Mathematical Modeling Flashcards

Grade 12 · Math · Flashcards

Core vocabulary, model types, and formulas for building and interpreting mathematical models of real-world phenomena.

Flashcards

The Modeling Cycle: Identify → Assume → Build → Solve → Interpret → Validate → Refine
Linear Model: y = mx + b; constant rate of change; appropriate when data show a steady trend
Exponential Growth Model: y = y₀e^(kt) or y = y₀b^t (b>1); grows at a constant percentage rate; population, compound interest
Exponential Decay Model: y = y₀e^(-kt); models radioactive decay, drug concentration, cooling
Doubling Time: t = ln(2)/k for continuous growth; time for a quantity to double
Half-Life: t₁/₂ = ln(2)/k; time for a decaying quantity to reduce to half its value
Logistic Growth: P(t)=K/(1+Ae^(-kt)); S-curve; approaches carrying capacity K asymptotically
Carrying Capacity K: Maximum sustainable population in logistic model; P → K as t → ∞
Inflection Point of Logistic: Occurs at P = K/2; maximum growth rate in logistic model
Separable ODE Solution: dP/dt = kP → P = P₀e^(kt); separate variables, integrate both sides
R² (Goodness of Fit): Proportion of variance explained by the model; R²=1 means perfect fit, R²=0 means model explains nothing
Residual: Residual = observed − predicted = y − ŷ; used to assess model fit and detect patterns
Newton's Law of Cooling: T(t) = Tₐ + (T₀ − Tₐ)e^(−kt); temperature approaches ambient temperature Tₐ exponentially