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One of the most important ideas of calculus is $$ f(x) \approx f(x_0) + f'(x_0)(x - x_0). $$ The approximation is good when $x$ is close to $x_0$. This approximation is very useful because the complicated, nonlinear function $f$ is approximated well by the very simple function on the right (at least when $x$ is near $x_0$), and we can use this approximation to simplify many calculations.

When teaching freshman calculus, I would really like to call the function $L(x) = f(x_0) + f'(x_0)(x - x_0)$ something like "the linear approximation to $f$ near $x_0$". I am afraid to use the word "affine" at this stage, because psychologically the word "affine" may sound unfamiliar and intimidating, and does not convey the fact that the graph of $L$ is a straight line and that $L$ is a very simple, easy, friendly and non-intimidating function.

I am considering using the term "linear" in this sense when teaching calculus, but emphasizing that I am using the "high school" definition of a linear function, and that in more advanced math (such as linear algebra) students will find that the word "linear" means something slightly different, and that in a more advanced class $L$ would be called "affine" rather than "linear".

What do you think is the best way to handle this issue? Do you use a term such as "local linear approximation" to $f$ when teaching calculus?

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    $\begingroup$ General advice: use the same terminology as the textbook. $\endgroup$ – Gerald Edgar Jul 9 '18 at 12:37
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    $\begingroup$ In calculus and analysis, "linear" means $ax+b$, because the graph makes a line. Calculus doesn't care where the origin is (the point $(0,0)$ is mostly only special in that it makes some calculations easier), so neither does the vocabulary. Algebra cares where the origin is (the point $(0,0)$ is the only one which is contained in all subspaces of the plane, for instance), which is reflected in the definitions of words like "linear" within that field. $\endgroup$ – Arthur Jul 9 '18 at 13:28
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    $\begingroup$ One could make the argument that it is linear algebra which is using the imprecise name, both generalizing and restricting the term. $\endgroup$ – Adam Jul 9 '18 at 19:01
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    $\begingroup$ Most textbooks I've seen call this a linear approximation. Unless the textbook you're using uses the term affine, I'd steer clear of it. $\endgroup$ – Sue VanHattum Jul 10 '18 at 0:36
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    $\begingroup$ Linear approximation all the way. Note there are parallels to physics and engineering where people do (e.g.) linear interpolation between data, even on nonlinear functions. $\endgroup$ – guest Jul 10 '18 at 2:15
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In calculus one calls $x \to ax + b$ a linear function. In linear algebra one calls $x \to ax + b$ an affine transformation, and says that it is linear only if $b = 0$.

The first order Taylor approximation to a differentiable map is reasonably called the linear approximation of the map. The terminology is reasonable because one is doing function theory, not linear algebra, and the meaning of the qualifier linear is context dependent. A formal way of giving content to the word "context" is to speak in terms of categories. Said more formally, what linear means when qualifying a morphism depends on the category to which the morphism pertains.

When working in the category of vector spaces (that is, doing linear algebra), a morphism is linear if it preserves the vector space structure, so $x \to ax + b$ is not linear, viewed as a map between one-dimensional vector spaces (it can be called affine if one works in the affine category).

When working in the category of smooth (to whatever degree) manifolds (that is, doing calculus), there is no notion of linear without some additional structure. A mapping ought to be called linear if it maps lines to lines, but for this to make sense there has to be a notion of line. The structure necessary to speak of lines is an affine connection - the lines are the geodesics of the connection. Properly speaking these are really lines only if the affine connection is torsion-free and flat (in this case there exist localy coordinates in which a geodesic really is a line). So, the additional structure that is used implicitly in speaking of a linear function is a flat torsion-free affine connection (on $\mathbb{R}$ it will be the usual derivative operator). A smooth mapping $f:M \to N$ between connected differentiable manifolds equipped with flat torsion-free affine connections can be defined to be linear if the covariant derivative of its differential vanishes (to make sense of this the differential $Df$ has to be viewed as a section of the bundle $T^{\ast}\otimes f^{\ast}TN$ equipped with the induced product connection; the condition is $\nabla Df = 0$). This is akin to defining a smooth map between connected differentiable manifolds to be constant if its differential vanishes, but is different in that it requires the extra structure of the flat connection (in fact the flatness is mostly irrelevant - dropping it one obtains the definition of a totally geodesic map - the flatness, which in the one-dimensional case is in any case automatic, is just there to justify using the word linear, because without flatness, there's no reason to think of geodesics as linear). If $M = N = \mathbb{R}$ with the usual flat connection, then this just means that $f^{\prime\prime} = 0$, so that $f$ has the form $f(x) = ax + b$.

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I think both "the linear approximation of $y=f(x)$ at $x=a$" and "local linear approximation" are fine. Now, it should be made clear that the linear approximation is a function. Next, what kind of function? A linear function! It has the form $y=mx+b$ after all. But what about $b$? I think you are right that it would be confusing to bring (the correct term) "affine" in this context. I have been trying to use "linear operator" for $y=mx$ in calculus 3 and linear algebra to distinguish the two.

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There are two unambiguous terms that you can use in any context: ‘homogeneous linear’ and ‘affine linear’; ‘linear’ by itself is ambiguous, and its meaning depends on the circumstances. It's not that one of these meanings is right and the other is wrong, so you should feel free to use whichever best fits your class (perhaps following the textbook). That said, you could certainly point out the ambiguity to your students (especially if they might read a different text later).

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The original post's formula specifies a "tangent line". In other words, the line that is tangent to the function f at the point (x, f(x)).

The students are familiar with the term "tangent" from their geometry classes: in particular, lines that are tangent to circles.

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