The hyperbolic functions are functions that have many applications to mathematics, physics, and engineering. Among many other applications, they are used to describe the formation of satellite rings around planets, to describe the shape of a rope hanging from two points, and have application to the theory of special relativity. This section defines the hyperbolic functions and describes many of their properties, especially their usefulness to calculus.
These functions are sometimes referred to as the “hyperbolic trigonometric functions” as there are many connections between them and the standard trigonometric functions. Figure 7.4.1 demonstrates one such connection. Just as cosine and sine are used to define points on the circle defined by , the functions hyperbolic cosine and hyperbolic sine are used to define points on the hyperbola .
We begin with their definitions.
(a)
(b)
(c)
(d)
(e)
(f)
The hyperbolic functions are graphed in Figure 7.4.2. In the graphs of and , graphs of and are included with dashed lines. As gets “large,” and each act like ; when is a large negative number, acts like whereas acts like .
Notice the domains of and are , whereas both and have vertical asymptotes at . Also note the ranges of these functions, especially : as , both and approach , hence approaches .
Watch the video:
Hyperbolic Functions — The Basics from https://youtu.be/G1C1Z5aTZSQ
The following example explores some of the properties of these functions that bear remarkable resemblance to the properties of their trigonometric counterparts.
Use Definition 7.4.1 to rewrite the following expressions.
Solution
So .
So .
Thus .
So
So
So
The following Key Idea summarizes many of the important identities relating to hyperbolic functions. Each can be verified by referring back to Definition 7.4.1.
Basic Identities
Derivatives
Integrals
We practice using Key Idea 7.4.1.
Evaluate the following derivatives and integrals.
(a)
(b)
(c)
Solution
Using the Chain Rule directly, we have .
Just to demonstrate that it works, let’s also use the Basic Identity found in Key Idea 7.4.1: .
Using another Basic Identity, we can see that . We get the same answer either way.
We employ substitution, with and . Applying Key Idea 7.4.1 we have:
We can simplify this last expression as is based on exponentials:
Just as the inverse trigonometric functions are useful in certain integrations, the inverse hyperbolic functions are useful with others. Figure 7.4.3 shows the restrictions on the domains to make each function one-to-one and the resulting domains and ranges of their inverse functions. Their graphs are shown in Figure 7.4.4.
Because the hyperbolic functions are defined in terms of exponential functions, their inverses can be expressed in terms of logarithms as shown in Key Idea 7.4.2. It is often more convenient to refer to than to , especially when one is working on theory and does not need to compute actual values. On the other hand, when computations are needed, technology is often helpful but many hand-held calculators lack a convenient button. (Often it can be accessed under a menu system, but not conveniently.) In such a situation, the logarithmic representation is useful. The reader is not encouraged to memorize these, but rather know they exist and know how to use them when needed.
Now let’s consider the inverses of the hyperbolic functions. We begin with the function . Since for all real , is increasing and must be one-to-one.
Finally, interchange the variable to find that
In a similar manner we find that the inverses of the other hyperbolic functions are given by:
;
;
;
;
;
The following Key Ideas give the derivatives and integrals relating to the inverse hyperbolic functions. In Key Idea 7.4.4, both the inverse hyperbolic and logarithmic function representations of the antiderivative are given, based on Key Idea 7.4.2. Again, these latter functions are often more useful than the former.
(a)
;
(b)
(c)
;
(d)
;
(e)
;
(f)
;
1. | ; | |
2. | ; | |
3. | ||
4. | ; | |
5. | ; |
We practice using the derivative and integral formulas in the following example.
Evaluate the following.
Solution
Applying Key Idea 7.4.3 with the Chain Rule gives:
Multiplying the numerator and denominator by gives a second integral can be solved with a direct application of item #3 from Key Idea 7.4.4, with . Thus
(7.4.3) | ||||
(7.4.4) |
This requires a substitution, then item #2 of Key Idea 7.4.4 can be applied.
Let , hence . We have
Note , hence Now apply the integral rule. | ||||
This section covers a lot of ground. New functions were introduced, along with some of their fundamental identities, their derivatives and antiderivatives, their inverses, and the derivatives and antiderivatives of these inverses. Four Key Ideas were presented, each including quite a bit of information.
Do not view this section as containing a source of information to be memorized, but rather as a reference for future problem solving. Key Idea 7.4.4 contains perhaps the most useful information. Know the integration forms it helps evaluate and understand how to use the inverse hyperbolic answer and the logarithmic answer.
The next section takes a brief break from demonstrating new integration techniques. It instead demonstrates a technique of evaluating limits that return indeterminate forms. This technique will be useful in Section 8.6, where limits will arise in the evaluation of certain definite integrals.
In Key Idea 7.4.1, the equation is given. Why is “” not used — i.e., why are absolute values not necessary?
The hyperbolic functions are used to define points on the right hand portion of the hyperbola , as shown in Figure 7.4.1. How can we use the hyperbolic functions to define points on the left hand portion of the hyperbola?
Suppose . Find the values of the other five hyperbolic functions at .
Suppose . Find the values of the other five hyperbolic functions at .
In Exercises 5–12., verify the given identity using Definition 7.4.1, as done in Example 7.4.1.
In Exercises 13–24., find the derivative of the given function.
In Exercises 25–30., find the equation of the line tangent to the function at the given -value.
at
at
at
at
at
at
In Exercises 31–38., evaluate the given indefinite integral.
(Hint: multiply by ; set .)
In Exercises 39–40., evaluate the given definite integral.