Calculus and Number Theory Fundamentals

Calculus

Limits

• A limit represents the behavior of a function as the input (or x-value) approaches a specific point
• 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 [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
  • Homogeneity: lim x→a [c * f(x)] = c * lim x→a f(x)

Derivatives

• A derivative represents the rate of change of a function with respect to its input
• Notation: f'(x) or (d/dx)f(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

Integrals

• A definite integral represents the area under a curve between two points
• Notation: ∫[a, b] f(x) dx
• Rules:
  • Fundamental Theorem of Calculus: ∫[a, b] f(x) dx = F(b) - F(a), where F(x) is the antiderivative of f(x)
  • Substitution method: ∫f(u) du = ∫f(g(x)) * g'(x) dx

Number Theory

Divisibility

• A dividend is divisible by a divisor if the remainder is 0
• Notation: a ≡ b (mod n) means "a is congruent to b modulo n"
• Properties:
  • Reflexive: a ≡ a (mod n)
  • Symmetric: if a ≡ b (mod n), then b ≡ a (mod n)
  • Transitive: if a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n)

Prime Numbers

• A prime number is a positive integer greater than 1 that is divisible only by 1 and itself
• Properties:
  • Prime factorization: every positive integer can be expressed as a product of prime numbers in a unique way
  • Euclid's lemma: if a prime number p divides a product ab, then p divides a or p divides b

Congruences

• A congruence is an equation of the form a ≡ b (mod n)
• Solving congruences:
  • Linear congruences: ax ≡ b (mod n) can be solved using the extended Euclidean algorithm
  • Systems of congruences: can be solved using the Chinese remainder theorem

मराठी मध्ये कॅल्कुलस आणि संख्या सिद्धान्ताच्या मूलभूत संकल्पना या क्विझमध्ये शिकल्या जात आहेत. कॅल्कुलसचे सीमा, व्युत्पन्न, आणि संपूर्ण संकल्पना आहेत, तर संख्या सिद्धान्ताचे विभाज्यता, मूळ संख्या, आणि समतुल्यता या संकल्पना आहेत.