Euler's method error

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I understand how the (local) error for euler's method of integration is derived from the perspective of the taylor expansion and inequality. However, I don't really see why taylor's equation needs to be invoked, since the euler method can also be derived as a tangent line approximation. How then is the order of the error estimated by interpreting euler's method as a tangent line approximation?
 

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Integral
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It is really the same thing. Using a Taylor series expansion, for the Euler method you truncate all but the linear terms, thus your approximation is assuming a local linear function. This is the exact same thing as using a tangent line approximation. You are assuming that the function is locally linear. When you say you are just making a tangent line approximation you are simply ignoring the fact that you are in reality just dropping the nonlinear terms of the Taylor series.
 

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