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1.7: Calculus of the Hyperbolic Functions - Mathematics


We were introduced to hyperbolic functions previously, along with some of their basic properties. In this section, we look at differentiation and integration formulas for the hyperbolic functions and their inverses.

Derivatives and Integrals of the Hyperbolic Functions

Recall that the hyperbolic sine and hyperbolic cosine are defined as

[sinh x=dfrac{e^x−e^{−x}}{2}]

and

[cosh x=dfrac{e^x+e^{−x}}{2}.]

The other hyperbolic functions are then defined in terms of (sinh x) and (cosh x). The graphs of the hyperbolic functions are shown in Figure (PageIndex{1}).

It is easy to develop differentiation formulas for the hyperbolic functions. For example, looking at (sinh x) we have

[egin{align*} dfrac{d}{dx} left(sinh x ight)&=dfrac{d}{dx} left(dfrac{e^x−e^{−x}}{2} ight) &=dfrac{1}{2}left[dfrac{d}{dx}(e^x)−dfrac{d}{dx}(e^{−x}) ight] &=dfrac{1}{2}[e^x+e^{−x}] &=cosh x. end{align*} ]

Similarly,

[dfrac{d}{dx} cosh x=sinh x.]

We summarize the differentiation formulas for the hyperbolic functions in Table (PageIndex{1}).

Table (PageIndex{1}): Derivatives of the Hyperbolic Functions
(f(x))(dfrac{d}{dx}f(x))
(sinh x)(cosh x)
(cosh x)(sinh x)
( anh x)( ext{sech}^2 ,x)
( ext{coth } x)(− ext{csch}^2, x)
( ext{sech } x)(− ext{sech}, x anh x)
( ext{csch } x)(− ext{csch}, x coth x)

Let’s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match:

[dfrac{d}{dx} sin x=cos x]

and

[dfrac{d}{dx} sinh x=cosh x. onumber]

The derivatives of the cosine functions, however, differ in sign:

[dfrac{d}{dx} cos x=−sin x, onumber]

but

[dfrac{d}{dx} cosh x=sinh x. onumber]

As we continue our examination of the hyperbolic functions, we must be mindful of their similarities and differences to the standard trigonometric functions. These differentiation formulas for the hyperbolic functions lead directly to the following integral formulas.

[ egin{align} int sinh u ,du &=cosh u+C [5pt] int ext{csch}^2 u , du &=−coth u+C [5pt] int coshu ,du &=sinh u+C [5pt] int ext{sech} u anh u ,du &=− ext{sech } u+C− ext{csch} u+C [5pt] int ext{sech }^2u ,du &= anh u+C [5pt] int ext{csch} u coth u ,du &=− ext{csch} u+C end{align}]

Example (PageIndex{1}): Differentiating Hyperbolic Functions

Evaluate the following derivatives:

  1. (dfrac{d}{dx}(sinh(x^2)))
  2. (dfrac{d}{dx}(cosh x)^2)

Solution:

Using the formulas in Table (PageIndex{1}) and the chain rule, we get

  1. (dfrac{d}{dx}(sinh(x^2))=cosh(x^2)⋅2x)
  2. (dfrac{d}{dx}(cosh x)^2=2cosh xsinh x)

Exercise (PageIndex{1})

Evaluate the following derivatives:

  1. (dfrac{d}{dx}( anh(x^2+3x)))
  2. (dfrac{d}{dx}left(dfrac{1}{(sinh x)^2} ight))
Hint

Use the formulas in Table (PageIndex{1}) and apply the chain rule as necessary.

Answer a

(dfrac{d}{dx}( anh(x^2+3x))=( ext{sech}^2(x^2+3x))(2x+3))

Answer b

(dfrac{d}{dx}left(dfrac{1}{(sinh x)^2} ight)=dfrac{d}{dx}(sinh x)^{−2}=−2(sinh x)^{−3}cosh x)

Example (PageIndex{2}): Integrals Involving Hyperbolic Functions

Evaluate the following integrals:

  1. ( displaystyle int xcosh(x^2)dx)
  2. ( displaystyle int anh x,dx)

Solution:

We can use u-substitution in both cases.

a. Let (u=x^2). Then, (du=2xdx) and

[egin{align*} int xcosh (x^2)dx &=int dfrac{1}{2}cosh u,du [5pt] &=dfrac{1}{2}sinh u+C [5pt] &=dfrac{1}{2}sinh (x^2)+C. end{align*}]

b. Let (u=cosh x). Then, (du=sinh x,dx) and

[egin{align*} int anh x ,dx=int dfrac{sinh x}{cosh x}dx &=int dfrac{1}{u}du [5pt] &=ln|u|+C [5pt] &= ln|cosh x|+C.end{align*}]

Note that (cosh x>0) for all (x), so we can eliminate the absolute value signs and obtain

[int anh x ,dx=ln(cosh x)+C. onumber]

Exercise (PageIndex{2})

Evaluate the following integrals:

  1. (displaystyle int sinh^3x cosh x ,dx)
  2. (displaystyle int ext{sech }^2(3x), dx)
Hint

Use the formulas above and apply u-substitution as necessary.

