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: 9
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 the fundamental concept of continuity in functions. Intuitively, a continuous function is one whose graph can be drawn without lifting your pen from the paper. There are no breaks, jumps, or holes. Discontinuous functions, on the other hand, have these interruptions. Understanding continuity is the first major step into the world of Calculus. It helps us understand how things change smoothly versus how they change abruptly. Think about the growth of a maize plant over time – it's a smooth, continuous process. Now think about the price of a mobile data bundle – it doesn't increase smoothly; it jumps from one price point to the next.
A. The Intuitive Idea of Continuity
Imagine you are drawing a graph on a piece of paper or on the chalkboard. If you can draw the entire graph of a function over a certain interval without lifting your chalk or pen, the function is continuous on that interval. If you must lift your chalk or pen to continue drawing, the function is discontinuous at that point.
Example 1: Continuous Function The graph of `f(x) = x²` is a smooth parabola. You can draw it in one continuous motion.
Example 2: Discontinuous Functions These graphs have breaks. A "hole": A single point is missing. A "jump": The graph suddenly jumps from one y-value to another. A vertical asymptote: The graph shoots off to infinity. B. The Formal Definition of Continuity (The Three-Condition Test)