# "Real world" examples of implicit functions

When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences.

So I'm looking for (relatively simple) examples from physics/engineering/life science/economics etc. of two quantities $x,y$ that depend on each other in such a way, that neither $x$ is a function of $y$, nor $y$ is a function of $x$. In other words: $x,y$ should fulfil some equation, such that there exists an $x$ value for which the equation has at least two solutions in $y$ and vice versa.

Equations are great, but if you have some examples where the relation between $x$ and $y$ is not easily expressed by an equation, but instead by a graph or some other data that is also fine.

My students are mainly first year engineers.

Edit: There is a related question here, but it's aimed for advanced analysis courses and doesn't ask for examples from applied areas. It has at least one applicable answer with isolines, but to explain it with equations in the firs semester single variable calculus, I would probably have to introduce functions of several variables.

## 8 Answers

Many linkages have this sort of behavior and are, perhaps, relevant to a mechanical engineer. For example, you could analyse a bicycle suspension --- a type of four-bar linkage.

For circular motion, you have x2+y2 = r2, so except for at the ends, each x has two y solutions, and vice versa. Harmonic motion is in some sense analogous to circular motion. For instance, for Hooke's law, if you set y = $\sqrt{k/m}v$, then x2+y2 = constant; if you know the position of a pendulum, you know the speed, but not the direction, and given a velocity, you know the magnitude of the displacement, but not the direction.

Economics has a ton of applications since they are interested in relationships between variables, not functions per se. Two simplish ones that come to mind are

• Price elasticity of demand: Supply curves are just a relation between supply and demand, it need not be functional. You can derive the price elasticity in this setting. Here's are some good notes from Y. Katznelsons course at UCSC.

• A typical problem looks like: Price $p$ and quantity $q$ are related by $p^2 - pq + q^2 = 400$. When the price is \$4, estimate the percent change in profit when the price is raised 1%. • To generalize the above, comparative statics uses implicit differentiation to study the effect of variable changes in economic models. Here's a decent introduction with example problems. • Preference bundles, utility and indifference curves. You have to gloss over some machinery but you're essentially doing calculus on level curves. If this looks interesting peek into the first few chapters of Hal Varian's Intermediate Microeconomics. • A sample problem would look like: Kim has \$10 to buy flour and GI Joe's. A bag of flour is \$.50 and a GI Joe is \$2. If the expected utility of $x_1$ bags of flour and $x_2$ GI Joes is $U(x_1,x_2) = \sqrt{x_1^2 + x_2^2}$, what point along the budget curve maximized utility? Sol: Find the point where tangent line to is the budget line.
• I really like this answer and would like to incorporate an example from economics in class. But in the notes you linked, I couldn't find a concrete and realistic implicit equation. The examples you mention (like $p^2-pq+q^2=400$) would be fine, if I could convince the students that I'm not just making up equations. Do you have a reference for the derivation of such an implicit equation in economics? Apr 1 '18 at 19:00
• Off the top of my head I don't really. In practice, price and demand relationships show up as a spreadsheet of prices and sales, and their fluctuations are dependent a lot of factors. Coming up with a way to compute the relationship for a single product in a single industry easily nets you a paper. So these problems are the theoretical background that we would hope to find in reality, if you look in the econ literature the calculus gives us some nice ways to interpret the spreadsheet. However, I've found that if you discuss these limitations with students they get that it's an idealization. Apr 2 '18 at 21:41
• I should say as well that the for most economists, the application is only local and never touches on a mathematical explanation: if we perturb the price slightly, what happens to sales? There's no hope of finding the actual function so why bother? That said here's another resource for some problems: math.ubc.ca/~maclean/math104/elasticity.pdf Even with the drawbacks, I've found students really like getting to talk about the limitations of these models. Apr 2 '18 at 21:50

In a conservative system in one dimension we have $E(x,v) = \frac{1}{2}mv^2+U(x)$ where $-\frac{dU}{dx} = F_{net}$. In other words, we have $E = KE+ PE$ where $PE=U$ and $KE = \frac{1}{2}mv^2$. Now, to study the rate of change in KE or PE we can relate it to the rate of change of the other, or $x$ or $v$. You can give a Poincare Plot of position verses velocity with energy level curves plotted. No equation for $E(x,v)$ need be given, but they could still reason various related rates depending on what you gave them.

How about Kepler's equation (has to do with orbits of planets) $$M = E - e\sin E$$ ($M$ is the mean anomaly, $E$ is the eccentric anomaly, and $e$ is the eccentricity)

For fixed $M$, this defines $E$ implicitly as a function of $e$, but we cannot solve it for $E$ explicitly.