Answer a

(displaystyle int sinh^3x cosh x ,dx=dfrac{sinh^4x}{4}+C)

Answer b

(displaystyle int ext{sech }^2(3x) , dx=dfrac{ anh(3x)}{3}+C)

Calculus of Inverse Hyperbolic Functions

Looking at the graphs of the hyperbolic functions, we see that with appropriate range restrictions, they all have inverses. Most of the necessary range restrictions can be discerned by close examination of the graphs. The domains and ranges of the inverse hyperbolic functions are summarized in Table (PageIndex{2}).

Table (PageIndex{2}): Domains and Ranges of the Inverse Hyperbolic Functions
FunctionDomainRange
(sinh^{−1}x)(−∞,∞)(−∞,∞)
(cosh^{−1}x)(1,∞)[0,∞)
( anh^{−1}x)(−1,1)(−∞,∞)
(coth^{−1}x)(−∞,1)∪(1,∞)(−∞,0)∪(0,∞)
( ext{sech}^{−1}x)(0,1)[0,∞)
( ext{csch}^{−1}x)(−∞,0)∪(0,∞)(−∞,0)∪(0,∞)

The graphs of the inverse hyperbolic functions are shown in the following figure.

To find the derivatives of the inverse functions, we use implicit differentiation. We have

[egin{align} y&=sinh^{−1}x sinh y&=x dfrac{d}{dx} sinh y&=dfrac{d}{dx}x cosh ydfrac{dy}{dx}&=1. end{align}]

Recall that (cosh^2y−sinh^2y=1,) so (cosh y=sqrt{1+sinh^2y}).Then,

[dfrac{dy}{dx}=dfrac{1}{cosh y}=dfrac{1}{sqrt{1+sinh^2y}}=dfrac{1}{sqrt{1+x^2}}.]

We can derive differentiation formulas for the other inverse hyperbolic functions in a similar fashion. These differentiation formulas are summarized in Table (PageIndex{3}).

Table (PageIndex{3}): Derivatives of the Inverse Hyperbolic Functions
(f(x))(dfrac{d}{dx}f(x))
(sinh^{−1}x)(dfrac{1}{sqrt{1+x^2}})
(cosh^{−1}x)(dfrac{1}{sqrt{x^2−1}})
( anh^{−1}x)(dfrac{1}{1−x^2})
(coth^{−1}x)(dfrac{1}{1−x^2})
( ext{sech}^{−1}x)(dfrac{−1}{xsqrt{1−x^2}})
( ext{csch}^{−1}x)(dfrac{−1}{|x|sqrt{1+x^2}})

Note that the derivatives of ( anh^{−1}x) and (coth^{−1}x) are the same. Thus, when we integrate (1/(1−x^2)), we need to select the proper antiderivative based on the domain of the functions and the values of (x). Integration formulas involving the inverse hyperbolic functions are summarized as follows.

[int dfrac{1}{sqrt{1+u^2}}du=sinh^{−1}u+C]

[int dfrac{1}{usqrt{1−u^2}}du=− ext{sech}^{−1}|u|+C]

[int dfrac{1}{sqrt{u^2−1}}du=cosh^{−1}u+C]

[int dfrac{1}{usqrt{1+u^2}}du=− ext{csch}^{−1}|u|+C]

[int dfrac{1}{1−u^2}du=egin{cases} anh^{−1}u+C & if|u|<1coth^{−1}u+C & if|u|>1end{cases}]

Example (PageIndex{3}): Differentiating Inverse Hyperbolic Functions

Evaluate the following derivatives:

  1. (dfrac{d}{dx}(sinh^{−1}(dfrac{x}{3})))
  2. (dfrac{d}{dx}( anh^{−1}x)^2)

Solution

Using the formulas in Table (PageIndex{3}) and the chain rule, we obtain the following results:

  1. (dfrac{d}{dx}(sinh^{−1}(dfrac{x}{3}))=dfrac{1}{3sqrt{1+dfrac{x^2}{9}}}=dfrac{1}{sqrt{9+x^2}})
  2. (dfrac{d}{dx}( anh^{−1}x)^2=dfrac{2(tanh^{−1}x)}{1−x^2})

Exercise (PageIndex{3})

Evaluate the following derivatives:

  1. (dfrac{d}{dx}(cosh^{−1}(3x)))
  2. (dfrac{d}{dx}(coth^{−1}x)^3)
Hint

Use the formulas in Table (PageIndex{3}) and apply the chain rule as necessary.

