Lesson Notes By Weeks and Term v4 - SHS 1

Lesson Video

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Performance objectives

• Explain the concept of a derivative and its relationship to limits.
• Calculate the derivative of a function using the limit definition.
• Identify points where a function is not differentiable.
• Apply calculus principles to real-life situations.

Lesson summary

Imagine you are travelling in a tro-tro from the 37 Station in Accra to Koforidua. The driver speeds up on the motorway, slows down in towns like Adenta, and stops to pick up passengers. If someone asks, "How fast were you going?", the answer depends on *when* they ask. Your average speed for the whole journey is different from your exact speed at the moment you passed the Peduase Lodge. Calculus gives us the tools to move from "average" change to "instantaneous" change. Today, we will learn the fundamental principle of finding this instantaneous rate of change, called the derivative, using a powerful idea called limits.

Lesson notes

Part 1: From Average Rate of Change to Instantaneous Rate of Change

Think about a curve representing a function `y = f(x)`. Average Rate of Change: If we pick two points on the curve, say at `x` and `x+h`, the slope of the line connecting them (a secant line) gives the *average rate of change* over that small interval `h`.

The coordinates are `P(x, f(x))` and `Q(x+h, f(x+h))`. The slope of the secant line PQ is: `Slope_avg = (Change in y) / (Change in x) = (f(x+h) - f(x)) / ((x+h) - x) = (f(x+h) - f(x)) / h` This expression is called the Difference Quotient. Instantaneous Rate of Change: Now, what if we want the rate of change at the single point `P`? We can imagine moving point `Q` closer and closer to `P`. As we do this, the interval `h` gets smaller and smaller (`h` approaches 0).

The secant line PQ pivots and becomes the tangent line at point `P`. The slope of this tangent line is the *instantaneous rate of change* at that exact point. Part 2: The Limit and the Definition of the Derivative

Evaluation guide

• Use the limits of a function to find its derivative.
• Collaborative learning, Experiential Learning, and initiate Talk for Learning.
• relates to its derivative.
• Learners working in mixed -ability groups investigate both algebraically and graphically how