Advanced Higher Maths Formula Sheet

Advanced Higher Maths Formula Sheet: A Comprehensive Guide

This article serves as a comprehensive guide to the key formulas encountered in Advanced Higher Mathematics. It's designed to be a valuable resource for students preparing for exams, providing a structured overview of essential concepts and their corresponding formulas. We'll cover a range of topics, explaining the formulas in a clear and accessible manner, making this the ultimate advanced higher maths formula sheet companion. This guide aims to not only provide a list of formulas but also enhance your understanding of their application and context within the broader mathematical framework.

I. Algebra and Functions

This section covers fundamental algebraic concepts and their applications to functions. Mastering these is crucial for success in advanced higher mathematics.

1. Polynomials

• Factor Theorem: If P(x) is a polynomial and P(a) = 0, then (x - a) is a factor of P(x).
• Remainder Theorem: If P(x) is divided by (x - a), the remainder is P(a).
• Roots of a Quadratic Equation: For the quadratic equation ax² + bx + c = 0, the roots are given by the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. The discriminant, b² - 4ac, determines the nature of the roots (real and distinct, real and equal, or complex).
• Partial Fractions: This technique is used to decompose rational functions into simpler fractions for easier integration and other manipulations. The specific method depends on the nature of the denominator (linear factors, repeated linear factors, quadratic factors).

2. Functions

• Composite Function: (f ∘ g)(x) = f(g(x)) – Applying function g then function f.
• Inverse Function: f⁻¹(x) is the inverse of f(x) if f(f⁻¹(x)) = f⁻¹(f(x)) = x. The graph of f⁻¹(x) is the reflection of f(x) in the line y = x.
• Even and Odd Functions: A function is even if f(-x) = f(x) (symmetric about the y-axis) and odd if f(-x) = -f(x) (symmetric about the origin).

3. Sequences and Series

• Arithmetic Sequence: aₙ = a₁ + (n - 1)d, where aₙ is the nth term, a₁ is the first term, and d is the common difference. The sum of an arithmetic series is Sₙ = n/2(a₁ + aₙ) or Sₙ = n/2(2a₁ + (n - 1)d).
• Geometric Sequence: aₙ = a₁rⁿ⁻¹, where aₙ is the nth term, a₁ is the first term, and r is the common ratio. The sum of a geometric series is Sₙ = a₁(1 - rⁿ)/(1 - r), where r ≠ 1. The sum to infinity of a convergent geometric series (|r| < 1) is S∞ = a₁/(1 - r).
• Binomial Theorem: (a + b)ⁿ = Σ (ⁿCk) aⁿ⁻ᵏ bᵏ, where k ranges from 0 to n, and ⁿCk is the binomial coefficient, calculated as n!/(k!(n - k)!).

II. Calculus

Calculus forms a significant part of Advanced Higher Mathematics. Understanding differentiation and integration thoroughly is paramount.

1. Differentiation

• Derivative of a Function: f'(x) = lim (h→0) [(f(x + h) - f(x))/h], representing the instantaneous rate of change of f(x).
• Rules of Differentiation:
  • Power Rule: d/dx(xⁿ) = nxⁿ⁻¹
  • Sum/Difference Rule: d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
  • Product Rule: d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)
  • Quotient Rule: d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²
  • Chain Rule: d/dx[f(g(x))] = f'(g(x))g'(x)
• Implicit Differentiation: Used to differentiate implicitly defined functions.
• Higher-Order Derivatives: Second derivatives (f''(x)), third derivatives (f'''(x)), etc., represent rates of change of the previous derivative.

2. Integration

• Indefinite Integral: ∫f(x)dx represents the family of functions whose derivative is f(x). Always includes a constant of integration (+C).
• Rules of Integration: Many integration rules mirror differentiation rules, but in reverse. Several standard integrals should be memorized, such as:
  • ∫xⁿdx = (xⁿ⁺¹)/(n+1) + C (n ≠ -1)
  • ∫eˣdx = eˣ + C
  • ∫1/x dx = ln|x| + C
  • ∫sin(x)dx = -cos(x) + C
  • ∫cos(x)dx = sin(x) + C
• Definite Integral: ∫ᵇₐf(x)dx represents the area under the curve of f(x) between x = a and x = b. The Fundamental Theorem of Calculus links differentiation and integration.
• Integration by Substitution: A technique used to simplify integrals by substituting a new variable.
• Integration by Parts: A technique used to integrate products of functions: ∫u dv = uv - ∫v du. Choosing appropriate 'u' and 'dv' is crucial.