Answer a

[dfrac{d}{dx}(cosh^{−1}(3x))=dfrac{3}{sqrt{9x^2−1}} onumber]

Answer b

[dfrac{d}{dx}(coth^{−1}x)^3=dfrac{3(coth^{−1}x)^2}{1−x^2} onumber]

Example (PageIndex{4}): Integrals Involving Inverse Hyperbolic Functions

Evaluate the following integrals:

  1. (displaystyle int dfrac{1}{sqrt{4x^2−1}}dx)
  2. (displaystyle int dfrac{1}{2xsqrt{1−9x^2}}dx)

Solution:

We can use u-substitution in both cases.

Let (u=2x). Then, (du=2dx) and we have

[egin{align*} int dfrac{1}{sqrt{4x^2−1}}dx &=int dfrac{1}{2sqrt{u^2−1}}du [5pt] &=dfrac{1}{2}cosh^{−1}u+C [5pt] &=dfrac{1}{2}cosh^{−1}(2x)+C. end{align*} ]

Let (u=3x.) Then, (du=3dx) and we obtain

[egin{align*} int dfrac{1}{2xsqrt{1−9x^2}}dx&=dfrac{1}{2}int dfrac{1}{usqrt{1−u^2}}du [5pt] &=−dfrac{1}{2} ext{sech}^{−1}|u|+C [5pt] &=−dfrac{1}{2} ext{sech}^{−1}|3x|+C end{align*}]

Exercise (PageIndex{4})

Evaluate the following integrals:

  1. (displaystyle int dfrac{1}{sqrt{x^2−4}}dx,x>2)
  2. (displaystyle int dfrac{1}{sqrt{1−e^{2x}}}dx)
Hint

Use the formulas above and apply u-substitution as necessary.

Answer a

(displaystyle int dfrac{1}{sqrt{x^2−4}}dx=cosh^{−1}(dfrac{x}{2})+C)

Answer b

( displaystyle int dfrac{1}{sqrt{1−e^{2x}}}dx=− ext{sech}^{−1}(e^x)+C)

Applications

One physical application of hyperbolic functions involves hanging cables. If a cable of uniform density is suspended between two supports without any load other than its own weight, the cable forms a curve called a catenary. High-voltage power lines, chains hanging between two posts, and strands of a spider’s web all form catenaries. The following figure shows chains hanging from a row of posts.

Hyperbolic functions can be used to model catenaries. Specifically, functions of the form (y=acosh(x/a)) are catenaries. Figure (PageIndex{4}) shows the graph of (y=2cosh(x/2)).

Example (PageIndex{5}): Using a Catenary to Find the Length of a Cable

Assume a hanging cable has the shape (10cosh(x/10)) for (−15≤x≤15), where (x) is measured in feet. Determine the length of the cable (in feet).

Solution

Recall from Section 6.4 that the formula for arc length is

[underbrace{int ^b_asqrt{1+[f′(x)]^2}dx}_{ ext{Arc Length}}. onumber]

We have (f(x)=10 cosh(x/10)), so (f′(x)=sinh(x/10)). Then the arc length is

[int ^b_asqrt{1+[f′(x)]^2}dx=int ^{15}_{−15}sqrt{1+sinh^2 left(dfrac{x}{10} ight)}dx. onumber]

Now recall that

[1+sinh^2x=cosh^2x, onumber]

so we have

[egin{align*} ext{Arc Length} &= int ^{15}_{−15}sqrt{1+sinh^2 left(dfrac{x}{10} ight)}dx [5pt] &=int ^{15}_{−15}cosh left(dfrac{x}{10} ight)dx [5pt] &= left.10sinh left(dfrac{x}{10} ight) ight|^{15}_{−15}=10left[sinhleft(dfrac{3}{2} ight)−sinhleft(−dfrac{3}{2} ight) ight]=20sinh left(dfrac{3}{2} ight) [5pt] &≈42.586,ft. end{align*}]

Exercise (PageIndex{5}):

Assume a hanging cable has the shape (15 cosh (x/15)) for (−20≤x≤20). Determine the length of the cable (in feet).

Answer

(52.95ft)

Key Concepts

  • Hyperbolic functions are defined in terms of exponential functions.
  • Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.
  • With appropriate range restrictions, the hyperbolic functions all have inverses.
  • Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.
  • The most common physical applications of hyperbolic functions are calculations involving catenaries.

Glossary

catenary
a curve in the shape of the function (y=acosh(x/a)) is a catenary; a cable of uniform density suspended between two supports assumes the shape of a catenary

Contributors

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


So you know how $sin$ and $cos$ relate the side lengths of right triangles on the unit circle to the interior angle? Well, $sinh$ and $cosh$ do the same but the instead of having right triangle which lie on the unit circle we have them lie on the unit hyperbola.

