calculus II

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Author:
mlalumia
ID:
23288
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calculus II
Updated:
2010-06-13 03:04:57
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hperbolic functions
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notes on chapter 7 section 4. calc II
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  1. 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)
  2. 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
  3. 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
  4. what is hyperbolic sine of x??
    sinhx = e^x - e^-x/2
  5. what is the hyperbolic cosine of x?
    coshx = (e^x + e^-x)/2
  6. tanh x?
    Sinhx/coshx = (e^x - e^-x)/e^x +e^-x
  7. coth x?
    = cosh x/sinh x = (e^x + e^-x)/e^x - e^-x
  8. Hyperbolic secant?
    sech x = 1/coshx = 2/(e^x + e^-x)
  9. hyperbolic csc?
    csch x + 1/sinhx = 2/(e^x - e^-x)
  10. 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=1
    • 2=2sinh x cosh x
    • 3= cosh^2 x + sinh^2 x
    • 4=(cosh 2x + 1)/2
    • 5=(cosh 2x -1)/2
    • 6 = 1 - sech^2 x
    • 7 = 1 + csch^2 x
  11. 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/dx
    • 2 sinh u du/dx
    • 3 sech ^2 u du/dx
    • 4 -csch^2 u du/dx
    • 5 -sech u tanh u du/dx
    • 6 -csc u coth u du/dx
  12. 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 + C
    • 2 sinh u + C
    • 3 tanh u + C
    • 4 -coth u + C
    • 5 -sech u + C
    • 6 -csch u + C
  13. 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/dx
    • 2 = 1/sqrt(u^2 -1) du/dx, u>1
    • 3 = 1/sqrt(1 - u^2) du/dx, |u| < 1
    • 4 = 1/sqrt(1- u^2) du/dx, |u| > 1
    • 5 = -du/dx/ u sqrt(1-u^2), 0< u< 1
    • 6 = -du/dx/ |u| sqrt (1 + u^2), u cant = 0
  14. 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 > 0
    • 2= cos^-1 (u/a) + C u > a > 0
    • 3= 1/a tanh^-1 (u/a) + C if u^2 < a^2 or 1/a coth^-1 (u/a) + C if u^2 > a^2
    • 4= -1/a sech^-1 (u/a) + C, 0 < u < a
    • 5=-1/a csch^-1 | u/a | + C, u cant = 0 and a>0
  15. three identities for inverse hyperbolic functions
    1 sech^-1 x
    2 csch^-1x
    3 coth^-1 x
    • 1= cosh^-1 1/x
    • 2= sinh^-1 1/x
    • 3=tanh^-1 1/x

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