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: 14
Grade code: 2.3.1.LI.3
Strand code: 3
Sub-strand code: 1
Content standard code: 2.3.1.CS.2
Indicator code: 2.3.1.LI.3
Theme: CALCULUS
Subtheme: PRINCIPLES OF CALCULUS
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Welcome, students! Today, we are beginning a new and powerful topic in mathematics called Calculus. So far, we have learned about differentiation, which helps us find the rate of change, like the speed of a car at a particular instant. But what if we know the speed at every moment and want to find the total distance travelled? What if we know the rate at which water is flowing into the Akosombo Dam and we want to find the total amount of water collected over a month? This is where integration comes in. Integration helps us to "add up" infinitely many small parts to find a whole. It is a tool used by engineers, economists, scientists, and even business owners to solve real-world problems.
This lesson is divided into three main parts. First, we will see how integration is the opposite of differentiation. Second, we will explore how to find the area of complex shapes. Finally, we will bring these two ideas together. Part 1: Integration as the Reverse of Differentiation (The Anti-derivative)
Let's start with a question. If I tell you that the derivative of a function `f(x)` is `f'(x) = 2x`, can you tell me what the original function `f(x)` could be?
Think about it... what function do we differentiate to get `2x`? You would correctly say `f(x) = x²`. Because `d/dx (x²) = 2x`.
Now, consider another function, `g(x) = x² + 5`. What is its derivative? `g'(x) = d/dx (x² + 5) = 2x + 0 = 2x`.