The equation for the unit circle is,

While the equation for the unit hyperbola is,

The catenary ( or chaînette) is the shape of the curve assumed by a hanging chain or cable with the two ends fixed, under its own weight. It happens its equation is $y=acosh frac xa$ where the constant $a$ depends on physical parameters (tension and mass per unit length). It is used in architecture and engineering for archs, bridges, &c.

You also find a derived curve in the shape of a skipping rope.

The imverse hyperbolic functions are particularly useful in integration, for example when dealing with positive quadratic functions inside square roots. Although such integrals can be done with trig functions, using hyperbolic functions makes them much easier.

Find $intsqrtdx$ for example

The hyperbolic functions $sinh$ and $cosh$ parameterize the hyperbola $x^2-y^2=1$ since $cosh^2 t-sinh^2 t=1$ for all $tinmathbb$.

Every real functions can be uniquely represented by the sum of the even function and the odd function. $f(x)=frac<2>+frac<2>$ Now let's put $f(x)=e^x$.


Derivatives of Hyperbolic Functions

The applet below shows the graphs of these functions and their derivatives.

This device cannot display Java animations. The above is a substitute static image
See About the calculus applets for operating instructions.

In the above applet, there is a pull-down menu at the top to select which function you would like to explore. The selected function is plotted in the left window and its derivative on the right.


The first thing you must bear in mind, is that for real functions of a real variable, anything you can do with hyperbolic functions, you can also do in other ways. In particular,an integral that can be worked out by an hyperbolic substitution can also be worked out by a trigonometric substitution. However, it is true that there are some analogues between circles and trigonometric functions and hyperbolas and hyperbolic functions, for which you should read the Wikipedia article on hyperbolic functions. But the real picture only comes into focus when you know about real and complex power series and understand $ e^ = cos( heta) + isin( heta) $ . Replace $ heta $ by $ - heta $ and solve for $ cos( heta) $ and $ sin( heta) $ . So hyperbolic functions are only a minor curiosity for real calculus but are vitally important for complex analysis.

My own answer to a question like this is to say that just as the ordinary trigonometric functions play a role in spherical trigonometry, so the hyperbolic functions play a role in hyperbolic geometry. For instance the Pythagorean Theorem on the surface of a sphere is $cos c=cos acos b$ , where $a$ and $b$ are the lengths of the legs of a right spherical triangle and $c$ is the length of the hypotenuse. On the hyperbolic plane, the corresponding form of Pythagoras is $cosh c=cosh acosh b$ .

Hardly the most important fact about the hyperbolic functions, but it does show a parallel with the circular functions.


Coordinated Calculus

What type of function describes the behavior of a line hanging between two poles?

What are the properties of a function that can be used to describe the behavior of a line between two poles?

What is the derivative of a function that can be used to describe the behavior of a line between two poles?

There is an important class of functions that show up in many real-life situations: the so-called . Hyperbolic functions can be used to describe the shape of electrical lines freely hanging between two poles or any idealized hanging chain or cable supported only at its ends and hanging under its own weight. Hyperbolic functions can also be used to describe the path of a spacecraft performing a gravitational slingshot maneuver.

Figure 2.95 Freely-hanging electric power cables can form a catenary.

Subsection The Hyperbolic Trigonometric Functions

There are two fundamental hyperbolic trigonometric functions, the ((sinh)) and ((cosh)). These functions are defined in terms of the functions (e^x) and (e^<-x> ext<.>) Graphs of the hyperbolic sine and hyperbolic cosine are given below in Figure2.96.

Hyperbolic Functions

Example 2.97

A cable hanging between two supports will form the shape of a hyperbolic cosine. In particular, the formula

where (T) is the tension at its lowest point and (w) is the weight of the cable per unit length, will yield the total cable sag when evaluated at (x=0 ext<.>) We can calculate the total sag in a powerline hanging between two poles spaced 400 feet apart where the mass per unit length is (50) lb/ft and the tension at the lowest point is (2025) lbs. Specifically, the total sag is given by

In addition to the hyperbolic sine and cosine, there is also a function which is defined as you might expect.

Hyperbolic Tangent

Subsection Identities and Properties

Similar to the usual trigonometric functions, the hyperbolic trigonometric functions have several important properties. While we will not take the time to directly show these properties are valid, we do encourage the reader to confirm these properties by using the formulas and by inspecting the graphs in Figure2.96 above.

Properties of Hyperbolic Functions

It is also useful to discuss the long-run behavior of the hyperbolic trigonometric functions. Again, inspection of Figure2.96 above suggests that as (x ightarrowinfty ext<,>) the graph of (cosh(x)) resembles the graph of (frac12e^x ext<.>) Similarly, it appears that as (x ightarrow-infty ext<,>) the graph of (cosh(x)) resembles the graph of (frac12e^<-x> ext<.>) This behavior is further explained by using the formulas for (cosh(x)) and (sinh(x)) and the facts that (e^<-x> ightarrow0) as (x ightarrowinfty ext<,>) and (e^ ightarrow0) as (x ightarrow-infty ext<.>)

Recall that the trig functions were defined on the unit circle, giving us the Pythagorean identity: if we set (x = cos( heta)) and (y=sin( heta) ext<,>) then the point ((x,y)) lies on the unit circle, and we have

In fact, an analogous identity holds for the hyperbolic trigonometric functions.

