\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

Section 1.2 Calculus 2

Hyperbolic Functions
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1.2 Hyperbolic Functions

1.2.1 Hyperbolic Sine and Cosine

  • Exponential functions are used to define the hyperbolic functions, which behave like trigonometric functions in many ways.
    • \(\sinh x = \frac{e^x-e^{-x}}{2}\)
    • \(\cosh x = \frac{e^x+e^{-x}}{2}\)
  • Example Evaluate \(\sinh(0)\) and \(\cosh(0)\).
  • Example Evaluate \(\cosh(\ln 4)\).
  • Example Prove that \(\sinh(2x)=2\sinh(x)\cosh(x)\).

\(\newcommand{\sech}{\mathrm{sech}\,}\) \(\newcommand{\csch}{\mathrm{csch}\,}\)

1.2.2 Other Hyperbolic Functions

  • The other hypberbolic functions are defined the same way their trig counterparts are, and have similar properties.
    • \(\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^x-e^{-x}}{e^x+e^{-x}}\)
    • \(\coth x = \frac{\cosh x}{\sinh x} = \frac{e^x+e^{-x}}{e^x-e^{-x}}\)
    • \(\sech x = \frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}\)
    • \(\csch x = \frac{1}{\sinh x}=\frac{2}{e^x-e^{-x}}\)
  • Example Evaluate \(\sech(-\ln 2)\).
  • Example Prove that \(\tanh^2(x)=1-\sech^2(x)\).

1.2.3 Derivatives and Integrals of Hyperbolic Functions

  • Their derivatives also behave similarly.
    • \(\frac{d}{dx}[\sinh x] = \cosh x\)
    • \(\frac{d}{dx}[\cosh x] = \sinh x\)
    • \(\frac{d}{dx}[\tanh x] = \sech^2 x\)
    • \(\frac{d}{dx}[\coth x] = -\csch^2 x\)
      • This is correct, but the video incorrectly leaves off the negative
    • \(\frac{d}{dx}[\sech x] = -\sech x\tanh x\)
    • \(\frac{d}{dx}[\csch x] = -\csch x\coth x\)
  • Example Use their definitions to prove that \(\frac{d}{dx}[\cosh x]=\sinh x\).
  • Example Use their definitions to prove that \(\frac{d}{dx}[\coth x]=\csch^2 x\).
  • Example Compute \(\frac{d}{dx}[\sinh(2x)+\coth(x^2)]\).
  • Their integral formulas may be found by just reversing the equations.
    • \(\int \cosh x\,dx = \sinh x + C \)
    • \(\int \sinh x\,dx = \cosh x + C \)
    • \(\int \sech^2 x\,dx = \tanh x + C \)
    • \(\int \csch^2 x\,dx = -\coth x + C \)
    • \(\int\sech x\tanh x\,dx = -\sech x + C \)
    • \(\int\csch x\coth x\,dx = -\csch x + C \)
  • Example Find \(\int (4\csch^2 x-3\sinh x)\,dx\).

Review Exercises

  1. Evaluate \(\sinh(\ln 6)\).
  2. Prove that \(\cosh (2x) = \cosh^2 x + \sinh^2 x\).
  3. Prove that \(\cosh^2 x - \sinh^2 x = 1\).
  4. Evaluate \(\tanh(\ln 3)\).
  5. Simplify \(\sinh(x)\coth(x)\cosh(x)-\frac{1}{\csch^2(x)}\). (Hint: convert everything to \(\sinh x\) and \(\cosh x\).)
  6. Prove that \(\frac{d}{dx}[\sinh x] = \cosh x\).
  7. Prove that \(\frac{d}{dx}[\sech x] = -\sech x\tanh x\). (Hint: use the fact that \(\frac{d}{dx}[\cosh x] = \sinh x\).)
  8. Compute \(\frac{d}{dx}[\tanh(3x)-\sech(\ln x)]\).
  9. Find \(\int (3\csch x\coth x - 2\sinh x)\,dx\).
  10. Let \(\sinh^{\leftarrow}(x)\) be the inverse function of \(\sinh(x)\). Use the facts \(\frac{d}{dx}[f^{\leftarrow}(x)]=\frac{1}{f’(f^{\leftarrow}(x))}\) and \(\cosh^2 x-\sinh^2 x = 1\) to prove that \(\frac{d}{dx}[\sinh^{\leftarrow}(x)]=\frac{1}{\sqrt{1+x^2}}\).
  11. Prove that \(\sinh^{\leftarrow}(x)=\ln(\sqrt{x^2+1}+x)\).
  12. Evaluate \(\cosh(\ln 2)\).
  13. Differentiate \(f(x)=\tanh(x^2)-\cosh(2x+1)\).

Solutions

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 7.3