Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

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Performance objectives

• Define a derivative and its link to the slope of a curve.
• Calculate the derivative of polynomial functions.
• Classify a curve's behavior (increasing, decreasing, or turning) at a point.
• Apply these principles to solve problems.

Lesson summary

In our study of linear functions (straight lines), we learned that the gradient (slope) is constant. However, the world around us is full of curves! Think about the journey from Accra up the Aburi mountains – the road is not a straight line with one constant steepness. It gets steeper, then less steep, and at the top, it becomes flat for a moment before going down. Calculus, specifically the derivative, gives us a powerful tool to describe this changing steepness at any exact point on a curve.

Lesson notes

Part 1: The Concept of a Changing Slope

For a straight line, `y = mx + c`, the slope `m` is the same everywhere. For a curve, like `y = x²`, the steepness changes. Near the bottom (the vertex), it's almost flat. As we move away from the vertex, it gets steeper and steeper.

The derivative of a function, written as `f'(x)` or `dy/dx`, is a new function that tells us the slope (or gradient) of the tangent line to the original function `f(x)` at *any* point `x`. Part 2: Recap of Differentiation Rules

Before we can analyse the behaviour, we must be able to find the derivative. Let's quickly review the essential rules: The Power Rule: If `f(x) = ax^n`, then `f'(x) = n * a * x^(n-1)`. *Example:* If `f(x) = 3x^4`, then `f'(x) = 4 * 3x^(4-1) = 12x^3`. The Product Rule: If `f(x) = u(x)v(x)`, then `f'(x) = u'(x)v(x) + v'(x)u(x)`. *Example:* If `f(x) = (x^2)(x+1)`, let `u=x^2` and `v=x+1`. Then `u'=2x` and `v'=1`. `f'(x) = (2x)(x+1) + (1)(x^2) = 2x^2 + 2x + x^2 = 3x^2 + 2x`. The Quotient Rule: If `f(x) = u(x) / v(x)`, then `f'(x) = [u'(x)v(x) - v'(x)u(x)] / [v(x)]^2`. *Example:* If `f(x) = (2x) / (x-3)`, let `u=2x` and `v=x-3`. Then `u'=2` and `v'=1`. `f'(x) = [(2)(x-3) - (1)(2x)] / (x-3)^2 = [2x - 6 - 2x] / (x-3)^2 = -6 / (x-3)^2`. The Chain Rule: If `f(x) = [g(x)]^n`, then `f'(x) = n[g(x)]^(n-1) * g'(x)`. *Example:* If `f(x) = (3x-4)^5`, let `g(x)=3x-4`. Then `g'(x)=3`. `f'(x) = 5(3x-4)^(5-1) * (3) = 15(3x-4)^4`. Part 3: Classifying the Behaviour of a Curve

Evaluation guide

• Generalise the behaviour with respect to the slope of a moving object along a curve.
• Talk for Learning, Think -pair- share, Experiential Learning, and Group
• Work/Collaborative Learning.
• Classify the behaviour of a curve