Calculus and Analytic Geometry Formulas and Theorems

Bolzano's Theorem

If f(x) is continuous on [a, b], there exists at least one value c in (a, b) such that f(c) = 0.

Consequences

If f(a) > g(a) and f(b) < g(b), there exists c in (a, b) such that f(c) = g(c).

Mean Value Theorem

If f(x) is continuously differentiable on [a, b], then there exists a point where the derivative f'(c) equals the average rate of change.

Linear Combinations and Geometry

A linear combination is defined as: a(x₁, y₁, z₁) + b(x₂, y₂, z₂) + c(x₃, y₃, z₃) = (x₄, y₄, z₄).

Distance Formulas

• Between two points: The magnitude of the vector.
• Point to line: dist(P, r) = |AP × Vd| / |Vd|, where A is a point on the line and Vd is the direction vector.
• Point to plane: |ax + by + cz + d| / √(a² + b² + c²).

Relative Positions

• Two lines: Check if parallel or coincident. If not parallel, use the determinant of the matrix formed by the direction vectors and the vector connecting points on the lines to check for intersection or skew lines.
• Two planes: Analyze the rank of the coefficient matrix and the augmented matrix to determine if they intersect in a line, are parallel, or coincide.
• Line and plane: Use the cross product of the direction vector and the normal vector to determine intersection or parallelism.
• Common perpendicular: Calculate the vector perpendicular to both lines using the cross product of their direction vectors, then solve the system of equations to find the common perpendicular line.

Derivative Rules

• Constant Multiple: D[k · f(x)] = k · f'(x)
• Product Rule: D[f(x) · g(x)] = f'(x)g(x) + f(x)g'(x)
• Quotient Rule: D[f(x)/g(x)] = (f'(x)g(x) - f(x)g'(x)) / [g(x)]²
• Power Rule: D(xᵏ) = k · xᵏ⁻¹; D[f(x)ᵏ] = k · f(x)ᵏ⁻¹ · f'(x)
• Trigonometric Functions:
  • D(sin x) = cos x; D(sin f(x)) = cos(f(x)) · f'(x)
  • D(cos x) = -sin x; D(cos f(x)) = -sin(f(x)) · f'(x)
  • D(tan x) = 1 + tan²x; D(tan f(x)) = (1 + tan²f(x)) · f'(x)
• Exponential Functions:
  • D(eˣ) = eˣ; D(eᶠ⁽ˣ⁾) = eᶠ⁽ˣ⁾ · f'(x)
  • D(aˣ) = aˣ · ln(a); D(aᶠ⁽ˣ⁾) = aᶠ⁽ˣ⁾ · ln(a) · f'(x)
• Logarithmic Functions:
  • D(ln x) = 1/x; D(ln f(x)) = (1/f(x)) · f'(x)
  • D(logₐx) = (1/x) · (1/ln a)
  • D(logₐf(x)) = (1/f(x)) · (1/ln a) · f'(x)