Transcription of Lecture 6 - UH
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Lecture 6 - UH
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.
Lecture 6 Section 7.7 Inverse Trigonometric Functions Section 7.8 Hyperbolic Sine and Cosine Jiwen He 1 Inverse Trig Functions 1.1 Inverse Sine Inverse Since sin−1 x (or arcsinx) 1
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