Calculus Fundamentals

Calculus

Limits

• A limit represents the behavior of a function as the input (x) approaches a specific value
• Notation: lim x→a f(x) = L, read as "the limit as x approaches a of f(x) is L"
• Properties:
  • Linearity: lim x→a [af(x) + bg(x)] = a lim x→a f(x) + b lim x→a g(x)
  • Homogeneity: lim x→a [f(x)g(x)] = [lim x→a f(x)] [lim x→a g(x)]
  • Sum and difference: lim x→a [f(x) ± g(x)] = lim x→a f(x) ± lim x→a g(x)

Derivatives

• A derivative measures the rate of change of a function with respect to its input
• Notation: f'(x) or (d/dx)[f(x)], read as "the derivative of f with respect to x"
• Rules:
  • Power rule: if f(x) = x^n, then f'(x) = nx^(n-1)
  • Product rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
  • Quotient rule: if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
  • Chain rule: if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)

Differentiation Applications

• Finding maximum and minimum values of a function
• Determining the rate at which a quantity changes over time
• Analyzing optimization problems

Integrals

• A definite integral represents the area under a curve between two limits
• Notation: ∫[a, b] f(x) dx, read as "the integral from a to b of f(x) with respect to x"
• Rules:
  • Constant multiple rule: ∫[a, b] k * f(x) dx = k * ∫[a, b] f(x) dx
  • Sum rule: ∫[a, b] [f(x) + g(x)] dx = ∫[a, b] f(x) dx + ∫[a, b] g(x) dx
  • Substitution method: ∫[a, b] f(g(x)) * g'(x) dx = ∫[a, b] f(u) du (u = g(x))

Integration Applications

• Finding the area between curves
• Evaluating volumes of solids
• Solving problems involving work and energy