Math 1431
Section 14819
TTh 10-11:30am 100 SEC
Bekki George
bekki@math.uh.edu
639 PGH
Office Hours:
11:30am – 12:30pm TTh and by appointment
Class webpage:
http://www.math.uh.edu/~bekki/Math1431.html
Hyperbolic Functions
Section 4.5
Primary Definitions
Hyperbolic cosine:
ex  ex
cosh  x  
2
Hyperbolic sine:
ex  ex
sinh  x  
2
tanh  x  
sinh  x 
cosh  x 
1
sech  x  
cosh  x 
coth  x  
cosh  x 
sinh  x 
1
csch  x  
sinh  x 
Why are these called Hyperbolic Functions?
Recall that sine and cosine are called circular functions.
Remember the unit circle?

















f x  ex
ex  ex
co s h  x  
2
g x  e
ex  ex
s inh  x  
2
x

Hyperbolic Identity
Show that
cosh 2  t   sinh 2  t   1
sinh (–x) = – sinh x
(odd)
cosh (–x) = cosh x (even)
cosh x + sinh x =
cosh x – sinh x =
d
cosh  x  
dx
d
sinh  x  
dx
Derivatives and Integral formulas:
d
sinh x  cosh x
dx
d
cosh  sinh x
dx
d
tanh x  sec h 2 x
dx
d
sec h x   tanh x sec h x
dx
d
coth x   csc h 2 x
dx
d
csc h x   coth x csc h x
dx
Examples:
( ( ))
d é
cosh ln x
1. dx êë
( ) ùú
( ) úû
é
d ê cosh x
2. dx ê 1 + sin h x
ë
ù
úû
3. y = sinh x , y' =
4. y = arctan(cosh x ), y' =
Popper18
d
cosh x  

1. dx
2.
a. sinh x
b. – sinh x
c. ex
a. cosh2 x
b. 2sinh x cosh x
c. 2ex
d
sinh 2 x =
dx
d
2
sinh
x
=
3. dx
a.
b.
c.
Some Max-Min Problems
Section 5.1
Optimization problems (to maximize or minimize):
1. Draw a picture, label it.
2. Determine the primary function (what is to be a max/min)
3. Use a secondary formula if necessary to get the primary function in
terms of one variable.
4. Determine a feasible domain.
5. Find the max/min.
6. SHOW that the answer is a max/min using the First or Second
Derivative test.
To maximize/minimize a function on a closed bounded interval, we
evaluate the function at the endpoints, and then evaluate the function at any
critical numbers in the interval.
1. Find the dimensions to minimize the perimeter of a rectangular garden
whose area is 48 square feet.
2. Find the point(s) on the graph of y = 4 – x2 closest to (0, 2).
3. Square corners are cut from a rectangular piece of tin that is 24 cm by
45 cm. The edges are folded up to form an open box. Find the length
of the side of the square corner removed in order to have a box with a
maximum volume.
5. A rectangle sits in the first quadrant with its base on the x-axis and its
left side on the y-axis. Its upper right hand corner is on the line passing
through the points (0, 4) and (3, 0). What is the largest possible area of
this rectangle?
6. Find A and B given the function y = Ax –1/2 + Bx1/2 has a minimum value of 6 at
x = 9.
7. Maximize the volume of a box, open at the top, which has a square base and
which is composed of 600 square inches of material. Let x represent each
dimension of the base and let y represent the height of the box.