See Also
Flashcards
$$\sinh x$$ 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??
$$ \sinh $$
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 $\cosh x$??
It is above $1$.
$$e^x - e^{-x} = 10$$ How would you rewrite this??
$$ e^2x - 1 = 10e^x $$
2021-03-16
$$\arcsinh x$$ 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 $\artanh x$??
$$ |x| < 1 $$
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 $\cosh x$??
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??
$$ -\sinhA\sinhB $$
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 $e^x$ defintitions of all the functions.
$$\sin^2 x + \cos^2 x = 1$$ 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??
$$ x = \sinh(y) $$
$$x = \sinh(y)$$ 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 $x$.
2021-03-25
$$\int \frac{1}{\sqrt{x^2 + 1}}dx$$ 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??
$$ 2\sqrt{x^2 + \frac{1}{2}} $$
Backlinks
- [[Further Maths - Syllabus]]S
- [[Further Maths - Integrating and Differentiating Inverse Trig Functions]]S
Metadata
date: 2021-03-15 12:53
tags:
- '@?further-maths'
- '@?school'
- '@?public'
title: Further Maths - Hyperbolic Functions
