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4

The term Hergert Numbers is sometimes used in my specific region of the US for the values of $x$ where $f''(x) = 0$ or $f''(x)$ is undefined. This is in reference to Rodger Hergert, an Illinois community college professor who sometime in the 1990s became very frustrated with the fact that these numbers had no good name. But it doesn't really matter what you ...

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The calculus textbook I'm currently using (Burzynski, Applied Calculus for Business, Life and Social Sciences, XYZ Textbooks) uses the term "hypercritical point". A quick web search indicates that a few other sources use it as well.

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Linear approximation is taught as standard method for interpolating tabular data in many engineering areas. For instance steam tables. Sort of a crutch to give you more value than the resolution of the table.

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For high-dimensional multivariable functions it's hard to imagine a geometric tangent, but the notion of linear approximation makes perfect sense. The derivative (usually called the Jacobian matrix https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant) still satisfies (for $\vec{x} \approx \vec{a}$): \vec{f}(\vec{x}) \approx J (\vec{x}-\vec{a}) + \...

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I would go with a graphical interpretation to start with. I remember learning this sort of thing in the USN sub force for power transients on reactors (rods, reactivity, power). It was especially useful for the enlisted men since they had less math. But even for officers, I felt that it gave huge insight to have to draw curves (even by eye, ...

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A good approach for engineers might be to connect the "slope of a graph" to sensitivity: the factor by which a small change in input gets multiplied to produce a change in the output. You can connect this to both understanding how changes propagate through a system and error analysis. You can either do this purely symbolically to derive the ...

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