Lesson Notes By Weeks and Term v4 - SHS 2
PRINCIPLES OF CALCULUS
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PRINCIPLES OF CALCULUS
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Subject: Additional Mathematics
Class: SHS 2
Term: 2nd Term
Week: 12
Grade code: 2.3.1.LI.5
Strand code: 3
Sub-strand code: 1
Content standard code: 2.3.1.CS.1
Indicator code: 2.3.1.LI.5
Theme: CALCULUS
Subtheme: PRINCIPLES OF CALCULUS
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This lesson explores one of the most powerful ideas in calculus: using the derivative to understand the behaviour of a function. Imagine walking along the Aburi hills. At some points, you are climbing upwards (a positive slope), at other points you are walking downwards (a negative slope), and at the very peak of a hill or the bottom of a valley, the ground is momentarily flat (a zero slope). In the same way, the derivative, `dy/dx`, tells us the slope of a function's graph at any point. By analysing the sign of the derivative, we can precisely describe whether the function is increasing, decreasing, or at a turning point.
Concept 1: The Derivative as a Slope Function
Recall that the derivative of a function `y = f(x)`, written as `dy/dx` or `f'(x)`, represents the instantaneous rate of change of `y` with respect to `x`. Geometrically, this is the gradient (slope) of the tangent line to the curve at any point `x`.
Think of `f'(x)` as a "slope-finding machine". You give it an x-value, and it tells you the exact slope of the curve at that point. Concept 2: Classifying the Behaviour of a Curve
The sign of the derivative at a point `x = c` tells us exactly what the curve is doing at that point. If `f'(c) > 0` (Positive): The slope of the tangent is positive. The curve is increasing or rising at `x = c`. As you move from left to right, the graph goes upwards. *Analogy:* You are climbing up a hill. If `f'(c) 0`, adding more fertilizer increases the yield. When `Y'(f) 0`, the point is a local minimum**. If `f''(c) < 0`, the point is a local maximum. If `f''(c) = 0`, the test is inconclusive. Task: Ask them to revisit Independent Practice Question #3 (`y = x³ - 3x + 2`) and use the second derivative test to classify the stationary points they found as either a local maximum or a local minimum.