WJEC FP3

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FP3 flashcards
Joshua Butterwor
Flashcards by Joshua Butterwor, updated more than 1 year ago
Joshua Butterwor
Created by Joshua Butterwor over 8 years ago
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Question Answer
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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