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Each lab pairs one idea with a place it's used. Pick by course, by field, or search.

Bézier Curves

Parametric curves and arc length

Calc IICS3D

Gradient Descent

Partial derivatives and the gradient

Calc IIICS3D

Edge Detection

Partial derivatives of an image

Calc IIICS3D

Robot Arm Kinematics

Matrix multiplication and rotations

Lin AlgRobotics3D

3D Transformations

Linear maps, determinants and eigenvectors

Lin AlgGame Dev3D

Forced Oscillations

Nonhomogeneous second-order ODEs

Diff EqPhysics3D

Vibrating Drumhead

The 2D wave equation

PDEPhysics3D

Projectile Motion

Quadratic functions

AlgebraPhysics3D

Compound Interest

Exponential functions

AlgebraEconomics

How Algorithms Scale

Logarithmic, polynomial and exponential growth

AlgebraCS

Drug Half-Life

Exponential decay and logarithms

AlgebraMedicine

Kepler's Third Law: A Planet's Year from Its Orbit

Rational exponents

AlgebraPhysics

Resistors in Parallel: A Rational Expression

Complex fractions and rational expressions

AlgebraEE

Reserved or On-Demand Servers?

Linear equations and inequalities

AlgebraCS

The Two Ways a Two-Story Building Sways

Equations reducible to quadratic form

AlgebraPhysics

How Fast a Tsunami Travels

Radical functions and radical equations

AlgebraPhysics

When Does Making More Get Cheap Enough?

Rational functions and rational inequalities

AlgebraEconomics

A Thermostat's Comfort Band

Absolute value equations and inequalities

AlgebraEE

Finding Your Phone with Three Circles

Equations of circles and nonlinear systems

AlgebraEE

Coupon First or Discount First?

Composition of functions, and why order matters

AlgebraEconomics

How a Ride-Share Fare Is Computed

Piecewise-defined functions

AlgebraEconomics

Breaking Kidney Stones with an Ellipse

Ellipses and their foci

AlgebraMedicine

Locating a Lightning Strike by Its Thunder

Hyperbolas: a constant difference of distances

AlgebraPhysics

Tuning a Game Character's Jump

Transformations of graphs, and symmetry

AlgebraGame Dev

Folding the Biggest Box from a Sheet of Cardboard

Factored polynomials: zeros and graphs

AlgebraEconomics

How Computers Evaluate Polynomials: Synthetic Division

Dividing polynomials, the remainder theorem and the factor theorem

AlgebraCS

How Long Can You Listen at That Volume?

Logarithms and logarithmic equations

AlgebraMedicine

Where Supply Meets Demand

Systems of two linear equations

AlgebraEconomics

Predicting a Ball's Landing from Three Snapshots

Systems of three linear equations

AlgebraRobotics

Position, Velocity, Acceleration

Derivatives as rates of change

Calc IPhysics3D

Maximizing Profit

Optimization with derivatives

Calc IEconomics

Newton's Method

Tangent lines and linear approximation

Calc ICS

Reading a Temperature Sensor

Functions and inverse functions

Calc IEE

Speed from GPS: The Limit

Limits

Calc IPhysics

Flipping a Switch: One-Sided Limits

One-sided limits and continuity

Calc IEE

Why Small Swings Keep Time

Computing limits: the squeeze theorem

Calc IPhysics

Projectors and Magnifiers

Infinite limits and vertical asymptotes

Calc IPhysics

Why Enzymes Max Out

Limits at infinity

Calc IBiology

Why Tax Brackets Don't Punish Raises

Continuity

Calc IEconomics

How Precise Must a Part Be?

The precise definition of a limit

Calc IPhysics

Why Wind Speed Matters So Much

Differentiation formulas: the power rule

Calc IEE

Power When Voltage and Current Both Change

Product rule

Calc IEE

The Shape of a Power Line

Derivatives of hyperbolic functions

Calc IEE

How Much Coffee for One More Tea?

