Good Examples of Equations Derived from Elementary Calculus

Question

I'm collecting additional enrichment content for my calculus students. I'm looking for examples of equations that are used in various fields, but which can be derived at least somewhat straightforwardly from elementary calculus. It's okay if there is a minor additional concept involved or requires some basic differential equation-like manipulation.

Here's my current set, and I would love to hear what others could be presented!

Geometry

• Volume of a cone: $v = \frac{1}{3}\pi\, r^2h$
• Volume of a sphere: $v = \frac{4}{3}\pi\,r^3$
• Grazing goat problem

Money

• Compound interest rate equation: $p = p_0\,e^{rt}$
• The Kelly Criterion: $r = \frac{bp + p - 1}{p}$

Biology

• The logistics curve: $p = \frac{Kp_0e^{rt}}{K + p_0\left(e^{rt} - 1\right)}$
• Collector's Curve: $y = -Ke^{-\frac{x}{K}} + K$

Physics

• Kinematic equations: $p = \frac{1}{2}At^2 + v_0t + p_0$
• Kinetic energy formula: $E_k = \frac{1}{2}mv^2$
• Escape velocity formula: $v = \sqrt{\frac{2Gm}{r}}$
• Rocket equation: $v = v_0 + -v_e\ln\left(\frac{m_0}{m}\right)$
• Newton's law of cooling: $q = q_\text{env} - \left(q_\text{env} - q_0\right)e^{-Kt}$

Electronics

• Capacitor-based timers: $v = v_\text{final}\left(1 - e^{-\frac{t}{RC}}\right)$
• Calculating RMS voltage: $v_\text{RMS} = \sqrt{\frac{1}{2}}A$

I updated this list to include the actual equations for most of them. I'm not looking for a "type of problem" but a specific equation that people in that field would recognize that can be derived. I would appreciate answers that included specific equations that could be derived through calculus. Some fields missing above that would be helpful include economics, chemistry, meteorology/climatology, astronomy, etc.

Answer 1

Electric field due to a charged rod of length $L$. If the charged rod is placed from $x=0$ to $x=L$ then we ask what the electric field due to the uniformly charged rod at $(x,0)$ for $x >L$. Coulomb's Law for $dQ$ at $(z,0)$ gives $$ dE = \frac{1}{4\pi \epsilon_o}\frac{dQ}{(x-z)^2}$$ for $0 \leq z \leq L$. Assuming uniform charge density $dQ = \frac{Q}{L}dz$ hence the total electric field directed in the $x$-direction at $(x,0)$ is: $$ E = \int dE = \frac{Q}{4\pi \epsilon_oL}\int_0^L\frac{dz}{(x-z)^2} = \frac{Q}{4\pi \epsilon_oL}\left[ \frac{1}{x-L}-\frac{1}{x}\right] = \frac{1}{4\pi \epsilon_o}\frac{Q}{x(x-L)}$$ When $x >> L$ notice $E \sim \frac{1}{4\pi \epsilon_o}\frac{Q}{x^2}$. There are many such examples to be gleaned from introductory electrostatics.

Answer 2

I think there is a lot with rates versus cumulative.

For example oil production. But also any buildup or depletion. Drugs. PFAS. Radioactivity. Etc. Pressire equalization through an orifice. The concept and derivation of half life.

Edit: responding to your edit.

N = N0e−λt

λ=ln(2)t1/2≈0.693t1/2

See also: https://courses.lumenlearning.com/physics/chapter/31-5-half-life-and-activity/ (but does not contain the derivation)

Note that Beers Law for UV-tranmittance (or radiaoctive shielding) are interesting examples where the variable is distance rather than time but the same effect of attenuation occurs. (People talk about "tenth thickness" instead of half life, but same concept.)