A Hyperbolic Identity

This identity shows us how the hyperbolic functions got their name. Suppose ((x,y)) is a point in the plane, and (x = cosh heta) and (y=sinh heta) for some ( heta ext<.>) Then the point ((x,y)) lies on the hyperbola (x^2-y^2 = 1 ext<.>)

Subsection Derivatives of Hyperbolic Functions

We now proceed to calculate the derivatives of each of the hyperbolic functions.

Example 2.98

Calculate the derivatives

Derivatives of Hyperbolic Trigonometric Functions

Subsection Applications of Hyperbolic Trigonometry

A company wishes to build a suspension bridge that stretches between the basketball arena and the baseball stadium on the other side of the railway lines in a particular city. The center part of the bridge will be suspended between two concrete pillars 280 feet apart and 80 feet high. The cable holding the bridge is to be exactly 30 feet above the railway tracks in the middle of the bridge, i.e. it sags exactly 50 feet.

In 1691, Gottfried Leibniz and Christian Huygens determined that any cable hanging under the force of gravity must have the shape of the graph of

This shape is known as a catenary. The parameter (a) is the ratio of cable tension to cable density. The only use of the parameter (b) is to provide a vertical shift, if needed.

We can ask two important questions. First, what values must (a) and (b) have in order for the catenary to fit the constraints provided by the placement of the concrete pillars and the low point of the cable?

In order to find (a) and (b) we need to solve two separate equations. We know that (y(0)=30) to ensure we have sufficient clearance above the railway tracks. We also know that (y(140)=80) since the cable attaches to a 80 foot tall pillar 140 feet from the lowest point (center). Therefore we have

Using the value of (a approx 203.82) together with (30=a+b) we have (b=-173.82 ext<.>) Therefore, the height of the bridge can be modeled by the equation

Subsection Summary

Hyperbolic functions are useful in modeling the shape of a cable hanging between two poles.

The hyperbolic functions are defined in terms of elementary exponential functions:

Hyperbolic sine and hyperbolic cosine satisfy an identity similar to the Pythagorean identity: (cosh^2(x)-sinh^2(x)=1) for any real number (x ext<.>)

The derivatives of the hyperbolic functions are also reminiscent of the regular trigonometric derivatives:


1.7: Calculus of the Hyperbolic Functions - Mathematics

Math 173: Calculus I Spring 2015
Section 02, CRN 395

Schedule: MWRF 1:00pm-1:50pm
Location: Fitzelle Hall 207
Text: Calculus Early Transcendentals, by Stewart (7th edition).

Professor: Jonathan Brown
Office: Fitzelle Hall 260
Email: [email protected]
Phone: 436-3720
Office Hours: Monday 11:00-12:00, Thursday 2:00-3:00, Friday 11:00-12:00, or by appointment
Course website: http://employees.oneonta.edu/brownjs/s15/173

Catalog Description: MATH 173 and 174 constitute the first two-thirds of the standard 12-credit calculus sequence, 173-174-276. Topics include functions and their graphs, limits, differentiation, integration, derivatives and integrals of the elementary functions, polar coordinates, parametric equations, infinite series.

Prerequisite: Pass Math 105 with a C or better.

Course Goals and Objectives: Math 173 provides an introduction to the calculus of one variable. The course has two primary goals:
(1) To analyze polynomial and transcendental functions of one variable with respect to the existence of limits, continuity, and differentiability.
(2) To demonstrate an understanding for the application of the derivative and the integral in maxima and minima points, increasing and decreasing functions, concavity, the Mean Value Theorem, estimation of zeros for transcendental and polynomial functions, related rates, velocity and acceleration, applying the Fundamental Theorem of Calculus, and calculating areas and volumes.

Course content: We will cover Chapter 2 (Limits), Chapter 3 (Derivatives), Chapter 4 (Applications of derivatives), and Chapter 5 (Integration).

SUNY Learning Outcomes:
Students will show competence in the following quantitative reasoning skills: arithmetic, algebra, geometry, data analysis, and quantitative reasoning.

Learning Outcome 1: Students will demonstrate the ability to interpret and draw inferences from mathematical models such as formulas, graphs, tables, and schematics.

Learning Outcome 2: Students will demonstrate the ability to represent mathematical information symbolically, visually, numerically, and verbally.

Learning Outcome 3: Students will demonstrate the ability to employ quantitative methods such as arithmetic, algebra,geometry, or statistics to solve problems.