Implicit differentiation

Calc IEconomics

Tracking a Car with a Drone Camera

Related rates

Calc IRobotics

Growth Rates Add When Quantities Multiply

Logarithmic differentiation

Calc IEconomics

Getting the Most Power from a Solar Panel

Critical points and absolute extrema

Calc IEE

Average-Speed Cameras

The Mean Value Theorem

Calc IPhysics

Light Takes the Fastest Path

Optimization

Calc IPhysics

Why 120 V AC Peaks at 170 V

Average function value

Calc IEE

Your Electricity Bill Is an Integral

The definite integral and Riemann sums

Calc IEE

The Odometer and the Speedometer

The Fundamental Theorem of Calculus

Calc IPhysics

Energy Released by a Discharging Capacitor

The substitution rule

Calc IEE

Volume of a Rocket Nozzle

Volumes by disks and washers

Calc IPhysics3D

Sculpting a Volcano with the Lathe Tool

Volumes by cylindrical shells

Calc IGame Dev3D

Taylor Series

Power series approximation

Calc IICS

Fourier Series

Infinite series of sines

Calc IIEE

Work Done by a Spring

Definite integrals as accumulation

Calc IIPhysics

Consumer Surplus

Area between curves

Calc IIEconomics

Calibrating a Fuel Gauge on a Horizontal Tank

Trig substitution

Calc IIPhysics

Filling a Tank That Leaks

Integrals involving roots

Calc IIPhysics

How Much Signal a Tuned Circuit Catches

Integrals involving quadratics

Calc IIEE

Why Fat Tails Break Averages

Comparison test for improper integrals

Calc IIEconomics

Rating a Model Rocket Motor

Approximating definite integrals

Calc IIPhysics

Heat Shield for a Nose Cone

Surface area of revolution

Calc IIPhysics3D

How Hard Water Pushes on a Dam

Hydrostatic pressure and force

Calc IIPhysics

The Path of a Pebble on a Rolling Wheel

Area and arc length with parametric equations

Calc IIPhysics

Microphone Pickup Patterns

Polar coordinates and tangents

Calc IIEE

How Focused Is an Antenna's Beam?

Area with polar coordinates

Calc IIEE

The 5-Kilometer Track on a CD

Arc length with polar coordinates

Calc IICS

A Ball That Bounces Forever (in Finite Distance)

Series and geometric series

Calc IIPhysics

Stack Blocks Past the Table's Edge

The divergence test and the harmonic series

Calc IIPhysics

Why a Square Wave Has Finite Power

The comparison test for series

Calc IIEE

Adding Up an Ionic Crystal's Energy

Absolute and conditional convergence

Calc IIPhysics

Retrying Packets on a Lossy Link

The ratio test

Calc IICS

Microphone Feedback: When the Echo Series Diverges

Power series as functions

Calc IIEE

Why ½mv² Is Only an Approximation

The binomial series

Calc IIPhysics

Potential Field Navigation

Gradients of scalar fields

Calc IIIRobotics3D

Best Choice on a Budget

Lagrange multipliers

Calc IIIEconomics

Electric Fields and Equipotentials

Level curves and gradients

Calc IIIPhysics

Hot Spots on a Circuit Board

Partial derivatives

Calc IIIEE3D

Which Way Is Uphill for a Rover?

Directional derivatives and the gradient

Calc IIIRobotics3D

Total Charge on a Plate

Double integrals

Calc IIIEE3D

Small-Signal Power in a Resistor

Tangent planes and linear approximation

Calc IIIEE3D

Saddle Points in a Neural Network's Loss

Relative extrema and the second-derivative test

Calc IIICS3D

Hottest and Coldest Points on a Plate

Absolute extrema on a closed region

Calc IIIPhysics3D

Strongest Signal on a Circular Track

Lagrange multipliers

Calc IIIRobotics3D

Wi-Fi Coverage Maps

Functions of several variables and level curves

Calc IIICS3D

Satellite Dishes and Saddles

Quadric surfaces

Calc IIIEE3D

Temperature Felt by a Drone

The multivariable chain rule

Calc IIIRobotics3D

Flying a Loop Through the Wind

Line integrals and conservative fields

Calc IIIRobotics3D

The Magnetic Field Around a Wire

Green's and Stokes' theorems

Calc IIIEE3D

Measuring Area by Walking the Edge

Green's theorem

Calc IIIPhysics3D

How Much Smoke Leaves a Region

Flux and the divergence theorem

Calc IIIPhysics3D

How Sharply a Car Must Steer

Curvature

Calc IIIRobotics

How a Camera Sees

Projection matrices and homogeneous coordinates

Lin AlgCS3D

Linear Regression

Least squares and projections

Lin AlgCS

Solving a Circuit with Two Equations

Systems of linear equations

Lin AlgEE

Steering a Hovercraft with Two Thrusters

Linear combinations, span and independence

Lin AlgRobotics

When a Transform Flattens Your Sprite

Determinants and invertibility

Lin AlgGame Dev

Pulling a Sled: The Part of the Force That Counts

Dot products and orthogonal projection

Lin AlgPhysics

Fixing a Skewed Motion Sensor

The Gram–Schmidt process

Lin AlgRobotics

Where Can a Robot Hand Move Fast?

