Ultimate NDA Calculus Cheat Sheet - Limits, Derivatives, Integrals, Series & Applications

Welcome to the comprehensive, LibreTexts-style chapter-wise study guide for NDA Calculus. Calculus is one of the highest-weightage sections in the Mathematics paper of the National Defence Academy (NDA) entrance examination. Designed specifically for NDA aspirants, this guide breaks down fundamental definitions, key theorems, differentiation and integration rules, application formulas, and infinite series.

1. Limits and Continuity

In calculus, a limit describes the value that a function approaches as the input approaches a specified value.

Definition: \(\lim_{x \to a} f(x) = L\) means that \(f(x)\) can be made arbitrarily close to \(L\) by making \(x\) sufficiently close to \(a\).

L'Hôpital's Rule

L'Hôpital's Rule is an essential shortcut in NDA time-management for evaluating indeterminate forms. If \(\lim_{x \to a} \frac{f(x)}{g(x)}\) yields an indeterminate form of \(\frac{0}{0}\) or \(\frac{\pm\infty}{\pm\infty}\), then:

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

Provided the limit on the right-hand side exists.

Squeeze Theorem (Sandwich Theorem)

If \(g(x) \le f(x) \le h(x)\) for all \(x\) near \(a\), and \(\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L\), then:

\[\lim_{x \to a} f(x) = L\]

2. Derivatives and Differentiation Rules

The derivative measures the instantaneous rate of change of a function with respect to its variable.

Limit Definition of Derivative:

\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]

Fundamental Rules of Differentiation

• Power Rule: \(\frac{d}{dx}[x^n] = n x^{n-1}\)
• Product Rule: \((fg)' = f'g + fg'\)
• Quotient Rule: \((\frac{f}{g})' = \frac{f'g - fg'}{g^2}\)
• Chain Rule: \(\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)\)

Standard Derivatives Table

• \(\frac{d}{dx}[e^x] = e^x\)
• \(\frac{d}{dx}[a^x] = a^x \ln a\)
• \(\frac{d}{dx}[\ln x] = \frac{1}{x}\)
• \(\frac{d}{dx}[\sin x] = \cos x\)
• \(\frac{d}{dx}[\cos x] = -\sin x\)
• \(\frac{d}{dx}[\tan x] = \sec^2 x\)
• \(\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}\)
• \(\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}\)

3. Indefinite and Definite Integrals

Integration acts as the inverse process of differentiation. The Fundamental Theorem of Calculus links definite integration to anti-derivatives:

\[\int_a^b f(x)\,dx = F(b) - F(a) \quad \text{where } F'(x) = f(x)\]

Integration by Parts

Derived from the product rule of differentiation:

\[\int u\,dv = uv - \int v\,du\]

Common Integrals Table

• \(\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n eq -1)\)
• \(\int \frac{1}{x}\,dx = \ln|x| + C\)
• \(\int e^x\,dx = e^x + C\)
• \(\int \sin x\,dx = -\cos x + C\)
• \(\int \cos x\,dx = \sin x + C\)
• \(\int \sec^2 x\,dx = \tan x + C\)
• \(\int \frac{1}{1+x^2}\,dx = \arctan x + C\)
• \(\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C\)

4. Applications of Integration

Integration provides key formulas to compute geometric properties of curves and 3D solids.

1. Arc Length of a Curve

The total length \(L\) of a smooth curve \(y = f(x)\) from \(x = a\) to \(x = b\):

\[L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx\]

2. Surface Area of Revolution

The surface area \(S\) generated by revolving \(y = f(x)\) around the x-axis from \(a\) to \(b\):

\[S = \int_a^b 2\pi f(x) \sqrt{1 + [f'(x)]^2}\,dx\]

3. Volume of Revolution (Disk Method)

The volume \(V\) of a solid generated by revolving the region under \(y = f(x)\) around the x-axis:

\[V = \pi \int_a^b [f(x)]^2\,dx\]

5. Sequences, Series, and Maclaurin Expansions

Geometric Series

An infinite geometric series converges if \(|r| < 1\):

\[\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\]

p-Series Test

The series \(\sum \frac{1}{n^p}\) converges if \(p > 1\) and diverges if \(p \le 1\).

Taylor and Maclaurin Series

The Taylor series expansion of a smooth function \(f(x)\) centered at \(x = a\) is:

\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n\]

Standard Maclaurin expansions (centered at \(a = 0\)):

• \(e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \dots\)
• \(\sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots\)
• \(\cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots\)

6. NDA Exam Analysis & Weightage

In the UPSC NDA Mathematics examination, Differential and Integral Calculus together account for approximately 25 to 30 questions out of 120 questions (representing nearly 20-25% of the total paper weightage).

• Limits, Continuity & Differentiability: Expect 5-7 direct questions involving L'Hôpital's Rule, functional continuity, and quick evaluation tricks.
• Derivatives & Applications: Features 8-10 questions covering tangent-normals, maxima-minima, and rate of change.
• Integrals & Area Under Curves: Features 10-12 questions split between standard indefinite integrals, definite integral properties, area computation, and basic differential equations.

7. Frequently Asked Questions (FAQ)

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