A function in mathematics is a relation that assigns each input from a set (domain) exactly one output from another set (codomain). It represents how one quantity depends on another. Functions are expressed algebraically graphically or through tables and are fundamental in modeling real-world phenomena solving equations and analyzing relationships in various fields of study.
Class 12 Maths Chapter 6 focuses on Applications of Derivatives for CBSE Exam 2024-25. It includes rate of change of quantities increasing and decreasing functions tangents and normals maxima and minima. These concepts are crucial for solving problems in physics economics and engineering while developing analytical skills and practical understanding of mathematical applications.
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The function f(x) = tan x – x increases or decreases throughout, and the only way to find this is by analyzing the derivative of f(x).
Now, differentiate f(x) with respect to x:
f'(x) = d/dx(tan x) – d/dx(x) = sec² x – 1
So,
f'(x) = sec² x – 1 = tan² x
Since tan² x is always non-negative for all real values of x, f'(x) ≥ 0 for all x where tan x is defined.
Therefore, f(x) is always increasing where it is defined, but has vertical asymptotes at x = (π/2) + nπ, where n is any integer.
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