2.5.3: Additional Exercises
- Page ID
- 116877
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Reading Questions
- What is the algebraic relation that defines the unit circle? What analogous relation defines the unit hyperbola (right branch)?
- How are \(\cosh(t)\) and \(\sinh(t)\) defined in terms of exponential functions?
- What is the Fundamental Hyperbolic Identity, and how does it relate to the Pythagorean identity for trigonometric functions?
- Express \(\tanh(t)\) in terms of \(\sinh(t)\) and \(\cosh(t)\), and then in terms of exponential functions.
- Which hyperbolic function represents the shape of a catenary (a hanging cable)?
- How does the graph of \(\cosh(x)\) differ from \(\sinh(x)\) in terms of symmetry and y-intercept?
- What are the horizontal asymptotes of \(y=\tanh(x)\)?
- Are all hyperbolic functions one-to-one? If not, which ones require domain restrictions to define their inverses, and what is a common restriction?
- How is \(\text{sinh}^{-1}x\) defined in terms of a natural logarithm?
- How is \(\text{cosh}^{-1}x\) defined in terms of a natural logarithm? What is the domain for this inverse function?
- Is \(\cosh(x)\) an even or odd function? How about \(\sinh(x)\)?
Homework
Evaluations
- Evaluate the given function. Round your answers to 3 decimal places, if necessary.
- \( \coth{(-7)} \)
- \( \operatorname{csch}{(\ln{(3)})} \)
- \( \cosh{(3)} \)
- \( \sinh{(0)} \)
- \( \tanh{(1)} \)
- \( \operatorname{sech}{(\ln{(7)})} \)
- \( \sinh^{-1}{(3)} \)
- \( \cosh^{-1}{\left(\frac{3}{2}\right)} \)
- \( \tanh^{-1}{\left(\frac{1}{2}\right)} \)
Simplifications
- Rewrite the following expressions in terms of exponentials and simplify.
- \(2\cosh(\ln x)\)
- \(\cosh 4x+\sinh 4x\)
- \(\cosh 2x−\sinh 2x\)
- \(\ln(\cosh x+\sinh x)+\ln(\cosh x−\sinh x)\)
Proofs
- Prove the identity. Your work should be legible, and all your logic should be clear and justified.
\[ \cosh{(x + y)} = \cosh{(x)} \cosh{(y)} + \sinh{(x)} \sinh{(y)} \nonumber \] - Prove the identity. Your work should be legible, and all your logic should be clear and justified.
\[ \sinh{(x + y)} = \sinh{(x)} \cosh{(y)} + \cosh{(x)} \sinh{(y)} \nonumber \] - Prove the identity. Your work should be legible, and all your logic should be clear and justified.
\[ \cosh{(2x)} = \cosh^2{(x)} + \sinh^2{(x)} \nonumber \] - Prove the identity. Your work should be legible, and all your logic should be clear and justified.
\[ \sinh{(2x)} = 2 \sinh{(x)} \cosh{(x)} \nonumber \] - Prove that \( f(x) = \cosh{(x)} \) is an even function.
- Prove that \( f(x) = \sinh{(x)} \) is an odd function.
- Prove the identitiy. Your work should be legible, and all your logic should be clear and justified.
\[\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)\nonumber\] - Prove the expression for \(\sinh^{−1}(x)\). Multiply \(x=\sinh(y)=\dfrac{e^y−e^{−y}}{2}\) by \(2e^y\) and solve for \(y\). Does your expression match the textbook?
- Prove the expression for \(\cosh^{−1}(x)\). Multiply \(x=\cosh(y)=\dfrac{e^y+e^{−y}}{2}\) by \(2e^y\) and solve for \(y\). Does your expression match the textbook?