The singular value decomposition

Lin AlgRobotics

Will the Owl Population Survive?

Eigenvalues and discrete dynamical systems

Lin AlgBiology

Why a Shadow Can't Tell You Depth

Null space, column space and rank

Lin AlgCS

Coordinates on a Crystal Lattice

Bases and coordinate systems

Lin AlgPhysics

Predator and Prey

Systems of ODEs and phase planes

Diff EqBiology3D

Epidemic Curves

Nonlinear systems of ODEs

Diff EqMedicine3D

The Real Pendulum

Nonlinear ODEs and phase portraits

Diff EqPhysics3D

RLC Circuits

Second-order linear ODEs

Diff EqEE

IV Drip Concentration

First-order linear ODEs

Diff EqMedicine

Logistic Population Growth

Separable ODEs

Diff EqBiology

Newton's Law of Cooling

Exponential decay ODEs

Diff EqPhysics

A Skydiver's Terminal Velocity

Differential equations and direction fields

Diff EqPhysics3D

Charging a Capacitor

Linear first-order equations

Diff EqEE3D

Draining a Tank

Separable equations

Diff EqPhysics3D

Air Flowing Around a Cylinder

Exact equations

Diff EqPhysics3D

How Economies Grow

Bernoulli equations

Diff EqEconomics3D

Thermal Runaway in a Battery

Intervals of validity

Diff EqEE3D

Paying Off a Loan

Modeling with first-order equations

Diff EqEconomics3D

How Much Fishing Is Too Much?

Equilibrium solutions

Diff EqBiology3D

Smooth Cameras and Euler's Method

Euler's method

Diff EqGame Dev3D

Why a Door Closer Never Slams

Second-order equations with real, distinct roots

Diff EqPhysics

Tuning a Robot Joint

Repeated roots and critical damping

Diff EqRobotics

An AC Circuit's Steady State

Undetermined coefficients

Diff EqEE

Beats and Resonance in a Footbridge

Mechanical vibrations

Diff EqPhysics

Transfer Functions for a Motor

The Laplace transform

Diff EqRobotics

Switching a Circuit On and Off

Step functions

Diff EqEE

Striking a Structure with a Hammer

The Dirac delta function

Diff EqPhysics

Reverb Is Convolution

Convolution integrals

Diff EqCS

A Pill's Path from Gut to Blood

Systems with real eigenvalues

Diff EqMedicine3D

An Oscillator's Phase Portrait

Systems with complex eigenvalues

Diff EqEE3D

Arms Races and Price Wars

Modeling with systems

Diff EqEconomics3D

Holding a Drone's Altitude

Systems with repeated eigenvalues

Diff EqRobotics3D

Stress in a Thick Pipe

Euler equations

Diff EqPhysics

The Bright Edge of a Rainbow

Series solutions

Diff EqPhysics

How Much a Beam Sags

Higher-order linear equations

Diff EqPhysics

When a Column Buckles

Boundary value problems and eigenvalues

Diff EqPhysics

Why Radio Stations Don't Interfere

Periodic and orthogonal functions

Diff EqEE

The Sound of a Plucked String

Fourier sine series

Diff EqPhysics

How JPEG Shrinks an Image

Fourier cosine series

Diff EqCS

Heat Diffusion

The heat equation

PDEPhysics3D

Pricing Options

Parabolic PDEs

PDEEconomics

Blur Is the Heat Equation

Diffusion and the heat kernel

PDECS

Why Traffic Jams Drift Backward

First-order PDEs and characteristics

PDEEconomics

Tracking a Chemical Spill Down a River

Advection–diffusion: transport plus the heat equation

PDEBiology

Heat Flowing from a Chip to Its Heatsink

Nonzero boundary conditions and the equilibrium temperature

PDEEE

A Hot Spot Spreading Around a Brake Disc

Periodic boundary conditions on a ring

PDEPhysics

Where a Piano Hammer Should Strike

The vibrating string: standing waves and harmonics

PDEPhysics

Finding a Cable Fault from Its Echo

D'Alembert's solution: waves traveling both ways, and reflections

PDEEE

Voltages Inside the Chest: Laplace on a Disk

Laplace's equation on a disk and the mean value property

PDEMedicine

Why Quantum Dots Glow Different Colors

Sturm–Liouville eigenvalue problems: eigenfunctions and orthogonality

PDEPhysics

Why Simulations Blow Up: The Stability Limit

Finite differences for the heat equation and their stability condition

PDECS

$$\mathbf{E} = \sum_i \frac{k\,q_i}{d_i^{2}}\,\hat{\mathbf{u}}_i, \qquad k = 8.99 \times 10^{9}\ \text{N·m}^2/\text{C}^2$$

