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: 10
Grade code: 2.3.1.LI.4
Strand code: 3
Sub-strand code: 1
Content standard code: 2.3.1.CS.1
Indicator code: 2.3.1.LI.4
Theme: CALCULUS
Subtheme: PRINCIPLES OF CALCULUS
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So far in our study of calculus, we have focused on differentiating algebraic functions like polynomials (e.g., `x³ + 2x - 5`) and rational functions. Today, we venture into a new, exciting category of functions called transcendental functions. These functions, specifically the natural exponential function (eˣ) and the natural logarithmic function (ln x), are fundamental to describing the world around us.
What are Transcendental Functions?
A transcendental function is a function that cannot be expressed as a finite combination of algebraic operations (addition, subtraction, multiplication, division, raising to a power, and taking roots).
The two most important transcendental functions for us today are: The Natural Exponential Function, `f(x) = e^x` The base 'e' is a special irrational number, approximately equal to 2.71828. It is often called Euler's number. This function describes processes of continuous growth or decay. The Natural Logarithmic Function, `f(x) = ln(x)` This is the inverse of the exponential function. `ln(x)` is the power to which 'e' must be raised to get `x`. It is used in scales that measure wide-ranging quantities, like the Richter scale for earthquakes or pH in chemistry. Core Differentiation Rules
Rule 1: Derivative of the Natural Exponential Function