Limits in Calculus

Limits

• Definition: A limit represents the value that a function approaches as the input (or x-value) gets arbitrarily close to a certain point.
• Notation: The limit of a function f(x) as x approaches a is denoted by lim x→a f(x) = L.
• Properties:
  • Linearity: The limit of a sum is the sum of the limits.
  • Homogeneity: The limit of a product is the product of the limits.
  • Chain Rule: The limit of a composite function is the composite of the limits.
• Types of Limits:
  • One-sided limits: The limit of a function as x approaches a from the left or right.
  • Two-sided limits: The limit of a function as x approaches a from both sides.
  • Infinite limits: The limit of a function as x approaches infinity or negative infinity.

Derivatives

• Definition: The derivative of a function f(x) represents the rate of change of the function with respect to x.
• Notation: The derivative of a function f(x) is denoted by 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.
  • Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
• Geometric Interpretation: The derivative of a function at a point represents the slope of the tangent line to the function at that point.
• Applications:
  • Finding the maximum and minimum values of a function.
  • Determining the rate of change of a quantity.

Integrals

• Definition: The definite integral of a function f(x) from a to b represents the area under the curve of the function between a and b.
• Notation: The definite integral of a function f(x) from a to b is denoted by ∫[a,b] f(x) dx.
• Types of Integrals:
  • Definite integrals: The integral of a function over a specific interval.
  • Indefinite integrals: The antiderivative of a function, denoted by ∫f(x) dx.
• Rules:
  • Substitution Method: Substitute u = f(x) to simplify the integral.
  • Integration by Parts: ∫udv = uv - ∫vdu.
  • Integration by Partial Fractions: Break down a rational function into simpler fractions.
• Applications:
  • Finding the area under curves.
  • Calculating volumes of solids.
  • Solving problems involving accumulation of quantities.

Limits

• A limit represents the value that a function approaches as the input gets arbitrarily close to a certain point.
• The limit of a function f(x) as x approaches a is denoted by lim x→a f(x) = L.
• The limit of a sum is the sum of the limits, demonstrating the linearity property.
• The limit of a product is the product of the limits, demonstrating the homogeneity property.
• The limit of a composite function is the composite of the limits, following the chain rule.
• One-sided limits involve the limit of a function as x approaches a from the left or right.
• Two-sided limits involve the limit of a function as x approaches a from both sides.
• Infinite limits involve the limit of a function as x approaches infinity or negative infinity.

Derivatives

• The derivative of a function f(x) represents the rate of change of the function with respect to x.
• The derivative of a function f(x) is denoted by f'(x) or (d/dx)f(x).
• The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).
• The product rule states that if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x).
• The quotient rule states that if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2.
• The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
• The derivative of a function at a point represents the slope of the tangent line to the function at that point.
• Derivatives have applications in finding the maximum and minimum values of a function and determining the rate of change of a quantity.

Integrals

• The definite integral of a function f(x) from a to b represents the area under the curve of the function between a and b.
• The definite integral of a function f(x) from a to b is denoted by ∫[a,b] f(x) dx.
• Definite integrals involve the integral of a function over a specific interval.
• Indefinite integrals involve the antiderivative of a function, denoted by ∫f(x) dx.
• The substitution method involves substituting u = f(x) to simplify the integral.
• Integration by parts involves the formula ∫udv = uv - ∫vdu.
• Integration by partial fractions involves breaking down a rational function into simpler fractions.
• Integrals have applications in finding the area under curves, calculating volumes of solids, and solving problems involving accumulation of quantities.