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Compare the relative severity of a magnitude 8.4 earthquake with a magnitude 7.4 earthquake.

The magnitude 8.4 earthquake is roughly 10 times as severe as the magnitude 7.4 earthquake.

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Hyperbolic functions

The hyperbolic functions are defined in terms of certain combinations of e x and e x . These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary ( [link] ). If we introduce a coordinate system so that the low point of the chain lies along the y -axis, we can describe the height of the chain in terms of a hyperbolic function. First, we define the hyperbolic functions    .

A photograph of a spider web collecting dew drops.
The shape of a strand of silk in a spider’s web can be described in terms of a hyperbolic function. The same shape applies to a chain or cable hanging from two supports with only its own weight. (credit: “Mtpaley”, Wikimedia Commons)

Definition

Hyperbolic cosine

cosh x = e x + e x 2

Hyperbolic sine

sinh x = e x e x 2

Hyperbolic tangent

tanh x = sinh x cosh x = e x e x e x + e x

Hyperbolic cosecant

csch x = 1 sinh x = 2 e x e x

Hyperbolic secant

sech x = 1 cosh x = 2 e x + e x

Hyperbolic cotangent

coth x = cosh x sinh x = e x + e x e x e x

The name cosh rhymes with “gosh,” whereas the name sinh is pronounced “cinch.” Tanh , sech , csch , and coth are pronounced “tanch,” “seech,” “coseech,” and “cotanch,” respectively.

Using the definition of cosh ( x ) and principles of physics, it can be shown that the height of a hanging chain, such as the one in [link] , can be described by the function h ( x ) = a cosh ( x / a ) + c for certain constants a and c .

But why are these functions called hyperbolic functions ? To answer this question, consider the quantity cosh 2 t sinh 2 t . Using the definition of cosh and sinh , we see that

cosh 2 t sinh 2 t = e 2 t + 2 + e −2 t 4 e 2 t 2 + e −2 t 4 = 1 .

This identity is the analog of the trigonometric identity cos 2 t + sin 2 t = 1 . Here, given a value t , the point ( x , y ) = ( cosh t , sinh t ) lies on the unit hyperbola x 2 y 2 = 1 ( [link] ).

An image of a graph. The x axis runs from -1 to 3 and the y axis runs from -3 to 3. The graph is of the relation “(x squared) - (y squared) -1”. The left most point of the relation is at the x intercept, which is at the point (1, 0). From this point the relation both increases and decreases in curves as x increases. This relation is known as a hyperbola and it resembles a sideways “U” shape. There is a point plotted on the graph of the relation labeled “(cosh(1), sinh(1))”, which is at the approximate point (1.5, 1.2).
The unit hyperbola cosh 2 t sinh 2 t = 1 .

Graphs of hyperbolic functions

To graph cosh x and sinh x , we make use of the fact that both functions approach ( 1 / 2 ) e x as x , since e x 0 as x . As x , cosh x approaches 1 / 2 e x , whereas sinh x approaches −1 / 2 e x . Therefore, using the graphs of 1 / 2 e x , 1 / 2 e x , and 1 / 2 e x as guides, we graph cosh x and sinh x . To graph tanh x , we use the fact that tanh ( 0 ) = 1 , −1 < tanh ( x ) < 1 for all x , tanh x 1 as x , and tanh x 1 as x . The graphs of the other three hyperbolic functions can be sketched using the graphs of cosh x , sinh x , and tanh x ( [link] ).

