NDA cheat sheet for Calculus! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Differential & Integral Calculus

Calculus forms a major portion of the NDA Mathematics paper, typically accounting for 25 to 30 questions. Mastery over limits, derivatives, and definite integrals is crucial for achieving high accuracy under strict time constraints.

1. Limits, Continuity, and Differentiability

A function \(f(x)\) is continuous at \(x = a\) if \(\lim_{x \to a} f(x) = f(a)\). L'Hôpital's Rule applies to indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\):

\[\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\]

Standard limits to memorize for rapid problem-solving include \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and \(\lim_{x \to 0} \frac{e^x - 1}{x} = 1\).

2. Differential Calculus & Applications

Key differentiation rules include the product rule \((uv)' = u'v + uv'\) and quotient rule \(\bigl(\frac{u}{v}\bigr)' = \frac{u'v - uv'}{v^2}\). For maxima and minima, set \(f'(x) = 0\):

• If \(f''(x) < 0\), the point is a local maximum.
• If \(f''(x) > 0\), the point is a local minimum.

3. Integral Calculus & Area Under Curves

Integration is the reverse process of differentiation. The Fundamental Theorem of Calculus links derivatives and integrals:

\[\int_{a}^{b} f(x) dx = F(b) - F(a)\]

Key properties of definite integrals useful for NDA questions:

\[\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a - x) dx\]