Learning Outcome 4: Students will demonstrate the ability to estimate and check mathematical results for reasonableness.

Learning Outcome 5: Students will demonstrate the ability to recognize the limits of mathematical and statistical methods.

Grades:
Homework: 10%
Quizzes: 15%
3 in class exams: 15% each
Final: 30%

Your letter grade will be determined from the following heuristic:
A : 93 - 100
A-: 90 - 92.99
B+: 87 - 89.99
B : 83 - 86.99
B-: 80 - 82.99
C+: 77 - 79.99
C :73 - 76.99
C-: 70 - 72.99
D+: 67 - 69.99
D : 63 - 66.99
D-: 60 - 62.99
E : 0 - 59.99

Quizzes: There will be a quiz during the last 10 minutes of class on most Fridays. I will drop your lowest quiz score.

Homework: Each week I will assign two types of homework in the course: practice problems from the text, and WeBWork problems. The link to the WebWork website for this course can be found on the course website. For the WeBWork problems, you will provide answers on the WeBWork website, and your work will be automatically graded.

The practice problems will be from the text, and they will not be collected or graded. The list of practice problems can be found below. Note that the odd numbered problems have answers in the back of the book. It is essential that you do these problems in addition to the WeBWork problems. Learning, development, and growth require time and sustained effort. Most of the knowledge and skills you obtain from this course will come from the time you spend with these problems. Furthermore the quiz problems will often closely resemble (or even be copied from) the problems from the text.

Tentative Exam Schedule:
Exam 1: 2/20
Exam 2: 3/27
Exam 3: 4/24

Final Exam:
The final exam will take place Friday May 8, 2:00pm - 4:30pm.
The final exam is cumulative.

Calculators: While calculators may occasionally be useful while doing the homework, they are not allowed to be used during quizzes and exams (unless you have a note from Student Disability Services).

Attendance Policy: Missing class is a very bad idea. Anyone who misses more than 25% of classes starting in week 2 may be removed from the course.

Getting Help
You are expected to spend anywhere between 6 and 12 hours per week outside of class on the material. The amount of time necessary to pass, or obtain the grade you expect, depends heavily from person to person. Typically, a student spending less than 6 hours per week outside the class should expect to earn a "C" or worse.

If you are struggling with the material, be aware that you have options:
(1) Struggle a bit. Part of the learning process is the internal struggle of trying to understand how all of the pieces fit together. It can be frustrating, but it can also be rewarding. If you make progress, however slow, then it is time well spent. If you feel that your time is not productive, then you should talk to me, or other students, about how you can make better use of your study time.
(2) Visit my office hours. It is part of my job to be available to help you understand the material. There is not enough time in the classroom to get everything across, and you are not expected to do it all on your own.
(3) Find alternate instructional material online. Different people have different styles, and while you might have found my way of discussing something to be confusing, there may be someone else out there who can discuss it in a away that makes sense to you. There are a number of free text books, lessons, and video lectures available online. Feel free to ask me about other sources.
(4) Visit the tutors in CADE. They have drop in tutoring for this course 3 nights a week.
(5) Talk to your fellow students who are taking or who have taken the class. Learning is a social activity you are not expected to do it all on your own.
(6) If after availing yourself of the above options, you still find that you are struggling regularly to understand the material, find a regular tutor. If you are in this situation, you will likely have to spend significant extra time on the material to ensure a passing grade.

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1.7: Calculus of the Hyperbolic Functions - Mathematics

Mathematics 1000 is an introduction to calculus. It develops crucial concepts like limits, continuity and derivatives, using a wide variety of functions: algebraic, trigonometric, exponential and logarithmic functions all feature heavily, while inverse trigonometric and hyperbolic functions are introduced. The course takes a chiefly computational approach, with an emphasis on applications to problems such as related rates, optimisation, kinematics and curve sketching. Students are expected to enter Mathematics 1000 with a good grasp of algebra and trigonometry, but are not assumed to have any prior experience with calculus.

On this page, you'll be able to download course handouts (including assignments, tests and solutions). If there is a disruption to the class schedule -- because of weather, for instance -- you should check this page for news relating to modified due dates and the like. Corrections to any errors on assignments or worksheets will also be posted here. although I'll try my best not to make any!


Derivatives of Hyperbolic Trigonometric Functions

Hyperbolic trig functions, although many people discredit them, can actually be very useful. True, there are few examples of explicit hyperbolic functions in the physical world. However, using hyperbolic functions can make exponentials appear to behave like trigonometric functions -- an analogy that can provide much intuition.

