4.11 Hyperbolic Functions

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Introduction

1 Analytic Geometry

  • 1. Lines
  • 2. Distance Between Two Points; Circles
  • 3. Functions
  • 4. Shifts and Dilations

2 Instantaneous Rate of Change: The Derivative

  • 1. The slope of a function
  • 2. An example
  • 3. Limits
  • 4. The Derivative Function
  • 5. Properties of Functions

3 Rules for Finding Derivatives

  • 1. The Power Rule
  • 2. Linearity of the Derivative
  • 3. The Product Rule
  • 4. The Quotient Rule
  • 5. The Chain Rule

4 Transcendental Functions

  • 1. Trigonometric Functions
  • 2. The Derivative of $\sin x$
  • 3. A hard limit
  • 4. The Derivative of $\sin x$, continued
  • 5. Derivatives of the Trigonometric Functions
  • 6. Exponential and Logarithmic functions
  • 7. Derivatives of the exponential and logarithmic functions
  • 8. Implicit Differentiation
  • 9. Inverse Trigonometric Functions
  • 10. Limits revisited
  • 11. Hyperbolic Functions

5 Curve Sketching

  • 1. Maxima and Minima
  • 2. The first derivative test
  • 3. The second derivative test
  • 4. Concavity and inflection points
  • 5. Asymptotes and Other Things to Look For

6 Applications of the Derivative

  • 1. Optimization
  • 2. Related Rates
  • 3. Newton's Method
  • 4. Linear Approximations
  • 5. The Mean Value Theorem

7 Integration

  • 1. Two examples
  • 2. The Fundamental Theorem of Calculus
  • 3. Some Properties of Integrals

8 Techniques of Integration

  • 1. Substitution
  • 2. Powers of sine and cosine
  • 3. Trigonometric Substitutions
  • 4. Integration by Parts
  • 5. Rational Functions
  • 6. Numerical Integration
  • 7. Additional exercises

9 Applications of Integration

  • 1. Area between curves
  • 2. Distance, Velocity, Acceleration
  • 3. Volume
  • 4. Average value of a function
  • 5. Work
  • 6. Center of Mass
  • 7. Kinetic energy; improper integrals
  • 8. Probability
  • 9. Arc Length
  • 10. Surface Area

10 Polar Coordinates, Parametric Equations

  • 1. Polar Coordinates
  • 2. Slopes in polar coordinates
  • 3. Areas in polar coordinates
  • 4. Parametric Equations
  • 5. Calculus with Parametric Equations

11 Sequences and Series

  • 1. Sequences
  • 2. Series
  • 3. The Integral Test
  • 4. Alternating Series
  • 5. Comparison Tests
  • 6. Absolute Convergence
  • 7. The Ratio and Root Tests
  • 8. Power Series
  • 9. Calculus with Power Series
  • 10. Taylor Series
  • 11. Taylor's Theorem
  • 12. Additional exercises

12 Three Dimensions

  • 1. The Coordinate System
  • 2. Vectors
  • 3. The Dot Product
  • 4. The Cross Product
  • 5. Lines and Planes
  • 6. Other Coordinate Systems

13 Vector Functions

  • 1. Space Curves
  • 2. Calculus with vector functions
  • 3. Arc length and curvature
  • 4. Motion along a curve

14 Partial Differentiation

  • 1. Functions of Several Variables
  • 2. Limits and Continuity
  • 3. Partial Differentiation
  • 4. The Chain Rule
  • 5. Directional Derivatives
  • 6. Higher order derivatives
  • 7. Maxima and minima
  • 8. Lagrange Multipliers

15 Multiple Integration

  • 1. Volume and Average Height
  • 2. Double Integrals in Cylindrical Coordinates
  • 3. Moment and Center of Mass
  • 4. Surface Area
  • 5. Triple Integrals
  • 6. Cylindrical and Spherical Coordinates
  • 7. Change of Variables

16 Vector Calculus

  • 1. Vector Fields
  • 2. Line Integrals
  • 3. The Fundamental Theorem of Line Integrals
  • 4. Green's Theorem
  • 5. Divergence and Curl
  • 6. Vector Functions for Surfaces
  • 7. Surface Integrals
  • 8. Stokes's Theorem
  • 9. The Divergence Theorem

17 Differential Equations

  • 1. First Order Differential Equations
  • 2. First Order Homogeneous Linear Equations
  • 3. First Order Linear Equations
  • 4. Approximation
  • 5. Second Order Homogeneous Equations
  • 6. Second Order Linear Equations
  • 7. Second Order Linear Equations, take two

18 Useful formulas

19 Introduction to Sage

  • 1. Basics
  • 2. Differentiation
  • 3. Integration

The hyperbolic functions appear with some frequency in applications, and are quite similar in many respects to the trigonometric functions. This is a bit surprising given our initial definitions.

