 ( a) t a. t The distane from T(at, at) to S(a, 0) is ( ) ( ) at a + at 0 a t a t + a + a t at + at + a a t + t + a( t + ) Q and R are where t a intersets a. Using a + t in a gives a(a + t) at ± 6a t + 6a at a 0. The fmula f the roots of a quadrati gives at ± a t +. Dislaimer: Ever efft has gone into ensuring the aura of this doument. However, the FM Netwk an aept no responsibilit f its ontent mathing eah speifiation eatl.

4 The responding odinates are the Further Mathematis netwk V 07 a at at t + ± +. The distane between the two points ( a+ at + at t +,at+ a t + ) and ( a at at t,at a t ) is ( + ) + ( + ) ( + )( + ) at ( + ) 6at t 6a t a t t So the distane from T to S is four times the distane from Q to R. The Ellipse The ellipse with equation + where b a ( e ) has foi at ( ± ae,0) and diretries with a equations ±. Parametriall the ellipse with equation + is given b aost, bsint, e h k 0 t < π. The tangent to this ellipse at (h, k) has equation +. The tangent to this ellipse at (aost,bsint) is ost+ sin t. The Hperbola The hperbola with equation where b a ( e ) has foi at ( ± ae,0) and diretries with a equations ±. Parametriall the hperbola with equation is given b aset, btant, e h k 0 t < π (where these are defined). The tangent to this hperbola at (h, k) has equation. The tangent to this ellipse at (aost,bsint) is set+ tan t. Note: The parabola, ellipse and hperbola are eah loi of a point P whih moves so that it s distane from a fied point (the fous) is in a onstant ration (e, the eentriit) to its distane from a fied line (the diretri). This fous-diretri propert is suh that e f the parabola, 0 < e < f the ellipse and e > f the hperbola. The Retangular Hperbola The retangular hperbola is a speial ase of a hperbola with e. You should know the Cartesian and parametri equations f a hperbola and the equations of tangents and nmals. Intrinsi Codinates If the length of the ar P on a urve is s, and the tangent to the urve at P makes an angle of ψ with the positive -ais, then (s, ψ) are alled the intrinsi odinates of the point P. In partiular d d d tan ψ, os ψ, sinψ d ds ds The radius of urvature ρ at point P(, ) on the urve is ds dψ 3 d 3 + d +. d d Dislaimer: Ever efft has gone into ensuring the aura of this doument. However, the FM Netwk an aept no responsibilit f its ontent mathing eah speifiation eatl.

5 the Further Mathematis netwk V 07 REVISION SHEET FP (Ed) HYPERBOLIC TRIG FUNCTIONS The main ideas are: Definitions of the hperboli trig funtions and their inverses. Wking with the hperboli trig funtions Identities involving hperboli trig funtions The Hperboli Trig Funtions These are defined as: sinh( ), osh( ), sinh( ) tanh( ). osh( ) ln0 ln0 0 F eample, sinh(ln0) Befe the eam ou should know: The definitions sinh( ), osh( ), sinh( ) tanh( ) osh( ) That ou an prove that arosh( ) ln( + ),arsinh( ) ln( + + ) + artanh( ) ln Your trig identities and hperboli funtion identities, eperiene will tell ou when it is best to wk in the eponential fm when dealing with equations. nd be able to prove hperboli identities from the definitions sinh( ),osh( ), it s wth pratiing indies f this. The Inverse Hperboli Trig Funtions Just as the hperboli trig funtions are defined in terms of e, their inverses an be epressed in term of logs. In + fat arosh( ) ln( + ), arsinh( ) ln( + + ), artanh( ) ln. You should be able to prove (and use) all of these. Here is the proof that arosh( ) ln( ) +. Let ar osh( ), then osh( ). Rearranging this gives 0 e + e. Multipling this b e gives 0 +. This is a quadrati in e and using the fmula f the roots of a quadrati gives ± e ±. Taking logs gives ar osh( ) ln( ± ). Do ou know wh the epression with the minus sign is rejeted here? These epressions an be used to give eat values of the inverse hperboli trig funtions in term of logs. F eample, arosh ln + ln + ln(3) Dislaimer: Ever efft has gone into ensuring the aura of this doument. However, the FM Netwk an aept no responsibilit f its ontent mathing eah speifiation eatl.

6 the Further Mathematis netwk V 07 Graphs of the Hperboli Trig Funtions osh( ) d sinh( ) d 8 tanh( ) d d seh ( ) - - sinh( ) d osh( ) d - Graphs of the Inverse Hperboli Trig Funtions - - You must also know the graphs of the inverse hperboli trig funtions, arsinh, arosh and artanh. s f an funtion these are obtained b refleting the respetive graphs of sinh, osh and tanh in. The eamples of arsinh and arosh are shown here. Notie that arosh() is onl defined f greater than equal to. Identities Involving Hperboli Trig Funtions Identities involving hperboli trig funtions inlude: osh u sinh u, osh( u) osh u+ sinh u, sin( u+ v) sinh( u)osh( v) + osh( u)sinh( v ) The onl differene between a hperboli trig identit and the responding standard trig identit is that the sign is reversed when a produt of two sines is replaed b a produt of two sinhs. This is alled Osbn s Rule. You an prove an hperboli trig identit using their definitions and should be able to do this f the eam. Equations Involving Hperboli Trig Funtions Eample Solve the equation 3 osh + 5sinh 0 giving our answer in terms of natural logarithms. Solution e e 3osh + 5sinh e + 8e 0 0 9e 0e e e 9 ln ln 9 ( )( ) Dislaimer: Ever efft has gone into ensuring the aura of this doument. However, the FM Netwk an aept no responsibilit f its ontent mathing eah speifiation eatl.