Lesson Notes By Weeks and Term v4 - SHS 1

Lesson Video

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Performance objectives

• Explain the concept of derivatives.
• Identify when a function is increasing or decreasing.
• Find the points where a function is momentarily at rest.
• Apply these concepts to analyse the behaviour of a function.

Lesson summary

Imagine you are driving a tro-tro from the Ayi Mensah tollbooth up the Aburi mountains. Sometimes you are accelerating and climbing (your altitude is increasing), sometimes you are descending (your altitude is decreasing), and at the very peak of a hill, for a brief moment, you are neither going up nor down. Calculus, specifically the derivative, gives us a powerful mathematical tool to describe this exact behaviour. By analysing the derivative of a function, we can determine precisely where it is increasing, decreasing, or momentarily at rest. This is fundamental in fields like physics (analysing motion), economics (maximising profit), and engineering (optimising designs).

Lesson notes

A. The Intuitive Idea: Looking at a Graph

Let's look at the graph of a curve, for instance, `y = x² - 4x + 1`. As we move from left to right, before we get to the bottom of the curve (`x=2`), the graph is going downwards. We say the function is decreasing. At the very bottom of the curve, the turning point, the graph is flat for an instant. It is neither going down nor up. We call this a stationary point or a point where the function is momentarily at rest. After the turning point, as we continue moving to the right, the graph is going upwards. We say the function is increasing. B. The Calculus Connection: The First Derivative

In our previous lessons, we learned that the first derivative, `dy/dx` or `f'(x)`, represents the gradient (slope) of the tangent line to the curve at any point `x`. When the function is increasing, the tangent line will slope upwards. This means its gradient is positive. When the function is decreasing, the tangent line will slope downwards. This means its gradient is negative. At a stationary point, the tangent line is horizontal. This means its gradient is zero.

This leads us to the three fundamental rules for analysing a function `f(x)`: Function is INCREASING when `f'(x) > 0` (The derivative is positive). Function is DECREASING when `f'(x) 0). Conclusion: Since `f'(x)` is always positive, the function `f(x) = 3x + 4` is always increasing for all values of `x`. This makes sense, as it's a straight line with a positive gradient.

Evaluation guide

• Generalise the behaviour of a moving object along a path or curve.
• Collaborative learning, Experiential Learning, and initiate Talk for Learning.
• object along a path or curve and provide their generalisations for whole class discussions.
• Activity 1