Lesson Notes By Weeks and Term v4 - SHS 1
PRINCIPLES OF CALCULUS
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PRINCIPLES OF CALCULUS
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Subject: Additional Mathematics
Class: SHS 1
Term: 2nd Term
Week: 10
Grade code: 1.3.1.LI.3
Strand code: 3
Sub-strand code: 1
Content standard code: 1.3.1.CS.1
Indicator code: 1.3.1.LI.3
Theme: CALCULUS
Subtheme: PRINCIPLES OF CALCULUS
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This lesson introduces a fundamental concept in calculus: continuity. Imagine drawing a line on a piece of paper. If you can draw the entire line without lifting your pen, the line represents a continuous function. If you have to lift your pen to jump to another spot, it's discontinuous. In Ghana, we can think of a journey on a smoothly paved road like the N1 highway as continuous. However, if you encounter a broken bridge or a large pothole that forces you to stop and find another path, that's a point of discontinuity.
A. The Intuitive Idea of Continuity
A function is continuous over an interval if its graph is a single, unbroken curve. You can trace the graph with your pen without lifting it from the paper.
A function is discontinuous at a point if there is a break, hole, or jump in the graph at that point. Hole (Removable Discontinuity): A single point is missing from the graph. Jump (Jump Discontinuity): The graph suddenly jumps from one y-value to another. Break/Asymptote (Infinite Discontinuity): The graph heads towards infinity at a certain x-value. B. The Formal Definition of Continuity at a Point
For a function `f(x)` to be continuous at a point `x = a`, it must satisfy all three of the following conditions. If even one condition fails, the function is discontinuous at that point.