Calculus BC - Formula Sheet Flashcards

Derivatives and Continuity

• The formal definition of the derivative is a limit that captures the instantaneous rate of change of a function.
• An alternate version of the derivative is linked to the secant line's slope approaching the tangent line's slope.
• A function f is continuous at point c if:
  • f(c) is defined,
  • the limit of f(x) as x approaches c exists, and
  • the limit equals f(c).

Theorems and Techniques

• The average rate of change of f(x) on interval [a, b] is calculated by (f(b) - f(a))/(b - a).
• The Intermediate Value Theorem states if f is continuous on [a, b] and k is in the interval [f(a), f(b)], there exists at least one c such that f(c) = k.
• Rolle's Theorem confirms there is at least one c in (a, b) where f'(c) = 0, given f is continuous on [a, b] and differentiable on (a, b) with f(a) = f(b).
• The Mean Value Theorem asserts that there exists at least one c in (a, b) where f’(c) equals the average rate of change over that interval.

Differentiation Rules

• Product Rule: d/dx [uv] = u(dv/dx) + v(du/dx).
• Quotient Rule: d/dx [u/v] = (v(du/dx) - u(dv/dx)) / v².
• Chain Rule: d/dx [f(g(x))] = f'(g(x)) * g'(x).
• Derivatives of trig functions:
  • d/dx [sin(u)] = cos(u) * (du/dx)
  • d/dx [cos(u)] = -sin(u) * (du/dx)
  • d/dx [tan(u)] = sec²(u) * (du/dx)
  • d/dx [sec(u)] = sec(u)tan(u) * (du/dx)
  • d/dx [cot(u)] = -csc²(u) * (du/dx)
  • d/dx [csc(u)] = -csc(u)cot(u) * (du/dx).

Critical Points and Extrema

• A critical number c occurs if f is defined at c and either f'(c) = 0 or f' is undefined.
• The First Derivative Test classifies critical points based on changes in sign:
  • From negative to positive indicates a relative minimum.
  • From positive to negative indicates a relative maximum.
• The Second Derivative Test confirms:
  • f''(c) > 0 implies a relative minimum.
  • f''(c) < 0 implies a relative maximum.
  • f''(c) = 0 is inconclusive.

Integrations and Theorems

• The Grund's theorem states that the integral of f'(x) over [a, b] equals F(b) - F(a).
• The second fundamental theorem of calculus states that d/dx ∫(from a to x) f(t)dt = f(x).
• The average value of a function f(x) on interval [a,b] is given by (1/(b-a))∫(from a to b) f(x)dx.

Integral Results

• Fundamental integrals:
  • ∫cos(u) du = sin(u) + C
  • ∫sin(u) du = -cos(u) + C
  • ∫sec²(u) du = tan(u) + C
  • ∫csc²(u) du = -cot(u) + C
  • ∫e^u du = e^u + C.

Series and Convergence Tests

• The nth term test for divergence states if lim (n→∞) a(n) ≠ 0, then the series diverges.
• A geometric series converges if |r| < 1, diverges if |r| ≥ 1.
• The ratio test helps determine convergence: lim (n→∞) |a(n+1)/a(n)| < 1 indicates convergence and > 1 indicates divergence.

Taylor Series

• Taylor series expansion for e^x: ∑ (x^n/n!) for all x.
• Taylor series for sin(x): ∑ (-1)^(n)(x^(2n+1)/(2n+1)!) for all x.
• Taylor series for cos(x): ∑ (-1)^(n)(x^(2n)/(2n)!) for all x.
• The series for ln(1+x) involves alternating terms based on x^(n)/n with the radius of convergence for -1 < x ≤ 1.