Hyperbolics Flashcards
cosh x =
1/2(e^x +e^-x)
sinh x =
1/2(e^x - e^-x)
tanh x
(e^x - e^-x)/(e^x +e^-x)
sinh x graph
cubic
sinh x domain
all real values of x
sinh x range
all real values of x
cosh x graph
quadratic through (1,0)
cosh x domain
all real values of x
cosh x range
y>=1
tanh x graph
asymtotes at x = 1 and x= -1 curves through (0,0)
tanh x domain
all real values of x
tanh x range
-1 <= y <= 1
cosh^2 x - sinh^2 x =
1
cos^2 x + sin^2 x =
1
cosh 2x =
2cosh^2 x -1
cos 2x =
2cos^2 x -1
derivative of sinh x
cosh x
derivative of cosh x
sinh x
derivative of sin x
cos x
derivative of cos x
- sin x
maclaurin series of sinh and cosh
same for sin and cos
why is the negative square root discarded when finding inverse hyerbolics
√(k^2 + 1) > k
inverse sinh x
ln(x+√(x^2+1))
inverse cosh x
ln(x+√(x^2-1))
x>= 1
inverse tanh x
1/2ln((1+x)/(1-x))
|x|<1
inverse sinh x graph
reflection of the cubic in y=x
inverse cosh x graph
√x graph
inverse tanhx graph
asymtotes at y = -1 and y = 1 and curves through 0
why is it significant that arcsinh and arctanh are 1-to-1 but arccosh isn’t
arccosh will have two roots to the equation
derivative of arcsinh x
1/ √(1+x^2)
derivative of arccosh x
1/√(x^2 -1)
derivative of arctanh x
1/ (1-x^2)
integral of a 1/ √(a^2+x^2)
arcsinh(x/a) + c
integral of 1/√(x^2 -a^2)
arccosh(x/a) + c
integral pf 1/√(a^2 - x^2)
arcsin(x/a) + c