3. Applications of Calculus

• Optimization: Finding maximum and minimum values of functions using derivatives. Involves finding critical points (where f'(x) = 0 or is undefined) and using the second derivative test to determine the nature of these points.
• Kinematics: Applying calculus to problems involving motion, relating displacement, velocity, and acceleration.
• Areas and Volumes of Revolution: Calculating areas and volumes using integration.

III. Vectors

Vectors are fundamental to many advanced mathematical concepts.

1. Vector Operations

• Vector Addition: Adding vectors using the triangle law or parallelogram law.
• Scalar Multiplication: Multiplying a vector by a scalar changes its magnitude but not direction.
• Dot Product: * a ⋅ b = |a||b|cosθ*, where θ is the angle between the vectors. The dot product results in a scalar.
• Cross Product: a x b = |a||b|sinθn, where n is a unit vector perpendicular to both a and b. The cross product results in a vector.

2. Vector Equations of Lines and Planes

• Equation of a Line: r = a + λb, where r is the position vector of a point on the line, a is a position vector of a point on the line, b is the direction vector of the line, and λ is a scalar parameter.
• Equation of a Plane: r ⋅ n = a ⋅ n, where r is the position vector of a point on the plane, n is the normal vector to the plane, and a is a position vector of a point on the plane.

3. Applications of Vectors

• Geometry: Solving problems involving lines, planes, angles, and distances.
• Physics: Representing forces, velocities, and accelerations.

IV. Matrices

Matrices are powerful tools used in various mathematical applications.

1. Matrix Operations

• Matrix Addition and Subtraction: Adding or subtracting matrices of the same dimensions.
• Scalar Multiplication: Multiplying a matrix by a scalar.
• Matrix Multiplication: Multiplying matrices (the number of columns in the first matrix must equal the number of rows in the second matrix).
• Determinant of a Matrix: A scalar value associated with a square matrix. Methods for calculating determinants vary depending on the matrix size.
• Inverse of a Matrix: A matrix that, when multiplied by the original matrix, yields the identity matrix. Only square matrices with non-zero determinants have inverses.

2. Applications of Matrices

• Solving Systems of Linear Equations: Using matrices to represent and solve systems of linear equations.
• Linear Transformations: Representing geometric transformations (rotation, scaling, shearing) using matrices.

V. Differential Equations

Differential equations relate a function to its derivatives.

1. Types of Differential Equations

• First-Order Differential Equations: Involve the first derivative of the function. Methods of solving include separation of variables and integrating factors.
• Second-Order Differential Equations: Involve the second derivative of the function. Solving techniques depend on the type of equation (homogeneous, non-homogeneous).

2. Solving Differential Equations

• Separation of Variables: Used for certain first-order differential equations where the variables can be separated.
• Integrating Factors: Used for first-order linear differential equations.
• Complementary Function and Particular Integral: Used for solving second-order linear differential equations.

VI. Numerical Methods

Numerical methods provide approximate solutions to problems that are difficult or impossible to solve analytically.

1. Numerical Integration

• Trapezoidal Rule: Approximates the definite integral using trapezoids.
• Simpson's Rule: Approximates the definite integral using parabolas.

2. Numerical Solution of Differential Equations

• Euler's Method: A simple iterative method for approximating solutions to differential equations.

VII. Further Topics (Depending on the Specific Curriculum)

This section may include topics such as:

• Complex Numbers: Numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit (i² = -1).
• Further Calculus: Topics such as Taylor series, Maclaurin series, and L'Hopital's rule.
• Probability and Statistics: Further statistical techniques.

Conclusion

This comprehensive guide serves as a valuable reference for students undertaking Advanced Higher Mathematics. While memorizing every formula is crucial, understanding their derivation and application is equally important. Consistent practice and problem-solving are key to mastering these concepts and achieving success in your studies. Remember to always refer to your course materials and consult with your teacher or tutor for clarification on any points. Good luck with your studies!