Taylor's Series Wave Equation Watch

NotNotBatman
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I'm looking at traffic flows and I have the first order wave equation \displaystyle \frac{\partial \rho}{\partial t} + c(\rho)\frac{\partial \rho}{\partial t} (partial derivatives)

I have a wave speed denoted by  c(\rho_{0}+ \tilde{\rho}(x,t)) So a small perturbation in the speed, by allowing  \rho(x,t) = \rho_0 + \tilde{\rho}(x,t).

and my notes say, using Taylor's theorem  c(\rho_{0}+ \tilde{\rho}(x,t)) = c(\rho_0)+c'(\rho_0)\tilde{\rho}  (x,t)

But I don't understand how this step is attained.
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DFranklin
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(Original post by NotNotBatman)
I'm looking at traffic flows and I have the first order wave equation \displaystyle \frac{\partial \rho}{\partial t} + c(\rho)\frac{\partial \rho}{\partial t} (partial derivatives)

I have a wave speed denoted by  c(\rho_{0}+ \tilde{\rho}(x,t)) So a small perturbation in the speed, by allowing  \rho(x,t) = \rho_0 + \tilde{\rho}(x,t).

and my notes say, using Taylor's theorem  c(\rho_{0}+ \tilde{\rho}(x,t)) = c(\rho_0)+c'(\rho_0)\tilde{\rho}  (x,t)

But I don't understand how this step is attained.
Taylor's theorem tells you that when h is small, f(a + h) \approx f(a) + h f'(a) ; I'd have thought the application here was obvious.
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NotNotBatman
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(Original post by DFranklin)
Taylor's theorem tells you that when h is small, f(a + h) \approx f(a) + h f'(a) ; I'd have thought the application here was obvious.
Thanks, honestly, it should be obvious, but I think I skipped learning this last year or just forgot it.
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