Two Charges and the Field Between Them

Electric fields and superposition

E&MPhysics

$$C = \frac{\kappa\,\varepsilon_0 A}{d}, \qquad Q = CV, \qquad U = \tfrac12 CV^2, \qquad E = \frac{V}{d}$$

Inside a Capacitor: Area, Gap and Energy

Capacitance and stored energy

E&MEE

$$H(\omega) = \frac{1}{1 + j\omega RC}, \qquad |H| = \frac{1}{\sqrt{1 + (\omega RC)^2}}, \qquad \phi = -\arctan(\omega RC), \qquad f_c = \frac{1}{2\pi RC}$$

Smoothing a Signal with an RC Filter

Impedance and frequency response

E&MEE

$$\mathbf{E} = E_0\cos(kz - \omega t)\,\hat{\mathbf{x}}, \qquad \mathbf{B} = \frac{E_0}{c}\cos(kz - \omega t)\,\hat{\mathbf{y}}, \qquad c = \lambda f = \frac{1}{\sqrt{\mu_0\varepsilon_0}}$$

Light Is a Traveling Wave

Electromagnetic waves

E&MPhysics

The Bell Curve from Randomness

Binomial distribution and the Central Limit Theorem

StatsCS3D

Reading a Medical Test

Bayes' theorem

StatsMedicine

Logistic Regression

The sigmoid and probability

StatsCS

Why a Bigger Poll Can't Fix a Biased One

Populations, samples and sampling bias

StatsEconomics

Reading a Latency Histogram

Histograms, cumulative frequency and percentiles

StatsCS

Mean vs Median: One Big Salary

Measures of center and spread, and resistance to outliers

StatsEconomics

Catching Glitches in a Distance Sensor

Quartiles, box plots and the 1.5 × IQR rule

StatsRobotics

Estimating π with Random Darts

Probability as long-run relative frequency

StatsCS

How Many Backups Do You Need?

Independent events: the multiplication and complement rules

StatsEE

Simpson's Paradox in a Kidney Stone Study

Contingency tables, conditional rates and confounding

StatsMedicine

d20 or 2d10? The Shape of a Dice Roll

Discrete distributions, expected value and standard deviation

StatsGame Dev

Will Your Opening Hand Have the Combo?

The hypergeometric distribution: sampling without replacement

StatsGame Dev

How Many Tries to Send a Packet?

The geometric distribution

StatsEE

Counting Clicks on a Geiger Counter

The Poisson distribution

StatsPhysics

Waiting for the Bus: Schedule vs Random

Continuous distributions: uniform and exponential densities

StatsEconomics

How Many Resistors Meet Spec?

The normal distribution and z-scores

StatsEE

Reading a Newborn's Growth Chart

Percentiles and inverse normal calculations

StatsMedicine

What 95% Confidence Really Means

Confidence intervals for a mean, with z and t

StatsPhysics

How Sure Is Your Model's Test Accuracy?

Confidence intervals for a proportion

StatsCS

Five Sigma: How Physicists Claim a Discovery

Hypothesis tests, p-values, Type I and II errors, and power

StatsPhysics

Does Each Tablet Really Hold 500 mg?

The one-sample t-test and the t distribution

StatsMedicine

Is the New Model Really Better? A Paired Test

Matched pairs vs two independent samples

StatsCS

A/B Testing a Signup Button

Comparing two proportions with a z-test

StatsEconomics

Catching Fake Numbers with Benford's Law

The chi-square goodness-of-fit test

StatsEconomics

Is “Free” a Spam Word? A Test of Independence

The chi-square test of independence

StatsCS

Residuals Reveal a Stretched Spring

Residuals, r² and influential points

StatsPhysics

Three Fertilizers, One F-Test

One-way ANOVA: between-group vs within-group variation

StatsBiology

A Naive Bayes Spam Filter

Bayes' rule with independent evidence

StatsCS

Choosing a Decision Tree Split with Entropy

Entropy as expected surprise, and information gain

StatsCS

k-Nearest Neighbors and Why Units Matter

Standardizing with z-scores

StatsCS

$$R(t) = e^{-(t/\eta)^{\beta}}, \qquad h(t) = \frac{\beta}{\eta}\left(\frac{t}{\eta}\right)^{\beta - 1}, \qquad \text{MTTF} = \eta\,\Gamma\!\left(1 + \frac{1}{\beta}\right)$$

