Lesson 4.8: Calculus Basics (Limits, Differentiation, Integration)

Calculus forms an important part of the Mathematics section for NDA, CDS, and AFCAT exams. Understanding basic limits, differentiation, and integration helps aspirants solve higher-level mathematical problems efficiently.

1. Limits

• Definition: Limit of f(x) as x → a is the value that f(x) approaches.

• Notation: lim⁡x→af(x)=L\lim_{x→a} f(x) = Llimx→a​f(x)=L

• Basic Rules:

  • lim⁡x→a[f(x)±g(x)]=lim⁡f(x)±lim⁡g(x)\lim_{x→a} [f(x) ± g(x)] = \lim f(x) ± \lim g(x)limx→a​[f(x)±g(x)]=limf(x)±limg(x)

  • lim⁡x→a[cf(x)]=c⋅lim⁡f(x)\lim_{x→a} [cf(x)] = c \cdot \lim f(x)limx→a​[cf(x)]=c⋅limf(x)

  • lim⁡x→a[f(x)⋅g(x)]=(lim⁡f(x))⋅(lim⁡g(x))\lim_{x→a} [f(x) \cdot g(x)] = (\lim f(x)) \cdot (\lim g(x))limx→a​[f(x)⋅g(x)]=(limf(x))⋅(limg(x))

2. Differentiation

• Definition: Derivative represents the rate of change of a function.

• Notation: f’(x) or dy/dx

• Basic Formulas:

  • (d/dx) xⁿ = nxⁿ⁻¹

  • (d/dx) sin x = cos x, (d/dx) cos x = –sin x

  • Sum/Difference Rule: (d/dx)[f(x) ± g(x)] = f’(x) ± g’(x)

• Applications:

  • Maxima and minima problems

  • Rate of change in physics or motion

3. Integration

• Definition: Integration is the reverse process of differentiation; finds area under a curve.

• Notation: ∫ f(x) dx

• Basic Formulas:

  • ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C, n ≠ –1

  • ∫ sin x dx = –cos x + C, ∫ cos x dx = sin x + C

  • ∫ (a dx) = ax + C

• Applications:

  • Area under curves

  • Simple physics and engineering problems

Exam Tips

• Memorize basic differentiation and integration formulas.

• Practice simple problems on limits, derivatives, and integrals.

• Solve previous year NDA, CDS, AFCAT calculus questions.

• Focus on step-by-step solving and formula application.