Applications of Calculus Flashcards

Grade 12 · Math · Flashcards

Key theorems, formulas, and procedures for applying calculus to real-world and theoretical problems.

Flashcards

Optimization: Closed Interval Method: Evaluate f at critical points and both endpoints; the largest value is the absolute max, smallest is the absolute min
Second Derivative Test: If f'(c)=0: f''(c) > 0 → local min; f''(c) < 0 → local max; f''(c)=0 → inconclusive
Related Rates Setup: Write an equation relating variables, differentiate implicitly with respect to t, then substitute known values
Velocity from Position: v(t) = s'(t); positive velocity = moving right/up; negative = moving left/down
Acceleration from Velocity: a(t) = v'(t) = s''(t)
Displacement Formula: Δs = ∫ₐᵇ v(t) dt = s(b) - s(a)
Total Distance Traveled: ∫ₐᵇ |v(t)| dt; split integral at zeros of v(t)
Mean Value Theorem: ∃ c in (a,b) where f'(c) = [f(b)-f(a)]/(b-a), provided f is continuous on [a,b] and differentiable on (a,b)
L'Hôpital's Rule: If lim produces 0/0 or ∞/∞ indeterminate form: lim[f/g] = lim[f'/g']
Disk Method (x-axis rotation): V = π ∫ₐᵇ [f(x)]² dx
Washer Method: V = π ∫ₐᵇ ([R(x)]² - [r(x)]²) dx where R is outer radius and r is inner radius
Newton's Method: Iterative root-finding: xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)