In any case, we still want to know the derivatives of the hyperbolic functions. We'll find the derivatives of sinh and cosh from their definitions in terms of exponentials:

OK, let's take a look at this. The derivatives of sinh and cosh seem to behave just like the derivatives of sin and cos, except that the annoying negative signs have gone away. (Remember: Dxsin(x) = cos(x), Dxcos(x) = -sin(x)) We can find the derivatives of the other hyperbolic trig functions using the quotient rule, the chain rule and the derivatives we've just derived:

d dx tanh (x) &sp=&sp d dx sinh (x) cosh (x) &sp=&sp cosh ^2^(x) &thinsp-&thinsp sinh ^2^(x) cosh ^2^(x) &sp=&sp 1 cosh ^2^(x) &sp=&sp sech ^2^(x)

d dx coth (x) &sp=&sp d dx cosh (x) sinh (x) &sp=&sp cosh ^2^(x) &thinsp-&thinsp sinh ^2^(x) sinh ^2^(x) &sp=&sp -1 sinh ^2^(x) &sp=&sp - csch ^2^(x)

d dx sech (x) &sp=&sp d dx ( cosh x)^-1^ &sp=&sp (-1)( cosh x)^-2^( sinh x) &sp=&sp - sech (x) tanh (x)

d dx csch (x) &sp=&sp d dx ( sinh x)^-1^ &sp=&sp (-1)( sinh x)^-2^( cosh x) &sp=&sp - csc (x) coth (x)

So, as with sinh and cosh, the derivatives of the other hyperbolic trig functions closely resemble those of the normal trig functions, with some discrepancies over negative signs. Be careful, though -- those negative signs can easily cause big errors!

Some examples:

This is the result we would expect since sinh ^2^x &thinsp-&thinsp cosh ^2^ &sp=&sp 1


The calculator allows you to use most hyperbolic functions, it is possible to calculate the hyperbolic cosine (noted ch or cosh), the hyperbolic sine (noted sh or sinh ), the hyperbolic tangent (noted th or tanh ), and the hyperbolic cotangent (noted coth or cotanh ).

The hyperbolic cotangent function is written coth, it is defined by the following formula :

The hyperbolic cotangent calculator allows through the coth function to calculate online the hyperbolic cotangent of a number.

To calculate the hyperbolic cotangent of a number, enter the number and to apply the coth function. For calculating the hyperbolic cotangent of the following number 2, enter coth(`2`) or directly 2, if the coth button already appears, the result 1.03731472073 is returned.

    The limits of the hyperbolic cotangent exist at `-oo` (minus infinity) and `+oo` (plus infinity): The hyperbolic cotangent function has a limit in `-oo` which is `-1`.
      `lim_(x->-oo)coth(x)=-1`
      `lim_(x->+oo)coth(x)=1`

    Syntax :

    coth(x), where x is a number.

    Other notation sometimes used : cotanh

    Examples :

    Derivative hyperbolic cotangent :

    To differentiate function hyperbolic cotangent online, it is possible to use the derivative calculator which allows the calculation of the derivative of the hyperbolic cotangent function

    Antiderivative hyperbolic cotangent :

    Antiderivative calculator allows to calculate an antiderivative of hyperbolic cotangent function.

    An antiderivative of coth(x) is antiderivative_calculator(`"coth"(x)`)=`ln(sh(x))`

    Limit hyperbolic cotangent :

    The limit calculator allows the calculation of limits of the hyperbolic cotangent function.


    40 Advanced Calculus Calculators

    Riemann Sums

    As MathOpenRef.com explains, a Riemann sum is &ldquoa method for approximating the total area underneath a curve on a graph, otherwise known as an integral.&rdquo Below is a collection of resources to help you better understand Riemann sums.

    MathWorld.Wolfram.com's Riemann Sum – Input your data to see your Riemann Sum on a graph. Play around with the inputted data to see how the graph changes.

    EMathHelp.net's Riemann Sum – Easy to use and includes a step-by-step explanation with your results.

    IntMath.com's Riemann Sums Applet – Tutorial information is provided. Choose a function from the drop down menu to see how it appears on the graph. Adjust the sliders to see how the graphed Riemann Sum changes on the graph.

    Trapezoidal Rule

    MathWords.com explains that the trapezoidal rule is &ldquoa method for approximating a definite integral using linear approximations of f.&rdquo The tools below will help you learn how to use the trapezoidal rule.

    NastyAccident.com's Trapezoidal Rule – Follow the instructions to enter your data. Results include a step-by-step explanation.

    EMathHelp.net's Trapezoidal Rule – Provides a step-by-step explanation with your results.

    EasyCalculation.com's Trapezoidal Rule – Learn more about the trapezoidal rule from the provided tutorial information. Follow the instructions provided to ensure you enter your data correctly.

    Partial Fraction Decomposition

    As PurpleMath.com explains, partial fraction decomposition is &ldquothe process of starting with the simplified answer and taking it back apart, or &lsquodecomposing' the final expression into its initial polynomial fractions.&rdquo Below is a collection of resources to help you better understand partial fraction decomposition.

