Calculus Techniques: Differential Calculus

Calculus Techniques

Differential Calculus

• Limits: concept of approaching a value, used to define derivatives and integrals
• Derivatives: measure of how a function changes as its input changes
  • Geometric interpretation: tangent line to a curve
  • Physical interpretation: rate of change, velocity, or acceleration
• Rules of differentiation:
  1. Power rule: if f(x) = x^n, then f'(x) = nx^(n-1)
  2. Product rule: if f(x) = u(x)v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)
  3. Quotient rule: if f(x) = u(x)/v(x), then f'(x) = (u'(x)v(x) - u(x)v'(x)) / v(x)^2
  4. Chain rule: if f(x) = g(h(x)), then f'(x) = g'(h(x)) \* h'(x)
• Implicit differentiation: finding derivatives of implicitly defined functions

Integral Calculus

• Definite integrals: area between a curve and the x-axis, bounded by a and b
  • Fundamental Theorem of Calculus: relates definite integrals to antiderivatives
• Integration techniques:
  1. Substitution method: substitute u for x to simplify the integral
  2. Integration by parts: integrate one function while differentiating the other
  3. Integration by partial fractions: break down a rational function into simpler fractions
  4. Trigonometric substitution: substitute trigonometric functions to simplify the integral
• Improper integrals: extend definite integrals to infinite or half-infinite intervals

Multivariable Calculus

• Partial derivatives: derivatives of a function with respect to one variable, while keeping others constant
• Gradient vectors: vector of partial derivatives, used to find direction of maximum change
• Double and triple integrals: integrate functions of multiple variables
• Line and surface integrals: integrate functions over curves and surfaces

Vector Calculus

• Vector fields: functions that assign vectors to points in space
• Gradient, divergence, and curl: operations on vector fields
  • Gradient: directional derivative of a scalar function
  • Divergence: measure of how a vector field spreads out or converges
  • Curl: measure of how a vector field rotates around a point
• Stokes' theorem and Gauss' divergence theorem: relate line and surface integrals to vector fields

Calculus Techniques

Differential Calculus

• Limits are essential to define derivatives and integrals, dealing with values that a function approaches.
• A derivative measures the rate of change of a function as its input changes, with a geometric interpretation as the tangent line to a curve, and a physical interpretation as rate of change, velocity, or acceleration.
• Rules of differentiation:
  • 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).
• Implicit differentiation is used to find derivatives of implicitly defined functions.

Integral Calculus

• A definite integral represents the area between a curve and the x-axis, bounded by a and b, and is related to antiderivatives by the Fundamental Theorem of Calculus.
• Integration techniques:
  • Substitution method: substitute u for x to simplify the integral.
  • Integration by parts: integrate one function while differentiating the other.
  • Integration by partial fractions: break down a rational function into simpler fractions.
  • Trigonometric substitution: substitute trigonometric functions to simplify the integral.
• Improper integrals extend definite integrals to infinite or half-infinite intervals.

Multivariable Calculus

• Partial derivatives are derivatives of a function with respect to one variable, while keeping others constant.
• Gradient vectors, composed of partial derivatives, are used to find the direction of maximum change.
• Double and triple integrals integrate functions of multiple variables.
• Line and surface integrals integrate functions over curves and surfaces.

Vector Calculus

• Vector fields are functions assigning vectors to points in space.
• Gradient, divergence, and curl are operations on vector fields:
  • Gradient: directional derivative of a scalar function.
  • Divergence: measure of how a vector field spreads out or converges.
  • Curl: measure of how a vector field rotates around a point.
• Stokes' theorem and Gauss' divergence theorem relate line and surface integrals to vector fields.