Q:

Tangent of a Straight Line

I just got back a math test in my EF Calc class, and I disagree with my teacher on this one problem. We are using derivatives to determine equations of lines tangent to a given equation. The equation given was: $f(x) = 27$
My response was "No Solution. Not a Curve." My teacher responded by saying that $y = 27$ was the correct answer. My argument is that these two lines have an infinite number of intersections, while my teacher says they only intersect once. So, my question is whether two lines with the same equation intersect once or infinitely.
I apologize if this question is of a lower degree of difficulty than the rest of the site, but this seemed the best place to ask. Thank you in advance for any help.

A:

Note that $f(x) = 27$ means that for every $x \in \Bbb R$, $f$ gives back the value $27$. This is a well defined function. It is not an injective function, though (do you remember the definition?). 
The tangent line to the graph of $f$ at $x_0$ is given by: $$y-f(x_0) = f'(x_0)(x-x_0).$$ If $f$ is constant, its derivative is zero at every point. And the value at every point is just a constant. Hence: $$y - 27 = 0\cdot (x-x_0) \implies y = 27.$$ So $y=27$ is the equation of the line you seek. At every point of the graph, the line is the same. Here we don't have a "tangent" line, as you have noticed. You can think of that line as the optimal linear approximation to the function near the point. So it is natural to conclude that for linear functions $f(x) = ax+b$ (our particular case is $a=0$ and $b=27$), the "tangent" line will be the function itself.

A:

The underlying problem that confuses you is to decide what it means for a line to be tangent to a curve.
In the particular case of circles, it happens that the tangents to a circle are precisely those straight lines that has exactly one point in common with the circle. It is tempting to make this into a definition, but it doesn't really work well for curves other than circles (and a group of sufficiently circle-like curves such as ellipses, namely smooth convex closed curves). The problem is that the global property of intersecting the circle exactly once doesn't really capture the essentially local intuition of what it means to be a tangent, namely that it "has the same direction of the curve" at the point of tangency.
For example the one-intersection-point definition would imply that the axis of a parabola is a tangent to it, which intuitively ought to be absurd -- the axis is perpendicular to the parabola, not tangent to it.
It turns out that it is not at all straightforward to define tangents for arbitrary curves -- essentially, figuring out a workable definition was the problem calculus was originally invented to do, and I can't suggest any working definition that doesn't involve calculus either openly or in disguise. (I'll offer to try to shoot down any other proposed definition on intuitive grounds, but that depends on us agreeing about what the intuition of "has the same direction as the curve" means).
In the end it seems that you just have to accept that the calculus based definition (or family of definitions) of "tangent" is what the mathematical community has collectively decided to use the word "tangent" for when we're talking about arbitrary curves. This definition gives the expected result for tangents to circles, and also produces tangents that agree with intuition for curves that it gives tangents for at all. It's also quite useful in practice, and utility plus consistency with the restricted case of circles is enough to make it a good definition.