Definition 4.11.1 The hyperbolic cosine is the function $$\cosh x ={e^x +e^{-x }\over2},$$ and the hyperbolic sine is the function $$\sinh x ={e^x -e^{-x}\over 2}.$$ $\square$

Notice that $\cosh$ is even (that is, $\cosh(-x)=\cosh(x)$) while $\sinh$ is odd ($\sinh(-x)=-\sinh(x)$), and $\ds\cosh x + \sinh x = e^x$. Also, for all $x$, $\cosh x >0$, while $\sinh x=0$ if and only if $\ds e^x -e^{-x }=0$, which is true precisely when $x=0$.

Lemma 4.11.2 The range of $\cosh x$ is $[1,\infty)$. $\square$

Proof. Let $y= \cosh x$. We solve for $x$: $$\eqalign{y&={e^x +e^{-x }\over 2}\cr 2y &= e^x + e^{-x }\cr 2ye^x &= e^{2x} + 1\cr 0 &= e^{2x}-2ye^x +1\cr e^{x} &= {2y \pm \sqrt{4y^2 -4}\over 2}\cr e^{x} &= y\pm \sqrt{y^2 -1}\cr} $$ From the last equation, we see $\ds y^2 \geq 1$, and since $y\geq 0$, it follows that $y\geq 1$.

Now suppose $y\geq 1$, so $\ds y\pm \sqrt{y^2 -1}>0$. Then $\ds x = \ln(y\pm \sqrt{y^2 -1})$ is a real number, and $y =\cosh x$, so $y$ is in the range of $\cosh(x)$. $\qed$

Definition 4.11.3 The other hyperbolic functions are $$\eqalign{\tanh x &= {\sinh x\over\cosh x}\cr \coth x &= {\cosh x\over\sinh x}\cr \sech x &= {1\over\cosh x}\cr \csch x &= {1\over\sinh x}\cr} $$ The domain of $\coth$ and $\csch$ is $x\neq 0$ while the domain of the other hyperbolic functions is all real numbers. Graphs are shown in figure 4.11.1 $\square$

0$, $\sinh x$ is increasing and hence injective, so $\sinh x$ has an inverse, $\arcsinh x$. Also, $\sinh x > 0$ when $x>0$, so $\cosh x$ is injective on $[0,\infty)$ and has a (partial) inverse, $\arccosh x$. The other hyperbolic functions have inverses as well, though $\arcsech x$ is only a partial inverse. We may compute the derivatives of these functions as we have other inverse functions.

Theorem 4.11.6 $\ds{d\over dx}\arcsinh x = {1\over\sqrt{1+x^2}}$.

Proof. Let $y=\arcsinh x$, so $\sinh y=x$. Then $\ds{d\over dx}\sinh y = \cosh(y)\cdot y' = 1$, and so $\ds y' ={1\over\cosh y} ={1\over\sqrt{1 +\sinh^2 y}} = {1\over\sqrt{1+x^2}}$. $\qed$

The other derivatives are left to the exercises.

Exercises 4.11

Ex 4.11.1 Show that the range of $\sinh x$ is all real numbers. (Hint: show that if $y=\sinh x$ then $\ds x =\ln (y+\sqrt{y^2+1})$.)

Ex 4.11.2 Compute the following limits:

    a. $\ds \lim_{x\to \infty } \cosh x$

    b. $\ds \lim_{x\to \infty } \sinh x$

    c. $\ds \lim_{x\to \infty } \tanh x$

    d. $\ds \lim_{x\to \infty } (\cosh x -\sinh x)$

(answer)

Ex 4.11.3 Show that the range of $\tanh x$ is $(-1,1)$. What are the ranges of $\coth$, $\sech$, and $\csch$? (Use the fact that they are reciprocal functions.)

Ex 4.11.4 Prove that for every $x,y\in\R$, $\sinh (x+y) =\sinh x \cosh y + \cosh x \sinh y$. Obtain a similar identity for $\sinh(x-y)$.

Ex 4.11.5 Prove that for every $x,y\in\R$, $\cosh (x+y) =\cosh x \cosh y + \sinh x \sinh y$. Obtain a similar identity for $\cosh(x-y)$.

Ex 4.11.6 Use exercises 4 and 5 to show that $\sinh(2x)=2\sinh x \cosh x$ and $\ds \cosh(2x)=\cosh^2 x +\sinh^2 x$ for every $x$. Conclude also that $\ds (\cosh (2x) -1)/2 = \sinh ^2 x$ and $\ds(\cosh (2x)+1)/2 = \cosh^2 x$.

Ex 4.11.7 Show that $\ds {d\over dx} (\tanh x) =\sech^2 x$. Compute the derivatives of the remaining hyperbolic functions as well.

Ex 4.11.8 What are the domains of the six inverse hyperbolic functions?

Ex 4.11.9 Sketch the graphs of all six inverse hyperbolic functions.

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