Transcription of Lecture 6 - UH
1 Lecture 6 Section Inverse Trigonometric Functions Hyperbolic Sine and CosineJiwen He1 Inverse Trig Inverse SineInverse Sincesin 1x(orarcsinx)1domain:[ 12 ,12 ] range:[ 1,1]2sin(sin 1x) =x34domain:[ 1,1]range:[ 12 ,12 ]Trigonometric Properties5sin(sin 1x) =xcos(sin 1x) = 1 x2tan(sin 1x) =x 1 x2cot(sin 1x) = 1 x2xsec(sin 1x) =1 1 x2csc(sin 1x) =1xDifferentiationTheorem 1x=1 1 sin 1x. Thenx= siny,ddxsin 1x=1ddysiny=1cosy=1cos(sin 1x)=1 1 1u=1 1 u2dudx, 1 1 u2du= sin 1u+CIntegration:u-Substitution6 Theorem 3. g (x) 1 (g(x))2dx= sin 1(g(x)) +CProofLetu=g(x). Thendu=g (x)dx, g (x) 1 (g(x))2dx= 1 1 u2du= sin 1u+C= sin 1(g(x)) +CExamples4. 1 4 x2dx= 1 1 u2du= sin 1u+C= sin 1x2+ 4 x2= 4(1 (x2)2). Letu=x2. Thendu=12dx. 1 2x x2dx= 1 1 u2du= sin 1u+C= sin 1(x 1) + that 2x x2= 1 (x2 2x+ 1) = 1 (x 1)2(complete the square). Letu=x 1. Thendu= Inverse TangentInverse Tangenttan 1x(orarctanx)7y= tanxdomain:( 12 ,12 )range:( , )8 Trigonometric Propertiestan(tan 1x) =xcot(tan 1x) =1xsin(tan 1x) =x 1 +x2cos(tan 1x) =1 1 +x2sec(tan 1x) = 1 +x2csc(tan 1x) = 1 +x2xDifferentiationTheorem 1x=11 + tan 1x.
2 Thenx= tany,ddxtan 1x=1ddytany=1(secy)2=1(sec(tan 1x))2=11 + 1u=11 +u2dudx, 11 +u2du= tan 1u+C9 Integration:u-SubstitutionTheorem 7. g (x)1 + (g(x))2dx= tan 1(g(x)) +CProofLetu=g(x). Thendu=g (x)dx, g (x)1 + (g(x))2dx= 11 +u2du= tan 1u+C= tan 1(g(x)) +CExamples8. 14 +x2dx=12 11 +u2du=12tan 1u+C=12tan 1x2+ that 4+x2= 4(1 +(x2)2). Letu=x2. Thendu=12dx. 12 + 2x+x2dx= 11 +u2du= tan 1(x+1)+ that 2+2x+x2= 1+(x2+2x+1) = 1+(x+1)2(complete the square). Letu=x+ 1. Thendu=dx. e x1 +e 2xdx= 11 +u2du= tan 1(e x) + that 1 +e 2x= 1 + (e x)2(completethe square). Letu=e x. Thendu= e (t) =kf(t).1. Forf(0) = 4,f(t) =:(a)kt+ 4, (b) 4ekt, (c) 4e Fork>0, double timeT=: (a)4k,(b)ln 2k(c) ln Inverse SecantInverse Secantsec 1x11y= secxdomain:[0,12 ) (12 , ]range:( , 1] [1, )12 Trigonometric Propertiessec(sec 1x) =xcsc(sec 1x) =x x2 1sin(sec 1x) = x2 1xcos(sec 1x) =1xtan(sec 1x) = x2 1cot(sec 1x) =1 x2 1 DifferentiationTheorem 1x=1|x| x2 sec 1x.
3 Thenx= secy,ddxsec 1x=1ddysecy=1(secytany)2=1|x| x2 1u=1|u| u2 1dudx, 1u u2 1du= sec 1|u|+C13 Integration:u-SubstitutionTheorem 11. g (x)g(x) (g(x))2 1dx= sec 1(|g(x)|) +CProofLetu=g(x). Thendu=g (x)dx, g (x)g(x) (g(x))2 1dx= 1u u2 1du= sec 1(|g(x)|) +CExamples12. 1x x 1dx= 2 1u u2 1du=12sec 1 x+ thatx 1 = ( x)2 1. Letu= x. Thenx=u2,dx= Other Trig InversesOther Trigonometric InversesOther Trigonometric Inverses14sin 1x+ cos 1x= 2or cos 1x= 2 sin 1xtan 1x+ cot 1x= 2or cot 1x= 2 tan 1xsec 1x+ csc 1x= 2or csc 1x= 2 sec 1xDifferentiationTheorem 1x= ddxsin 1x= 1 1 x2ddxcot 1x= ddxtan 1x= 11 +x2ddxcsc 1x= ddxsec 1x= 1|x| x2 1 Quiz (cont.)The value, at the end of the 4 years, of a principle of $100 invested at 4%compounded3. annually: (a) 400(1 + ), (b) 100(1 + )4, (c) 100(1 + ).4. continuously: (a) , (b) , (c) 100(1 + ) Hyperbolic Sine and DefinitionHyperbolic Sine and Cosine15 Definition (ex e x),coshx=12(ex+e x)Theorem cosh,ddxcoshx= sinh,Identities1617cosh2x sinh2x= 1sinh(x+y) = sinhxcoshy+ coshxsinhycosh(x+y) = coshxcoshy+ sinhxsinhycos2x+ sin2x= 1sin(x+y) = sinxcosy+ cosxsinycos(x+y) = cosxcosy sinxsinyOutlineContents1 Inverse Trig Inverse Sine.
4 Inverse Tangent .. Inverse Secant .. Other Trig Inverses ..142 Hyperbolic Sine and Definition ..1519
