Standard form

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Two young mathematicians discuss the standard form of a line.

Check out this dialogue between two calculus students (based on a true story):

Devyn: Riley, I think we’ve been too explicit with each other. We should try to be more implicit.
Riley: I. Um. Don’t really…
Devyn: I mean when plotting things!
Riley: Okay, but I still have no idea what you are talking about.
Devyn: Remember when we first learned the equation of a line, and the “standard form” was $ax+by = c$ or something, which is totally useless for graphing. Also a circle is $x^2 + y^2 = r^2$ or something, and here $y$ isn’t even a function of $x$ .
Riley: Ah, I’m starting to remember. We can write the same thing in two ways. For example, if you write $y = mx + b,$ then $y$ is explicity a function of $x$ but if you write $ax + by = c,$ then $y$ is implicitly a function of $x$ .
Devyn: What I’m trying to say is that we need to learn how to work with these “implicit” functions.

Consider the unit circle $x^2 + y^2 = 1.$ The point $P=(0,1)$ is on this circle. Reason geometrically to determine the slope of the line tangent to $x^2 + y^2 = 1$ at $P$ .

Draw a picture.

The slope is $\answer {0}$ .

Consider the unit circle $x^2 + y^2 = 1.$ The point $P=\left (\frac {\sqrt {2}}{2},\frac {\sqrt {2}}{2}\right )$ is on this circle. Reason geometrically to determine the slope of the line tangent to $x^2 + y^2 = 1$ at $P$ .

Draw a picture.

The slope is $\answer {-1}$ .