Higher Advanced Mathematics SQA

This subject is broken down into 35 topics in 3 modules:

  1. Calculus 14 topics
  2. Algebra, Proof and Number Theory 11 topics
  3. Matrices, Vectors and Complex Numbers 10 topics
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This page was last modified on 28 September 2024.

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Advanced Mathematics

Calculus

Differentiating Exponential and Natural Logarithmic Functions

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Differentiating Exponential and Natural Logarithmic Functions

Basics of Differentiation

  • Understand that differentiation is the process by which we find the rate at which a quantity is changing. This concept is fundamental to calculus.
  • Master the basic rules of differentiation such as power rule, quotient rule, product rule and chain rule.

Exponential Differentiation

  • Understand the concept of an exponential function. These are functions where a constant base is raised to a variable power(

    y = a^x

    ).
  • Know that the derivative of an exponential function

    y = e^x

    is simply

    y' = e^x

    i.e., the derivative of e^x with respect to x is itself.
  • For other bases (where 'a' is the base, other than e), use the formula

    y' = a^x * ln(a)

Logarithmic Differentiation

  • Recognise that natural logarithmic functions are inverse of the exponential function with base e (

    y = ln x

    ).
  • Understand that the derivative of

    y = ln x

    is

    y' = 1/x

    .
  • Study how to differentiate logarithmic functions with different bases using the change of base formula.

Differentiating More Complex Functions

  • Learn to apply the chain rule to differentiate more complex exponential and logarithmic functions. This rule involves differentiating a function of a function.
  • Master the application of product rule and quotient rule where appropriate.

Practice and Application

  • Constantly practice the application of these rules in increasingly complex scenarios.
  • Apply these principles in real world problems, such as exponential growth and decay, and elasticity in economics.

Course material for Advanced Mathematics, module Calculus, topic Differentiating Exponential and Natural Logarithmic Functions

Advanced Mathematics

Algebra, Proof and Number Theory

Finding the General Term and Summing Arithmetic and Geometric Progressions

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Finding the General Term and Summing Arithmetic and Geometric Progressions

Finding the General Term of Arithmetic and Geometric Progressions

Basic Concept of Arithmetic and Geometric Progressions

  • A sequence is a list of numbers in a specific order.

  • An arithmetic progression (AP) is a sequence of numbers in which the difference between any two consecutive terms is constant.

  • A geometric progression (GP) on the other hand, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number known as the common ratio.

General Term of an Arithmetic Progression

  • The nth term or the general term of an arithmetic progression where 'a' is the first term and 'd' is the common difference is given by a + (n - 1)d.

  • The expression a + (n - 1)d gives the nth term in terms of the first term a, the common difference d, and the position n.

General Term of a Geometric Progression

  • The nth term or the general term of a geometric progression where 'a' is the first term and 'r' is the common ratio is given by a * r^(n - 1).

  • The expression a * r^(n - 1) gives the nth term in terms of the first term a, the common ratio r, and the position n.

Summing Arithmetic and Geometric Progressions

Sum of Terms in an Arithmetic Progression

  • The sum 'S' of an arithmetic progression of 'n' terms or arithemetic series, where 'a' is the first term and 'l' is the last term is given by S = n/2 * (a + l) or S = n/2 * (2a + (n - 1)d)

Sum of Terms in a Geometric Progression

  • The sum of a finite geometric progression or geometric series is given by S = a * (1 - r^n) / (1 - r) for r ≠ 1.

  • The sum of the first 'n' terms, S_n, is given by S_n = a * (1 - r^n) / (1 - r) for r ≠ 1, where 'n' is the number of terms to be added, 'a' is the first term and 'r' is the common ratio.

  • For an infinite geometric progression where the absolute value of the common ratio 'r' is less than 1, the sum S of terms can be calculated using the formula S = a / (1 - r).

Course material for Advanced Mathematics, module Algebra, Proof and Number Theory, topic Finding the General Term and Summing Arithmetic and Geometric Progressions

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