Calculus: Understanding Limits

• Calculus is a branch of mathematics focused on continuous change, covering topics like limits, derivatives, integrals, and infinite series
• Provides tools and techniques to model and analyze dynamic systems in science, engineering, economics, and more

Limits

• Describes the value that a function approaches as the input approaches a certain value
• Foundational to calculus, defining continuity, derivatives, and integrals
• Notation: lim (x→c) f(x) = L, meaning as x approaches c, the function f(x) approaches L
• Precise definition (ε-δ definition): For every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε
• Limit Laws: Rules for computing limits of combinations of functions (sum, product, quotient, etc.)
• Squeeze Theorem: If g(x) ≤ f(x) ≤ h(x) near c and lim (x→c) g(x) = lim (x→c) h(x) = L, then lim (x→c) f(x) = L
• One-sided Limits: Limits as x approaches c from the left (x→c⁻) or right (x→c⁺)
• A limit exists only if both one-sided limits exist and are equal
• Indeterminate Forms: Expressions like 0/0, ∞/∞, 0⋅∞, ∞-∞, 1^∞, 0⁰, ∞⁰ that require further analysis to evaluate the limit

Continuity

• A function f(x) is continuous at x = c if lim (x→c) f(x) = f(c)
• For continuity, the limit must exist, the function must be defined at the point, and the limit must equal the function value
• Types of Discontinuities: Removable (a hole), Jump (a sudden jump in value), Infinite (approaches infinity), Oscillating
• Intermediate Value Theorem: If f is continuous on [a, b] and N is between f(a) and f(b), then there exists a c in (a, b) such that f(c) = N

Derivatives

• The derivative measures the instantaneous rate of change of a function
• Represents the slope of the tangent line to the function's graph at a point
• Definition: f'(x) = lim (h→0) [f(x+h) - f(x)] / h
• Differentiation Rules:
  • Power Rule: d/dx (xⁿ) = nx^(n-1)
  • Constant Multiple Rule: d/dx [cf(x)] = cf'(x)
  • Sum/Difference Rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x)
  • Product Rule: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
  • Quotient Rule: d/dx [f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]²
  • Chain Rule: d/dx [f(g(x))] = f'(g(x)) ⋅ g'(x)
• Derivatives of Trigonometric Functions:
  • d/dx (sin x) = cos x
  • d/dx (cos x) = -sin x
  • d/dx (tan x) = sec² x
• Implicit Differentiation: Used when y is not explicitly defined as a function of x; differentiate both sides of the equation with respect to x and solve for dy/dx
• Higher-Order Derivatives: Derivatives of derivatives (e.g., second derivative f''(x), third derivative f'''(x))
• Applications of Derivatives:
  • Finding critical points (where f'(x) = 0 or is undefined)
  • Determining intervals of increasing/decreasing behavior
  • Finding local maxima and minima
  • Determining concavity (using the second derivative)
  • Finding inflection points (where concavity changes)
  • Optimization problems (finding maximum or minimum values subject to constraints)
  • Related Rates (finding rates of change of related quantities)
  • Linear Approximation: Approximating function values using the tangent line: f(x) ≈ f(a) + f'(a)(x-a)
• L'Hôpital's Rule: If lim (x→c) f(x)/g(x) is of the form 0/0 or ∞/∞, then lim (x→c) f(x)/g(x) = lim (x→c) f'(x)/g'(x) (provided the limit on the right exists)