• Thanks, but in this case $M$ is a function of $E$. If a student would care for $dE/dM$ he could also apply the formula for "derivatives of inverse functions", which is also covered earlier in the course. So he might object that one doesn't need implicit differentiation in this case. Feb 23 '18 at 15:02

Probably the simplest algebraic examples of what you're looking for are given by the elliptic curves. In Weierstrass, Jacobian quartic, and Jacobian intersection form (respectively) such curves are described by the equations \begin{align}(x,y)&:\mathbb{R}^2 &y^2 &=4x^3-g_2x-g_3 &\mathrm{\delta}t&\equiv \frac{\mathrm{d}x}{y}\\ (x,y)&:\mathbb{R}^2 &y^2 &=(1-x^2)(1-mx^2)&\mathrm{\delta}t&\equiv \frac{\mathrm{d}x}{y}\\ (s,c,d)&:\mathbb{R}^3 &s^2+c^2=1&\quad ms^2+d^2=1&\mathrm{\delta}t&\equiv \frac{\mathrm{d}s}{cd} \end{align} In physics, these are the nondimensionalized constraint equations for a particle in a cubic potential, a particle in a certain quartic potential, and the pendulum, respectively; $\mathrm{\delta}t$ is a nondimensionalized time increment that features in the respective physical problem. (For more details on the physics, see the references in the DLMF below.)

I admit that I've abstracted a bit from the physics and that these examples fit into situations that other answers have described. Still, I think these implicit equations deserve special mention for a few reasons:

• Unlike the implicit equations that determine conic sections, it is provably impossible to describe these curves using a rational-function parametrization—you can't "cheat" and use an elementary substitution.
• The special structures that such curves exhibit make them prototypes for extensive families of exactly solvable models in physics, generalizing the way we use conic sections (see the references in the DLMF for the Jacobian and Weierstrass elliptic functions).
• For special values of the parameter $m$ or $(g_2,g_3)$, these algebraic curves have rational-number solutions that have been extensively catalogued—useful if you want to "plug in numbers".
• Outside of physics, tangent lines to elliptic curves arise in a practical setting many students are unaware of: cryptography. Security for ATM cards and other smart cards is often based on elliptic curve cryptography (ECC). The math of ECC uses the elliptic curve group law, which involves tangent lines when doubling a point. Admittedly, Weierstrass equations used in ECC are defined over finite fields rather than $\mathbf R$ (often fields of characteristic 2, so the equations are more like $y^2 + xy =$ cubic in $x$ than $y^2 =$ cubic in $x$), but the idea of implicit diff. is still used.
– KCd
Feb 16 '19 at 10:36
• When teaching calculus I completely ignore the technicality of the field of definition and just say that the method of computing tangent lines by implicit differentiation is used in the security for smart cards and tell them that they can look up "elliptic curve cryptography" if they're interested.
– KCd
Feb 16 '19 at 10:37

I want to mention $$\tag1x^2+y^2=1,$$that has already appeared in other answers. We were all led to think (maybe involuntarily) by our teachers that this can easily be solved as $$\tag2y=\sqrt {1-x^2} ,$$ or the same with a minus in front.

My point is to question the "easily" above. We all became familiar with the square root from an early age, but when you think about it our only access to it was in the case of some easy numbers, or through a magic black box (the calculator).

The equation $(1)$ allows us to analyze the function $(2)$ in an easier and more direct way (for instance, think about sketching the graph).

Building off the circle example, you can actually work out the centripetal acceleration formula by implicitly differentiating twice. If your students aren't familiar with vectors you can just plug in x = 0 and y = 1:

$$x^2+y^2=r^2$$ Differentiate with respect to t: $$2x\cdot x' + 2y \cdot y' = 0$$ Plugging in x = 0, y = r, you can solve $$y' = 0$$.

Differentiating again, we get: $$2x'\cdot x' + 2x\cdot x'' + 2y' \cdot y' +2 y \cdot y'' = 0$$ Plugging in everything we know ($$x=0, y=r, y'=0$$): $$2(x')^2 + 2 r y'' = 0$$ Which we can solve to get: $$y'' = -\frac{(x')^2}{r}$$

I also throw in the example of the squircle ($$x^4+y^4 = 1$$) and an elliptic curve (https://en.wikipedia.org/wiki/Elliptic_curve) for fun. A key fact about elliptic curves that is used in applications is that any tangent line at a point with intersects the curve at exactly one other point, and if the point of tangency has rational coordinates, so does the intersection point.