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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)
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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
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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
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what is hyperbolic sine of x??
sinhx = e^x - e^-x/2
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what is the hyperbolic cosine of x?
coshx = (e^x + e^-x)/2
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tanh x?
Sinhx/coshx = (e^x - e^-x)/e^x +e^-x
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coth x?
= cosh x/sinh x = (e^x + e^-x)/e^x - e^-x
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Hyperbolic secant?
sech x = 1/coshx = 2/(e^x + e^-x)
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hyperbolic csc?
csch x + 1/sinhx = 2/(e^x - e^-x)
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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
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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
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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
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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
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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
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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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