Problem 99 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)=\tan x\), then \(f^{n}(0)-{ }^{n} C_{2} f^{n-2}(0)+{ }^{n} C_{4} f^{n-4}(0)-\ldots=\) (A) \(\sin \frac{n \pi}{2}\) (B) \(\cos \frac{n \pi}{2}\) (C) \(\tan \frac{n \pi}{2}\) (D) None of these

Step-by-Step Solution

Answer

(B) \( \cos \frac{n \pi}{2} \)

Step 1: Understanding the Problem

We are given a function \( f(x) = \tan x \) and need to evaluate the expression \( f^{n}(0) - { }^{n} C_{2} f^{n-2}(0) + { }^{n} C_{4} f^{n-4}(0) - \ldots \). This involves understanding the behavior of \( \tan x \) and its derivatives evaluated at \( x = 0 \).

Step 2: Evaluate Derivatives of \( \tan x \)

The derivatives of \( \tan x \) evaluated at \( x = 0 \) should be determined as they are part of the expression. Note:\( f'(x) = \sec^2 x \). Hence, \( f'(0) = 1 \), \( f''(x) = 2 \tan x \sec^2 x \), which results in \( f''(0) = 0 \) because \( \tan 0 = 0 \), and similarly for higher derivatives evaluated at \( x = 0 \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in the study of mathematics, especially calculus. They include sine, cosine, and tangent functions. These functions relate the angles and sides of a right-angled triangle. Among these, the tangent function, denoted as \(\tan x\), is very potent, owing to its properties and applications.

• Definition: The tangent of an angle is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle.
• Periodicity: The function \(\tan x\) has a periodicity of \(\pi\), meaning it repeats its values every \(\pi\) units.
• Behavior at Zero: For \(x = 0\), \(\tan x\) equals zero. This fact is significant when evaluating derivatives at \(x = 0\).

Understanding these characteristics helps when dealing with complex expressions involving trigonometric functions. This foundational knowledge also aids in grasping more advanced concepts like calculating derivatives.

Higher Order Derivatives

Higher-order derivatives refer to derivatives of functions that are derived multiple times. In calculus, these are critical for understanding the intricate behavior of functions.

• First Derivative: For \(f(x) = \tan x\), the first derivative is \(f'(x) = \sec^2 x\). At \(x = 0\), it evaluates to 1 since \(\sec(0) = 1\).
• Second Derivative: The second derivative of \(\tan x\), \(f''(x) = 2 \tan x \sec^2 x\), simplifies to 0 at \(x = 0\) because \(\tan 0 = 0\).
• Subsequent Derivatives: For higher derivatives, the pattern often involves increasingly complex forms of trigonometric identities involving \(\tan x\) and \(\sec x\). These too simplify to 0 at \(x = 0\) because they contain terms like \(\tan 0\), which is zero.

These derivatives are the building blocks for constructing Taylor series and are essential in various fields of science and engineering for analyzing motion and change.

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra used to expand expressions that are raised to any power. It is succinctly expressed as:\[(a + b)^n = \sum_{k=0}^{n} { }^{n}C_{k} a^{n-k} b^k\]

• Binomial Coefficients: The terms \({ }^{n}C_k\) denote the binomial coefficients, calculated as \(\frac{n!}{k!(n-k)!}\), which determine the number of ways to choose \(k\) items from \(n\) without regard to the order.
• Expansion and Patterns: The theorem reveals a pattern of expanding polynomial expressions, which is crucial when working with expressions like those in our exercise.
• Application in Calculus: In calculus, integrating binomial expansions with derivative calculations allows for solving more complex problems involving higher-order derivatives and trigonometric functions.

Understanding the Binomial Theorem aids in grasping the mathematical structure and symmetries involved in calculus operations, making it easier to analyze and solve calculus problems involving series expansions and derivative patterns.