Calculus Basics

Calculus

• Branches of Calculus:
  • Differential Calculus: studies rates of change and slopes of curves
  • Integral Calculus: studies accumulation of quantities

Key Concepts

• Limits: the behavior of a function as the input (or x-value) approaches a specific point
• Derivatives: measure the rate of change of a function with respect to its input
• Integrals: measure the area between a curve and the x-axis

Derivatives

• 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)
  • Chain Rule: if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x)
• Applications of Derivatives:
  • Finding maximum and minimum values of a function
  • Determining the rate at which a quantity changes over time

Integrals

• Definite Integral: the area between a curve and the x-axis over a specific interval
• Indefinite Integral: the antiderivative of a function, represented as ∫f(x)dx
• Methods of Integration:
  • Substitution Method: substituting u = f(x) or x = f(u) to simplify the integral
  • Integration by Parts: ∫udv = uv - ∫vdu
  • Integration by Partial Fractions: breaking down a rational function into simpler fractions

Applications of Calculus

• Physics and Engineering: modeling real-world phenomena, such as motion, force, and energy
• Economics: modeling economic systems, including supply and demand curves
• Computer Science: algorithms for optimization, machine learning, and data analysis

Three Dimensional Geometry

• Direction Cosines and Ratios:
  • l, m, n are the direction cosines of a line
  • a, b, c are the direction ratios of a line
• Angle between Skew Lines:
  • cos θ = l1l2 + m1m2 + n1n2
  • cos θ = (a1a2 + b1b2 + c1c2) / (√(a1² + b1² + c1²) * √(a2² + b2² + c2²))

Vector Equation of a Line

• Line through a Point:
  • r = a + λb, where a is the position vector and b is the direction vector
  • r = a + (b - a), where a and b are the position vectors of two points
• Cartesian Equation of a Line:
  • (x - x1) / l = (y - y1) / m = (z - z1) / n

Shortest Distance between Two Skew Lines

• Shortest Distance:
  • (b1 x b2) / |b1 x b2|
  • √((bc - b2c)² + (ca - c1a)² + (ab2 - ab1)²)

Distance between Parallel Lines

• Distance:
  • |b x (a1 - a2)| / |b|

Equation of a Plane

• Vector Form:
  • r . n = d, where n is the unit vector normal to the plane and d is the distance from the origin
• Cartesian Form:
  • lx + my + nz = d, where l, m, n are the direction cosines of the normal to the plane
• Perpendicular to a Line:
  • (r - a) . N = 0, where a is the position vector and N is the normal vector
• Through a Point and Perpendicular to a Line:
  • A(x - x1) + B(y - y1) + C(z - z1) = 0