Year 11 Maths Advanced Module 4 15 Lessons

Further Calculus

From the bacteria that doubles every 20 minutes to the archaeologist dating 50,000-year-old bones. From the decibel scale spanning a trillion-fold range to the pH scale keeping your blood alive. This module explores exponential and logarithmic functions, their derivatives, and their extraordinary power to model growth, decay, and change in the natural world.

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Know

Key Facts

Exponential and logarithmic functions and their properties
Laws of logarithms and change of base
Derivatives of $e^x$, $a^x$, $\ln x$, and $\log_a x$

Understand

Concepts

Exponentials and logs as inverse functions
Why $e$ is the natural base for calculus
How derivatives quantify rates of change

Can Do

Skills

Sketch and transform exponential and log graphs
Solve exponential equations using logarithms
Differentiate and optimise exponential functions

Lessons

Introduction to Exponential Functions

~35 min · Growth, decay, evaluating expressions

Graphs of Exponential Functions

~35 min · Asymptotes, intercepts, transformations

The Number $e$ and Natural Exponentials

~35 min · Continuous compounding, $A = Pe^{rt}$

Exponential Modelling and Equations

~35 min · Solving $a^x = b$, carbon-14 dating

Introduction to Logarithmic Functions

~35 min · Inverse of exponentials, domain, pH scale

Checkpoint 1 — Lessons 1–5

~25 min · 8 MC + 3 SA

Graphs of Logarithmic Functions

~35 min · Asymptotes, inverse relationship, decibels

Laws of Logarithms

~35 min · Product, quotient, power laws

Change of Base and Logarithmic Equations

~35 min · Change of base, solving, extraneous solutions

Differentiating $e^x$

~35 min · Chain rule, tangents, Newton's cooling

Differentiating $a^x$

~35 min · General exponential, Moore's Law

Checkpoint 2 — Lessons 6–10

~25 min · 8 MC + 3 SA

Differentiating $\ln x$

~35 min · Chain rule, utility theory

Differentiating $\log_a x$

~35 min · General log derivative, pH sensitivity

Exponential Growth and Decay

~35 min · Half-life, doubling time, C-14 dating

Optimisation with Exponentials

~35 min · Stationary points, drug concentration

Module Synthesis

~35 min · Consolidation, connections, common mistakes

Checkpoint 3 — Lessons 11–15

~25 min · 8 MC + 3 SA

Module Quiz — Full Module

~40 min · 10 MC + 5 SA

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