WJEC FP3

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FP3 flashcards
Joshua Butterwor
Karteikarten von Joshua Butterwor, aktualisiert more than 1 year ago
Joshua Butterwor
Erstellt von Joshua Butterwor vor mehr als 8 Jahre
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Express \(\sinh{x}\) in terms of exponentials. \[\sinh{x}=\frac{e^{x}-e^{-x}}{2}\]
Express \(\cosh{x}\) in terms of exponentials. \[\cosh{x}=\frac{e^{x}+e^{-x}}{2}\]
Express \(\tanh{x}\) in terms of exponentials. \[\tanh x=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\] or \[\tanh x=\frac{e^{2x}-1}{e^{2x}+1}\]
Express \(\cosh^{-1}{x}\) in terms of natural logs \[\cosh^{-1}{x}=\ln({x+\sqrt{x^{2}-1}})\]
Express \(\sinh^{-1}{x}\) in terms of natural logs \[\sinh^{-1}{x}=\ln({x+\sqrt{x^{2}+1}})\]
Express \(\tanh^{-1}{x}\) in terms of natural logs \[\tanh^{-1}{x}=\frac{\ln(\frac{1+x}{1-x})}{2}\]
Differentiate \(\cosh{x}\) \[\frac{\partial \cosh{x}}{\partial x}=\sinh{x}\]
Differentiate \(\sinh{x}\) \[\frac{\partial \sinh{x}}{\partial x}=\cosh{x}\]
Differentiate \(\tanh{x}\) \[\frac{\partial \tanh{x}}{\partial x}=\sech^{2}{x}\]
Differentiate \(\sinh^{-1}{x}\) \[\frac{\partial sinh^{-1}{x}}{\partial x}=\frac{1}{\sqrt{1+{x}^2}}\]
Differentiate \(\cosh^{-1}{x}\) \[\frac{\partial cosh^{-1}{x}}{\partial x}=\frac{1}{\sqrt{{x}^{2}-1}}\]
Differentiate \(\tanh^{-1}{x}\) \[\frac{\partial tanh^{-1}{x}}{\partial x}=\frac{1}{1-{x}^{2}}\]
Integrate \(\tanh^{-1}{x}\) \[\int \tanh x = \ln{\cosh{x}}\]
What is Osborne's Rule? The idea that trigonometric equations can be changed to hyperbolic equations by exchanging the trig functions for their hyperbolic counterparts and changing the sign wherever there is a product of two sines.
Use Osborne's rule to find the corresponding hyperbolic equation to: \[\tan^{2}{x}+1\equiv \sec^{2}{x}\] \[1=sech^{2}{x} + \tanh^{2}{x}\]
Use Osborne's rule to find the corresponding hyperbolic equation to: \[\cot^{2}{x}+1\equiv \cosec^{2}{x}\] \[cosech^{2}{x}=1+ \coth^{2}{x}\]
What is the integral of\( f'(x)[f(x)]^{n}\) \[\frac{1}{n+1}[f(x)]^{n+1}+c\]
What is the integral of \(\frac{f'(x)}{f(x)}\) \[\ln\left | f(x) \right |+c\]
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