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Calculus 1 Lecture Notes
Section 0.5
Page 1 of 4
Section 0.5: Exponential and Logarithmic Functions
Big Idea: The hyperbolic functions are very useful in certain applications, and are defined rather
simply in terms of the exponential function.
Big Skill: You should be able to perform basic calculations with hyperbolic functions.
 1
e  lim  1  
n 
 n
n
1. Uses of the hyperbolic functions:
a. Engineering applications (catenary curves)
b. Finding certain antiderivatives.
c. Solving differential equations, such as
y(x) = y(x), which has solution y(x) = Asinh(x) + Bcosh(x)
vs. y(x) = -y(x), which has solution y(x) = Asin(x) + Bcos(x)
2. Definition of the hyperbolic functions:
e x  ex
2
sinh x  e x  e  x e 2 x  1 


tanh x 

cosh x  e x  e  x e 2 x  1 
1
2
sech x 
 x
cosh x e  e  x
e x  ex
2
cosh x  e x  e  x e 2 x  1 


coth x 

sinh x  e x  e  x e 2 x  1 
1
2
csch x 
 x
sinh x e  e  x
sinh x 
cosh x 
3. Some identities involving hyperbolic functions:
cosh 2 u  sinh 2 u  1
Note: if x = cosh u and y = sinh u, then
x 2  y 2  1 is the equation of a
hyperbola
sinh 2u  2 sinh u cosh u
cosh (s + t) = cosh s cosh t + sinh s sinh t
sinh  i  

ei  ei
2
 cos  i sin     cos     i sin    
2
 i sin 
cosh 2u  2 cosh 2 u  1
sinh (s + t) = sinh s cosh t + cosh s sinh t
4. Osbornes’s Rule: a trigonometry identity can be converted to an analogous identity for
hyperbolic functions by expanding, exchanging trigonometric functions with their hyperbolic
counterparts, and then flipping the sign of each term involving the product of two hyperbolic
sines. For example, given the identity
Osborne's rule gives
the corresponding identity
Calculus 1 Lecture Notes
Section 0.5
Page 2 of 4
Citation: Eric W. Weisstein. "Osborne's Rule." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/OsbornesRule.html
Calculus 1 Lecture Notes
Section 0.5
5. Graphs of the hyperbolic functions:
Page 3 of 4
Calculus 1 Lecture Notes
Section 0.5
Page 4 of 4
6. Inverses of the hyperbolic functions and their formulae:
y  sinh 1 x iff sinh y  x
y  cosh 1 x iff cosh y  x

sinh 1 x  ln x  x 2  1
y  tanh
1
tanh 1 x 


cosh 1 x  ln x  x 2  1
1
x iff tanh y  x
y  coth
1 1 x
ln
2 1 x
coth 1 x 

x iff coth y  x
1 x 1
ln
2 x 1
y  sech 1 x iff sech y  x
1
Also: coth 1  x   tanh 1  
 x
1
y  csch x iff csch y  x
1
sech 1 x  ln  
x
1
csch 1 x  ln  
x

1
 1 
2
x

1
Also: sech 1  x   cosh 1  
 x
7. Proof of the formula for the inverse hyperbolic sine:
y  sinh 1 x
x  sinh y 
e y  ey
2
2x  e y  e y
2 xe y  e 2 y  1
e 2 y  2 xe y  1  0
e 
 2x e y  1  0
ey 
2 x  4 x 2  4(1)( 1)
 x  x2 1
2(1)
y 2
 

y  sinh 1 x  ln x  x 2  1


1
 1 
2
x

1
Also: csch 1  x   sinh 1  
 x