Integrals

• Represents the area under a curve
• Reverse process of differentiation (antidifferentiation)
• Indefinite Integral: ∫ f(x) dx = F(x) + C, where F'(x) = f(x) and C is the constant of integration
• Basic Integration Rules:
  • Power Rule: ∫ xⁿ dx = (x^(n+1)) / (n+1) + C (for n ≠ -1)
  • ∫ (1/x) dx = ln |x| + C
  • ∫ eˣ dx = eˣ + C
  • ∫ sin x dx = -cos x + C
  • ∫ cos x dx = sin x + C
• Substitution Rule (u-substitution): ∫ f(g(x))g'(x) dx = ∫ f(u) du, where u = g(x)
• Integration by Parts: ∫ u dv = uv - ∫ v du (used for integrating products of functions)
• Trigonometric Substitution: Using trigonometric identities to simplify integrals involving square roots of quadratic expressions
• Partial Fractions: Decomposing rational functions into simpler fractions to facilitate integration
• Definite Integral: ∫ₐᵇ f(x) dx represents the area under the curve of f(x) from x = a to x = b
• Fundamental Theorem of Calculus (FTC):
  • Part 1: If F(x) = ∫ₐˣ f(t) dt, then F'(x) = f(x)
  • Part 2: ∫ₐᵇ f(x) dx = F(b) - F(a), where F'(x) = f(x)
• Applications of Integrals:
  • Finding areas between curves
  • Calculating volumes of solids of revolution (disk method, washer method, shell method)
  • Determining average value of a function over an interval
  • Calculating arc length of a curve
  • Finding work done by a force
  • Calculating center of mass and moments of inertia
• Improper Integrals: Integrals where either the interval of integration is infinite or the function has a discontinuity within the interval
• Techniques for evaluating improper integrals involve limits

Infinite Series

• An infinite series is the sum of an infinite sequence of terms
• Notation: Σₙ₌₁^∞ aₙ = a₁ + a₂ + a₃ + ...
• Partial Sums: Sₙ = a₁ + a₂ + ... + aₙ is the sum of the first n terms of the series
• Convergence/Divergence:
  • A series converges if the sequence of partial sums approaches a finite limit
  • A series diverges if the sequence of partial sums does not approach a finite limit
• Tests for Convergence/Divergence:
  • Divergence Test (n-th Term Test): If lim (n→∞) aₙ ≠ 0, then the series diverges
  • Integral Test: If f(x) is positive, continuous, and decreasing for x ≥ 1, then Σ aₙ and ∫₁^∞ f(x) dx either both converge or both diverge
  • Comparison Test: If 0 ≤ aₙ ≤ bₙ for all n, and Σ bₙ converges, then Σ aₙ converges. If aₙ ≥ bₙ ≥ 0 for all n, and Σ bₙ diverges, then Σ aₙ diverges
  • Limit Comparison Test: If lim (n→∞) (aₙ/bₙ) = c, where 0 < c < ∞, then Σ aₙ and Σ bₙ either both converge or both diverge
  • Ratio Test: If lim (n→∞) |aₙ₊₁ / aₙ| = L, then:
    • If L < 1, the series converges absolutely
    • If L > 1, the series diverges
    • If L = 1, the test is inconclusive
  • Root Test: If lim (n→∞) |aₙ|^(1/n) = L, then:
    • If L < 1, the series converges absolutely
    • If L > 1, the series diverges
    • If L = 1, the test is inconclusive
  • Alternating Series Test: If Σ (-1)ⁿ aₙ satisfies:
    • aₙ > 0 for all n
    • aₙ is decreasing
    • lim (n→∞) aₙ = 0
    • Then the series converges
• Absolute vs. Conditional Convergence:
  • A series Σ aₙ converges absolutely if Σ |aₙ| converges
  • A series Σ aₙ converges conditionally if Σ aₙ converges, but Σ |aₙ| diverges
• Power Series: A series of the form Σ cₙ(x - a)ⁿ, where cₙ are coefficients and a is the center
• Radius and Interval of Convergence:
  • The radius of convergence R is a non-negative real number or ∞ such that the series converges if |x - a| < R and diverges if |x - a| > R
  • The interval of convergence is the set of all x for which the series converges
• Taylor and Maclaurin Series:
  • Taylor Series: Representation of a function as an infinite sum of terms involving its derivatives at a single point: f(x) = Σ [fⁿ(a) / n!] (x - a)ⁿ
  • Maclaurin Series: Taylor series centered at a = 0: f(x) = Σ [fⁿ(0) / n!] xⁿ
• Common Taylor/Maclaurin Series:
  • eˣ = Σ (xⁿ / n!)
  • sin x = Σ [(-1)ⁿ x^(2n+1) / (2n+1)!]
  • cos x = Σ [(-1)ⁿ x^(2n) / (2n)!]
  • 1 / (1 - x) = Σ xⁿ (geometric series)
• Applications of Series:
  • Approximating function values
  • Solving differential equations
  • Representing functions that do not have elementary closed-form expressions