Further Maths - Hyperbolic Functions
See Also
Flashcards
2021-03-15
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\[\sinh x\]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??
\[\frac{e^{2x} - 1}{e^{2x} + 1}\]
What function is this?
What function is this?
What function is this?
What function is this?
What function is this?
What function is this?#####
\[y = \sinh x\]What does the graph look like??
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\[y = \cosh x\]What does the graph look like??
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\[y = \tanh x\]What does the graph look like??
What is true about any value of $\cosh x$?
It is above $1$.
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\[e^x - e^{-x} = 10\]How would you rewrite this??
\[e^2x - 1 = 10e^x\]2021-03-16
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\[\arcsinh x\]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 $\arcosh x$?
What is the domain for $\artanh x$??
\[|x| < 1\]
What function is this?
What function is this?
What function is this?
What function is this?
What function is this?
What function is this?#####
\[y = \arsinh x\]What does the graph look like??
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\[y = \arcosh x\]What does the graph look like??
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\[y = \artanh x\]What does the graph look like??
What is true about any value of $\cosh x$?
It is above $1$.
2021-03-17
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?
By Osborn’s Rule, what is $\tan^2 x$ in hyperbolic functions?
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 $e^x$ defintitions of all the functions.
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\[\sin^2 x + \cos^2 x = 1\]What is the hyperbolic equivalent??
\[\cos^2 x - \sin^2 x = 1\]#####
\[\frac{d}{dx} \sinh x\]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??
\[\frac{1}{\frac{x^2 + 1}}\]#####
\[\frac{d}{dx} (\cosh^{-1} x)\]What is the equal to??
\[\frac{1}{\frac{x^2 - 1}}\]#####
\[\frac{d}{dx} (\tanh^{-1} x)\]What is the equal to??
\[\frac{1}{\frac{1 - x^2}}\]2021-03-24
If $y = \sinh^{-1}(x)$, what is $x$ equal to?
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\[x = \sinh(y)\]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 $x$.
2021-03-25
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\[\int \frac{1}{\sqrt{x^2 + 1}}dx\]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??
\[2\sqrt{x^2 + \frac{1}{2}}\]