When Will a Bearing Fail? The Weibull Model

Weibull reliability and failure rate

Eng StatsRobotics

$$\text{UCL} = \mu_0 + \frac{3\sigma}{\sqrt{n}}, \qquad \text{LCL} = \mu_0 - \frac{3\sigma}{\sqrt{n}}, \qquad \bar x_i \sim N\!\left(\mu, \frac{\sigma^2}{n}\right)$$

Catching a Drifting Tablet Press

X̄ control charts

Eng StatsMedicine

$$C_p = \frac{\text{USL} - \text{LSL}}{6\sigma}, \qquad C_{pk} = \min\!\left(\frac{\text{USL} - \mu}{3\sigma},\ \frac{\mu - \text{LSL}}{3\sigma}\right)$$

Can the Line Hold the Spec? Cp and Cpk

Process capability

Eng StatsEE

$$P_a(p) = \sum_{k=0}^{c}\binom{n}{k}p^{k}(1-p)^{n-k}$$

Accept the Shipment? Operating Characteristic Curves

Acceptance sampling

Eng StatsEconomics

$$s_i = a_i \oplus b_i \oplus c_i, \qquad c_{i+1} = a_ib_i + c_i\,(a_i \oplus b_i)$$

Adding Numbers with Logic Gates

Boolean algebra: XOR, AND and the full adder

DiscreteEE

$$N = k^L, \qquad t = \frac{N}{2g}$$

How Long Would It Take to Guess Your Password?

The product rule

DiscreteCS

$$P(\text{no collision}) = \prod_{k=0}^{n-1}\left(1 - \frac{k}{N}\right) \approx e^{-n^2/(2N)}$$

The Birthday Problem and Hash Collisions

The product rule and the pigeonhole principle

DiscreteCS

$$D_n = n!\sum_{k=0}^{n}\frac{(-1)^k}{k!}, \qquad \frac{D_n}{n!} \to \frac{1}{e}$$

Will Someone Draw Their Own Name in Secret Santa?

Inclusion–exclusion and derangements

DiscreteGame Dev

$$H_n = 2H_{n-1} + 1,\;\; H_1 = 1 \quad\Longrightarrow\quad H_n = 2^n - 1$$

The Tower of Hanoi: Recursion and 2ⁿ − 1

Recurrence relations and induction

DiscreteGame Dev

$$\theta_k = k\alpha, \quad \rho_k = c\sqrt{k}, \qquad \alpha = \frac{360^\circ}{\varphi^2} \approx 137.5^\circ, \qquad \frac{F_{n+1}}{F_n} \to \varphi$$

Fibonacci Spirals in a Sunflower

The Fibonacci recurrence and the golden ratio

DiscreteBiology

$$c = m^e \bmod n, \qquad m = c^d \bmod n, \qquad d = e^{-1} \bmod (p-1)(q-1)$$

Sending a Secret with RSA

Modular exponentiation and inverses

DiscreteCS

$$d(v) = \min_{u}\bigl(d(u) + w(u, v)\bigr), \qquad d(S) = 0$$

Planning a Robot's Route Through a Warehouse

Shortest paths in weighted graphs

DiscreteRobotics

$$\text{one loop covers every street once} \iff \deg(v) \text{ is even at every intersection } v$$

Planning a Snowplow Route

Euler circuits and odd-degree vertices

DiscreteEconomics

$$c(u) \ne c(v) \text{ for every edge } \{u, v\}, \qquad \chi(G) = \text{fewest colors possible}$$

Assigning Radio Frequencies with Graph Coloring

Graph coloring and the chromatic number

DiscreteEE

$$T_{\text{simple}}(n) \approx \frac{n^2}{2}, \qquad T_{\text{merge}}(n) \approx n\log_2 n, \qquad \text{time} = \frac{T(n)}{v}$$

Why Fast Sorting Matters: n² vs n log n

Growth rates and big-O

DiscreteCS