Transcript Calc06_10_Briggsx

Georg Friedrich Bernhard Riemann
1826 – 1866
Riemann's ideas concerning geometry of space had a profound
effect on the development of modern theoretical physics. He
clarified the notion of integral by defining what we now call the
Riemann integral.
Lets examine the exponential functions:
ex
e x
y
and y 
.
2
2
Now, lets look at the combination of these two functions with addition.
e x e x e x  e x
y +

.
2 2
2
e x  e x
 cosh x.
We call this function hyperbolic cosine. y 
2
x
 
y

c

b
cosh
Any curve of the form
  is called a catenary curve.
a
(Note: x is not an angle!)
Next, lets look at the combination of these two functions with subtraction.
e x e x e x  e x
y 

.
2
2
2
e x  e x
We call this function hyperbolic sine. y 
 sinh x.
2
above.
Here are some identities.
To name a few!!
As you can see, they are very analogous to the trigonometric functions.
I can read your minds…what are the derivatives of the family of
hyperbolic functions.
e x  e x
Lets find the derivative of y 
 sinh x.
2
d
sinh x 
Hence
dx
d  e x  e x 
d
 
sinh x   cosh x


dx  2 
dx
1 d x x
  e  e 
2 dx
1
  e x  (e  x ) 
2
e x  e x

2
 cosh x
So the generalized derivative is
d
du
sinh u   cosh u 
dx
dx
How about the derivatives of the inverse hyperbolic functions? You can
study those this summer!!
Idealized Ocean Wave
Examples
1) Find y, if y  cosh 5

.
2) Evalute
.
3 x2  x 5