# calculus II

 The flashcards below were created by user mlalumia on FreezingBlue Flashcards. definition of even and odd functions from 1.2 An even function f satisfies f(-x)=f(x), while an odd function satisfies f(-x)=-f(x) Every functin f that is defined on an interval centered at the origin can be written in a unique way as the sum of one even functin and one odd functin.. WHAT is the composition?? What is the composition written e^x? f(x)=f(x)+f(x)/2 + f(x) - f(-x)/2 e^x = e^x + e^-x/2 + e^x - e^-x/2 what is the definition of hyperbolic funtions? The even and odd parts of e^x, and are called hyerbolic cosine and hyperbolic sine of x what is hyperbolic sine of x?? sinhx = e^x - e^-x/2 what is the hyperbolic cosine of x? coshx = (e^x + e^-x)/2 tanh x? Sinhx/coshx = (e^x - e^-x)/e^x +e^-x coth x? = cosh x/sinh x = (e^x + e^-x)/e^x - e^-x Hyperbolic secant? sech x = 1/coshx = 2/(e^x + e^-x) hyperbolic csc? csch x + 1/sinhx = 2/(e^x - e^-x) know the identities for these 7 hyperbolic functions 1. cosh^2 x - sinh^2 x 2. sinh 2x 3. cosh 2x 4. cosh^2 x 5. sinh^2 x 6.tanh^2 x 7coth^2 x 1=12=2sinh x cosh x3= cosh^2 x + sinh^2 x4=(cosh 2x + 1)/25=(cosh 2x -1)/26 = 1 - sech^2 x7 = 1 + csch^2 x what are the derivatives of the six hyperbolic functions? d/dx of? 1 sinh u? 2 cosh u? 3 tanh u? 4 coth u? 5 sech u? 6 csch u? 1 cosh u du/dx2 sinh u du/dx3 sech ^2 u du/dx4 -csch^2 u du/dx5 -sech u tanh u du/dx6 -csc u coth u du/dx the derivatives of the hyperbolic functions lead to the formulas for the integrals of hyperbolic fnctions. what are the integrals of these six hyperbolic functions?? 1 sinh u du? 2 cosh u du? 3 sech^2 u du? 4 csch^2 u du? 5sech u tan u du? 6 csch u coth u du? 1 cosh u + C2 sinh u + C3 tanh u + C4 -coth u + C5 -sech u + C6 -csch u + C There are also six derivatives of inverse hyperbolic functins d (____^-1 u)/dx. What are they? 1 sinh? 2 cosh ? 3 tanh ? 4 coth ? 5sech? 6 csch? 1= 1/ sqrt(1 + u^2) du/dx2 = 1/sqrt(u^2 -1) du/dx, u>13 = 1/sqrt(1 - u^2) du/dx, |u| < 14 = 1/sqrt(1- u^2) du/dx, |u| > 15 = -du/dx/ u sqrt(1-u^2), 0< u< 16 = -du/dx/ |u| sqrt (1 + u^2), u cant = 0 now shit gets a little crazy. there are 5 integrals that lead to inverse hyperbolic functions. 1 fdu/ sqrt(a^2 + u^2) 2 fdu/ sqrt(u^2 - a^2) 3 fdu/ sqrt(a^2 - u^2) 4 fdu/ usqrt(a^2 - u^2) 5 fdu/ usqrt(a^2 + u^2) what are the inverse hyperbolic functions? 1 = sinh^ -1 (u/a) + C a > 02= cos^-1 (u/a) + C u > a > 03= 1/a tanh^-1 (u/a) + C if u^2 < a^2 or 1/a coth^-1 (u/a) + C if u^2 > a^24= -1/a sech^-1 (u/a) + C, 0 < u < a5=-1/a csch^-1 | u/a | + C, u cant = 0 and a>0 three identities for inverse hyperbolic functions 1 sech^-1 x 2 csch^-1x 3 coth^-1 x 1= cosh^-1 1/x2= sinh^-1 1/x3=tanh^-1 1/x Authormlalumia ID23288 Card Setcalculus II Descriptionnotes on chapter 7 section 4. calc II Updated2010-06-13T07:04:57Z Show Answers