See Also
Flashcards
Definitions of the hyperbolic functions
\[\sinh x\]
What is the definition?
\[\frac{e^x - e^{-x}}{2}\]
\[\cosh x\]
What is the definition?
\[\frac{e^x + e^{-x}}{2}\]
\[\tanh x\]
What is the definition?
\[\frac{e^{2x} - 1}{e^{2x} + 1}\]
Recognising and sketching the graphs
What function is this?
\[\sinh\]
What function is this?
\[\cosh\]
What function is this?
\[\tanh\]
\[y = \sinh x\]
What does the graph look like?
\[y = \cosh x\]
What does the graph look like?
\[y = \tanh x\]
What does the graph look like?
What is true about any value of $\cosh x$?
\[e^x - e^{-x} = 10\]
How would you rewrite this?
\[e^{2x} - 1 = 10e^x\]
Inverse hyperbolic functions and their domains
\[\arcsinh x\]
What is the definition?
\[\ln(x + \sqrt{x^2 + 1})\]
\[\arcosh x\]
What is the definition?
\[\ln(x + \sqrt{x^2 - 1})\]
\[\artanh x\]
What is the definition?
\[\frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right)\]
What is the domain for $\arcosh x$?
\[x \ge 1\]
What is the domain for $\artanh x$?
\[ \vert x \vert < 1\]
What function is this?
\[\arsinh\]
What function is this?
\[\arcosh\]
What function is this?
\[\artanh\]
\[y = \arsinh x\]
What does the graph look like?
\[y = \arcosh x\]
What does the graph look like?
\[y = \artanh x\]
What does the graph look like?
What is true about any value of $\cosh x$?
Osborn’s Rule and converting identities
What is Osborn’s Rule?
Replace any product of two $\sin$ terms by minus the products of two $\sin$ terms.
By Osborn’s Rule, what is $\sinA\sinB$ in hyperbolic functions?
\[-\sinhA\sinhB\]
By Osborn’s Rule, what is $\tan^2 x$ in hyperbolic functions?
\[-\tanh^2 x\]
How do you convert a trig identity to a hyperbolic trig identity?
- Replace all normal functions with their hyperbolic equivalents
- Use Osborn’s Rule
If you’re not allowed to use Osborn’s Rule when converting a hyperbolic trig identity, what can you do?
Use the $e^x$ definitions of all the functions.
\[\sin^2 x + \cos^2 x = 1\]
What is the hyperbolic equivalent?
\[\cos^2 x - \sin^2 x = 1\]
Derivatives of hyperbolic and inverse hyperbolic functions
\[\frac{d}{dx} \sinh x\]
What is this equal to?
\[\cosh x\]
\[\frac{d}{dx} \cosh x\]
What is this equal to?
\[\sinh x\]
\[\frac{d}{dx} \tanh x\]
What is this equal to?
\[\sech^2 x\]
\[\frac{d}{dx} (\sinh^{-1} x)\]
What is the equal to?
\[\frac{1}{\sqrt{x^2 + 1}}\]
\[\frac{d}{dx} (\cosh^{-1} x)\]
What is the equal to?
\[\frac{1}{\sqrt{x^2 - 1}}\]
\[\frac{d}{dx} (\tanh^{-1} x)\]
What is the equal to?
\[\frac{1}{1 - x^2}\]
Deriving an inverse derivative from first principles
If $y = \sinh^{-1}(x)$, what is $x$ equal to?
\[x = \sinh(y)\]
\[x = \sinh(y)\]
What do you get if you differentiate both sides?
\[\frac{dx}{dy} = \cosh(y)\]
\[\frac{dx}{dy} = \cosh(y)\]
The aim here is to get $\frac{dy}{dx}$. How could you write $\cosh(y)$ made out of something you already know?
\[\frac{dx}{dy} = \sqrt{1 + \sinh^2(y)}\]
\[\frac{dx}{dx} = \sqrt{1 + \sinh^2(x)}\]
How could you rewrite this in terms of what you already know?
\[\frac{dx}{dy} = \sqrt{1 + x^2}\]
\[\frac{dx}{dy} = u\]
How could you rewrite this so it’s $\frac{dy}{dx}$?
\[\frac{dy}{dx} = \frac{1}{\sqrt{1 + x^2}}\]
When finding the derivative of an inverse function, what’s the trick?
Rewriting some $f(y)$ in terms of $x$.
Integrals giving inverse hyperbolic functions
\[\int \frac{1}{\sqrt{x^2 + 1}}dx\]
What is this equal to?
\[\sinh^{-1} x\]
\[\int \frac{1}{\sqrt{x^2 - 1}}dx\]
What is this equal to?
\[\cosh^{-1} x\]
\[\frac{d}{dx}\left(\sinh^{-1}\left(\frac{x}{a}\right)\right)\]
What is this equal to?
\[\frac{1}{\sqrt{x^2 + a^2}}\]
\[\frac{d}{dx}\left(\cosh^{-1}\left(\frac{x}{a}\right)\right)\]
What is this equal to?
\[\frac{1}{\sqrt{x^2 - a^2}}\]
\[\int\frac{1}{\sqrt{x^2 + a^2}}dx\]
What is this equal to?
\[\sinh^{-1}\left(\frac{x}{a}\right) \pmb{+ c}\]
\[\int\frac{1}{\sqrt{x^2 - a^2}}dx\]
What is this equal to?
\[\cosh^{-1}\left(\frac{x}{a}\right) \pmb{+ c}\]
\[\int\frac{1}{\sqrt{x^2 - 16}}dx\]
What is this equal to?
\[\cosh^{-1}\left(\frac{x}{4}\right) \pmb{+ c}\]
\[\int\frac{1}{\sqrt{x^2 + 8}}dx\]
What is this equal to?
\[\sinh^{-1}\left(\frac{x}{2\sqrt{2}}\right) \pmb{+ c}\]
\[\sqrt{4x^2 + 1}\]
How could you rewrite this to aid with integrating?
\[2\sqrt{x^2 + \frac{1}{2}}\]