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I'm tutoring a first-semester calculus student, and we were looking over the slides the teacher has used.

After teaching (or rather, repeating, for those who completed AP high school math) basic derivatives, there's a couple slides about "linear approximations" and how to find them. The student had a hard time grasping the idea until the end, when she realized "oh, this is just the tangent to the function at that point", and realized she already knew this from high school.

This, after going through the whole idea of $f(x) \approx f(a) + (x-a)f'(a)$ for some constant $a$.

In HS, she learned how to find a linear approximation by applying the point-slope formula, where the slope is the derivative at this point, and the point is the point at which the tangent touches the curve.

What I'm wondering is, will there come a time further down the line where the idea of a "linear approximation" is a better way of looking at it than as "just" a tangent?

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Thinking about tangent lines as linear approximations is a great way to foreshadow Taylor series. After linear approximations in calc 1, I usually give students a few prompts to get them thinking about quadratic approximations, and then I talk briefly about how it appears in calc 2.

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  • $\begingroup$ Why can’t it be both? I mean, the derivative is going to pop up in various guises depending on the student’s chosen path through mathematics, local linearity is basic to continuity proofs, and numerical approximations are, well, they’re what we do. $\endgroup$ – Sciolism Apparently Feb 7 '19 at 20:11
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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 $\vec{x} \approx \vec{a}$):

$$\vec{f}(\vec{x}) \approx J (\vec{x}-\vec{a}) + \vec{f}(a)$$

Where $J$ is the Jacobian matrix of $\vec{f}$ at $\vec{a}$.

Linear approximations are also useful in solving differential equations. Solving for the position of a pendulum as a function of time requires approximating $\sin(x) \approx x$ for small values of $x$ if one wants an analytic solution (https://en.wikipedia.org/wiki/Pendulum_(mathematics)#Small-angle_approximation)

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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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