Module 2 · GATE Engineering Mathematics

Calculus

Limits, continuity, differentiation, integration, multivariable calculus.

Limits · Differentiation · Integration · Multivariable Calculus

🎯 Take Mock Test

Learning Objectives

Understand the fundamental concepts of Calculus
Apply key formulas and techniques to solve problems
Practice with exam-level questions to build speed and accuracy

Key Concepts

What is Calculus?

Calculus is the mathematical study of continuous change, divided into differential calculus (rates of change and slopes) and integral calculus (accumulation and areas). For JEE Main & Advanced, calculus carries the highest weightage (~35% of the maths paper). Key topics include limits, continuity, differentiability, differentiation, integration, differential equations, and area under curves.

Key Concepts

1. Limits & Continuity: A function f(x) has limit L as x approaches a if f(x) gets arbitrarily close to L. Standard limits: lim(x→0) sin x / x = 1, lim(x→0) (e^x - 1)/x = 1, lim(x→0) (1+x)^(1/x) = e. A function is continuous at x = a if lim(x→a) f(x) = f(a). L'Hôpital's rule: if limit gives 0/0 or ∞/∞, differentiate numerator and denominator separately.

2. Differentiation: Derivative f'(x) = dy/dx = lim(h→0) [f(x+h) - f(x)]/h. Standard derivatives: d/dx(x^n) = nx^(n-1), d/dx(sin x) = cos x, d/dx(e^x) = e^x, d/dx(log x) = 1/x. Chain rule: dy/dx = (dy/du)(du/dx). Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v - uv')/v^2.

3. Application of Derivatives: Tangent slope = f'(x₀). Normal slope = -1/f'(x₀). For maxima/minima: f'(x) = 0 (critical points), f''(x) > 0 → local minima, f''(x) < 0 → local maxima. Rate of change: dy/dt = (dy/dx)(dx/dt). Increasing function if f'(x) > 0, decreasing if f'(x) < 0.

4. Integration: ∫x^n dx = x^(n+1)/(n+1) + C (n ≠ -1). ∫(1/x) dx = log|x| + C. ∫sin x dx = -cos x + C. ∫e^x dx = e^x + C. Methods: substitution, integration by parts (∫u dv = uv - ∫v du), partial fractions. Definite integral: ∫(a to b) f(x) dx = F(b) - F(a). Properties: ∫(a to b) f(x) dx = ∫(a to b) f(a+b-x) dx, ∫(-a to a) f(x) dx = 0 if f is odd, 2∫(0 to a) f(x) dx if f is even.

5. Area Under Curve: Area bounded by y = f(x), x-axis, x = a, x = b is ∫(a to b) |f(x)| dx. Area between two curves = ∫(a to b) |f(x) - g(x)| dx.

6. Differential Equations: Order = highest derivative. Degree = power of highest derivative. Variable separable: dy/dx = f(x)g(y) → ∫dy/g(y) = ∫f(x)dx. Linear DE: dy/dx + Py = Q → integrating factor IF = e^(∫P dx), solution: y(IF) = ∫Q(IF) dx.

Solved Examples

Example 1 (Limit using L'Hôpital)

Evaluate: lim(x→0) (sin²x - x²)/(x⁴)

a) -1/3 b) 1/3 c) -1/6 d) 1/6

Solution: a) -1/3. Apply L'Hôpital's rule 4 times. First: (2 sin x cos x - 2x)/(4x³) = (sin 2x - 2x)/(4x³). Second: (2 cos 2x - 2)/(12x²) = -2(1 - cos 2x)/(12x²) = -(2·2 sin²x)/(12x²) = -4 sin²x/(12x²). Third: -8 sin x cos x/(24x) = -4 sin 2x/(24x). Fourth: -8 cos 2x/24 = -1/3.

Example 2 (Maxima-Minima)

Find the maximum value of f(x) = x³ - 6x² + 9x + 15 in [0,5].

a) 15 b) 19 c) 35 d) 29

Solution: c) 35. f'(x) = 3x² - 12x + 9 = 3(x² - 4x + 3) = 3(x-1)(x-3). Critical points: x = 1, 3. f''(x) = 6x - 12. At x=1: f'' = -6 < 0 → local max, f(1) = 1-6+9+15 = 19. At x=3: f'' = 6 > 0 → local min, f(3) = 27-54+27+15 = 15. At endpoints: f(0) = 15, f(5) = 125-150+45+15 = 35. Global maximum in [0,5] is 35 at x=5.

Example 3 (Integration by Parts)

Evaluate: ∫e^x sin x dx

a) (e^x/2)(sin x - cos x) + C b) (e^x/2)(sin x + cos x) + C c) e^x(sin x - cos x)/2 + C d) Both a and c

Solution: d) Both a and c are the same. Let I = ∫e^x sin x dx. Using integration by parts: I = e^x sin x - ∫e^x cos x dx = e^x sin x - [e^x cos x + ∫e^x sin x dx] = e^x sin x - e^x cos x - I. So 2I = e^x(sin x - cos x) → I = (e^x/2)(sin x - cos x) + C.

Shortcut Techniques

For 0/0 or ∞/∞ limits, always try L'Hôpital's rule first — differentiate numerator and denominator separately until you get a determinate form.
Memorise standard expansions: e^x = 1 + x + x²/2! + ..., sin x = x - x³/3! + ..., cos x = 1 - x²/2! + ... for limits with series.
For maxima-minima word problems, write the function to optimise in one variable using the given constraint, then differentiate.
Integration shortcut: ∫e^{ax} sin(bx) dx = e^{ax}(a sin bx - b cos bx)/(a²+b²) + C.
Area shortcut: area between parabola y² = 4ax and its latus rectum x = a is (8a²/3).

Pro Tip

For JEE Advanced, calculus questions often combine multiple concepts — e.g., area under curve with differential equations, or limits with expansion series. Practice mixed-concept problems from previous years. Maintain a formula sheet for quick revision of all standard limits, derivatives, and integration formulas.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Advanced Applications

Modern exams feature questions combining multiple quantitative concepts. Master these by practicing previous year papers.

Shortcut Methods

Percentages: Use fraction table (12.5%=1/8, 20%=1/5, 25%=1/4). SI/CI: For 2 years, CI-SI = P(r/100)^2. Time Work: Use LCM of time periods. Average Speed: 2ab/(a+b) for equal distances.

Pro Tip

Write key formulas on rough sheet immediately after the exam bell while memory is fresh.

Practice Questions

← Linear Algebra Differential Equations →