Lesson Notes By Weeks and Term v4 - SHS 1

Lesson Video

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Performance objectives

• Explain the concept of the derivative as a rate of change.
• Calculate the derivative of simple polynomial functions using limits.
• Use technology to analyze the rate of change of a function.
• Apply calculus principles to real-life situations.

Lesson summary

This lesson introduces the fundamental concept of Calculus: the rate of change. We will move from understanding the average speed of a journey to the instantaneous speed at a specific moment. Think about a tro-tro travelling from Accra to Kumasi. Its *average speed* might be 80 km/h, but at any given moment, its speedometer might read 100 km/h, 60 km/h, or even 0 km/h in traffic. This "speedometer reading" is the instantaneous rate of change. We will investigate this idea graphically and using technology, laying the foundation for understanding the derivative, a powerful tool used in science, engineering, and economics to analyse how things change.

Lesson notes

A. Average Rate of Change

The average rate of change measures how much a function changes, on average, over an interval. It is the same as the slope of the line connecting two points on the function's graph. This line is called a secant line. Formula: For a function `h(u)`, the average rate of change between `u = a` and `u = b` is: ``` Average Rate of Change = (Change in h) / (Change in u) = (h(b) - h(a)) / (b - a) ``` This is the familiar slope formula `(y2 - y1) / (x2 - x1)`.

Evaluation guide

• Use technology or any innovative ways to investigate the rate of change of a function, h
• (u), with respect to u.
• Collaborative learning, Experiential Learning, and initiate Talk for Learning.
• function relates to its derivative.