An image of six graphs. Each graph has an x axis that runs from -3 to 3 and a y axis that runs from -4 to 4. The first graph is of the function “y = cosh(x)”, which is a hyperbola. The function decreases until it hits the point (0, 1), where it begins to increase. There are also two functions that serve as a boundary for this function. The first of these functions is “y = (1/2)(e to power of -x)”, a decreasing curved function and the second of these functions is “y = (1/2)(e to power of x)”, an increasing curved function. The function “y = cosh(x)” is always above these two functions without ever touching them. The second graph is of the function “y = sinh(x)”, which is an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is “y = (1/2)(e to power of x)”, an increasing curved function and the second of these functions is “y = -(1/2)(e to power of -x)”, an increasing curved function that approaches the x axis without touching it. The function “y = sinh(x)” is always between these two functions without ever touching them. The third graph is of the function “y = sech(x)”, which increases until the point (0, 1), where it begins to decrease. The graph of the function has a hump. The fourth graph is of the function “y = csch(x)”. On the left side of the y axis, the function starts slightly below the x axis and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the x axis, which it never touches. The fifth graph is of the function “y = tanh(x)”, an increasing curved function. There are also two functions that serve as a boundary for this function. The first of these functions is “y = 1”, a horizontal line function and the second of these functions is “y = -1”, another horizontal line function. The function “y = tanh(x)” is always between these two functions without ever touching them. The sixth graph is of the function “y = coth(x)”. On the left side of the y axis, the function starts slightly below the boundary line “y = 1” and decreases until it approaches the y axis, which it never touches. On the right side of the y axis, the function starts slightly to the right of the y axis and decreases until it approaches the boundary line “y = -1”, which it never touches.
The hyperbolic functions involve combinations of e x and e x .

Identities involving hyperbolic functions

The identity cosh 2 t sinh 2 t , shown in [link] , is one of several identities involving the hyperbolic functions, some of which are listed next. The first four properties follow easily from the definitions of hyperbolic sine and hyperbolic cosine. Except for some differences in signs, most of these properties are analogous to identities for trigonometric functions.

Rule: identities involving hyperbolic functions

  1. cosh ( x ) = cosh x
  2. sinh ( x ) = sinh x
  3. cosh x + sinh x = e x
  4. cosh x sinh x = e x
  5. cosh 2 x sinh 2 x = 1
  6. 1 tanh 2 x = sech 2 x
  7. coth 2 x 1 = csch 2 x
  8. sinh ( x ± y ) = sinh x cosh y ± cosh x sinh y
  9. cosh ( x ± y ) = cosh x cosh y ± sinh x sinh y

Questions & Answers

find the domain and range of f(x)= 4x-7/x²-6x+8
Nick Reply
find the range of f(x)=(x+1)(x+4)
Jane Reply
-1, -4
Marcia
That's domain. The range is [-9/4,+infinity)
Jacob
If you're using calculus to find the range, you have to find the extrema through the first derivative test and then substitute the x-value for the extrema back into the original equation.
Jacob
Good morning,,, how are you
Harrieta Reply
d/dx{1/y - lny + X^3.Y^5}
mogomotsi Reply
How to identify domain and range
Umar Reply
hello
Akpevwe
He,,
Harrieta
hi
Dr
hello
velocity
I only talk to girls
Dr
women are smart then guys
Dr
Smarter
Adri
sorry
Dr
hi adri ana
Dr
:(
Shun
was up
Dr
hello
Adarsh
is it chatting app?.. I do not see any calculus here. lol
Adarsh
Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
mukul Reply
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
ade
show that lim f(x) + lim g(x)=m+l
BARNABAS Reply
list the basic elementary differentials
Chio Reply
Differentiation and integration
Okikiola Reply
yes
Damien
proper definition of derivative
Syed Reply
the maximum rate of change of one variable with respect to another variable
Amdad
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
Inembo Reply
what is calculus?
BISWAJIT Reply
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
Geoffrey Reply
what is x and how x=9.1 take?
Pravin Reply
what is f(x)
Inembo Reply
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
Adri
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
Adri
I dont understand what you wanna say by (A' n B^c)^c'
Adri
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Adri
Ok so the set is formed by vectors and not numbers
Adri
A vector of length n
Adri
But you can make a set out of matrixes as well
Adri
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
Wait what's your math level?
Adri
High-school?
Adri
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
Adri
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
Adri
I would say 24
Adri
Offer both
Adri
Sorry 20
Adri
Actually you have 40 - 4 =36 who offer maths or physics or both.
Adri
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
Adri
56-36=20 who give both courses... I would say that
Adri
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
Practice Key Terms 7

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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