    WolframAlpha.com's Partial Fraction Decomposition – Simple and straightforward, just enter the numerator and denominator to get your result.

    Calc101.com's Step-by-Step Partial Fractions – Enter your expression (or use the example provided) and then a step-by-step explanation for finding the partial fraction will be provided.

    QuickMath.com's Partial Fractions – Quick and easy to use, just enter your function to find the partial fraction. A Basic and Advanced version are provided.

    Symbolab.com's Partial Fractions – Enter your expression or use one of the examples provided. A step-by-step explanation will be provided with the results.

    Inverse Functions

    As Wikipedia.org explains, an inverse function &ldquois a function that &lsquoreverses' another function.&rdquo Below is a collection of tools to help you strengthen your understanding of inverse functions.

    Symbolab.com's Inverse Function – Cleanly designed, easy to use, and provides a step-by-step explanation with results. Click &ldquoGraph&rdquo to see your inverse function on a graph.

    WolframAlpha.com's Inverse Function – Simple enough to illustrate the fundamentals, the results include your graphed inverse function.

    NumberEmpire.com's Inverse Function – Choose one of the four examples or enter your own function to get the inverse function.

    AnalyzeMath.com's Inverse Function – Click the &ldquoShow Me&rdquo button and this resource will guide you through the four-step process for finding the inverse function.

    CalculatorSoup.com's Inverse Function – Use the drop down menu to choose which function you'd like to find. Then, enter the value of &ldquox&rdquo to get your results.

    Keisan.Casio.com's Inverse Function – Enter the value of &ldquox&rdquo and the inverse hyperbolic functions will be provided.

    Gyplan.com's Inverse Function – Use the drop down menu to choose the type of inverse function you'd like to find and then enter the value of &ldquox&rdquo to get your results.

    Differential Equation

    As Wolfram MathWorld explains, a differential equation is &ldquoan equation that involves the derivatives of a function as well as the function itself.&rdquo Below are several tools to help you learn more about differential equations:

    WolframAlpha.com's Differential Equations – Use to solve several different kinds of differential equations. Results include the solution, plots of sample individual solutions, the graphed sample solution family, and more.

    Symbolab.com's Ordinary Differential Equations – Cleanly designed and easy to use, the results include a step-by-step explanation. Enter your own equation or experiment using the provided examples.

    MathScoop.com's Euler Method – Uses the Euler Method to solve your equation. The results include a Euler Table and a graph of the Euler points.

    Keisan.Casio.com's Euler's Method – The needed formula is included, and an Euler table is created with your results.

    Had2Know.com's Second Order Differential Equation Solver – Learn more about solving differential equations from the provided tutorial information and explained cases.

    Arc Length

    MathWords.com teaches that arc length is thelength of a curve or line. Below is a collection of resources to help you find arc length.

    1728.org's Arc Length – Choose what you'd like to solve for then enter your known values. Your result will be provided.

    HandyMath.com's Complete Circular Arc – Input two known values to find the radius, length, width, height, apothem, angle, and area of an arc or circle segment.

    AJDesigner.com's Circle Arc – A labeled circle diagram and the arc length formula are given. Enter the radius and central angle to get your result.

    WolframAlpha.com's Arc Length – This resource will perform several functions related to finding arc length, and it provides an example for each to help you get started.

    TutorVista.com's Arc Length – A step-by-step explanation for how to find the arc length and examples with explanations and results are provided.

    MathOpenRef.com's Interactive Arc Length – Drag point A or point B to see how the arc length adjusts.

    Flexibility.com's Arc Length – Choose which option to use for finding arc length based on your known values. A labeled circle diagram is given as a visual aid.

    PlanetCalc.com's Arc Length – A labeled circle diagram and formulas are provided. Enter the radius and angle to find the arc length and other properties, such as area, chord length, and perimeter.

    EasyCalculation.com's Arc Length – Enter your radius and angle and the arc length will be provided.

    Center of Mass

    MathWords.com provides the formulas for finding Center of Mass. Below is a collection of tools to help strengthen your understanding of center of mass.

    TutorVista.com's Center of Mass – Enter the &ldquodifferent value of masses&rdquo and the &ldquodistance of the respective masses&rdquo to find the center of mass.

    Calculator.Swiftutors.com's Center of Mass – Provides tutorial information, very uncomplicated and easy for any student to navigate.

    LearningAboutElectronics.com's Center of Mass – Tutorial information, a labeled diagram and instructions on using the tool are provided. Enter all known masses and their respective distances to find the center of mass.

    Sequences

    As Paul's Online Math Notes explains, a sequence is &ldquoa list of numbers written in a specific order.&rdquo The tool below will help you learn more about sequences:


    Watch the video: Hyperbolic trig functions. MIT Single Variable Calculus, Fall 2010 (November 2021).