📘 Calculus: GATE Engineering Mathematics – Section 2

📘 Calculus: GATE Engineering Mathematics – Section 2

🧠 Prepared with Inspiration from GrandMaster Bikram Sutradhar
🏆 5-Time World Record Holder | Best Teacher Awardee 2023 | IIT JAM AIR 392
🌐 Website: sirbikramsutradhar.com | 📞 Contact: 9863002294
🕔 05:03 PM IST, Sunday, July 06, 2025

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🔍 Introduction to Calculus

Calculus is one of the most essential and scoring topics in the GATE Engineering Mathematics syllabus.
It forms Section 2 and includes:

✅ Functions of a single variable
✅ Limits & Continuity
✅ Differentiability & Mean Value Theorems
✅ Taylor Series Expansions
✅ Maxima and Minima
✅ Definite and Indefinite Integrals
✅ Applications of Integrals (Area, Volume, etc.)

This masterclass presents each concept with formulas, visuals, and problem-solving approaches — all inspired by the bAstronautWay learning method of Sir Bikram Sutradhar.

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📐 1. Limits and Continuity

🔹 Limit of a function f(x) as x → a:
lim x→a f(x) exists if both left-hand and right-hand limits exist and are equal.

🔹 Continuity at a point x = a:
f(x) is continuous at x = a ⇔

• lim x→a⁻ f(x) = lim x→a⁺ f(x) = f(a)

🧠 Shortcut: Discontinuity often arises from modulus, step, or piecewise functions!

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📈 2. Differentiability and Mean Value Theorems

🔹 f(x) is differentiable at x = a if the derivative from both sides exists and is equal.
Differentiability ⇒ Continuity, but not vice versa!

📌 Rolle’s Theorem:
If f(a) = f(b) and f is continuous and differentiable on [a, b], then ∃ c ∈ (a, b) such that f′(c) = 0

📌 Lagrange’s Mean Value Theorem (MVT):
∃ c ∈ (a, b) such that f′(c) = [f(b) − f(a)] / (b − a)

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🧮 3. Taylor’s and Maclaurin Series

🧠 Used for approximating functions around a point.

Taylor’s Theorem:
f(x) = f(a) + f′(a)(x−a) + f″(a)(x−a)²/2! + … + fⁿ(a)(x−a)ⁿ/n!

📌 If a = 0 → Maclaurin Series

🧪 Application: Find expansion of sin(x), cos(x), eˣ, log(1+x) up to 4 terms for numerical methods in GATE.

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📉 4. Maxima and Minima

To find extrema:

1. Compute f′(x) and solve f′(x) = 0
2. Use f″(x):
   1. If f″(x) > 0 → local minima
   1. If f″(x) < 0 → local maxima

✨ Trick: Maxima/minima without boundary check = dangerous in GATE!

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∫ 5. Integral Calculus

✅ Definite Integrals:
∫ₐᵇ f(x) dx = Area under the curve from x = a to x = b
– Additive, symmetric & substitution-friendly!

✅ Indefinite Integrals:
∫f(x) dx = F(x) + C

📌 Important Tricks:

• Use reduction formulas for powers of sin, cos, etc.
• Apply integration by parts when f(x) = u·v (choose u wisely using ILATE rule!)

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📊 6. Applications of Definite Integrals

Used for:

• Area between curves
• Volume of solids of revolution
• Length of curves
• Surface area

🧠 Common GATE problem: Area between y = f(x) and x-axis from a to b = ∫ₐᵇ |f(x)| dx

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🎯 50 Most Important Questions – GATE Level

Here are sample practice questions curated with GATE trends in mind:

1. Evaluate lim x→0 (sin x) / x
2. Prove continuity of f(x) = |x| at x = 0
3. Use MVT for f(x) = x² + 3x in [1,3]
4. Expand eˣ using Maclaurin series up to x⁴
5. Find local maxima of f(x) = x³ − 6x² + 9x
6. Evaluate ∫₀^π sin²x dx
7. Area under f(x) = |x| from −1 to 1
8. ∫ x log x dx using integration by parts
9. Show f(x) = x³ is continuous and differentiable
10. Find volume generated by revolving y = √x from 0 to 4 about x-axis

📩 Get full 50-question PDF → DM “GATE Calculus 50” to 9863002294

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💫 Learn the bAstronautWay – Only with Sir Bikram Sutradhar

✨ 5× World Record Holder
🥇 Best Teacher Awardee 2023
🎯 IIT JAM AIR 392
🚀 Visionary Educator with Visual + Conceptual + Competitive Mastery

👨‍🏫 Learn more: sirbikramsutradhar.com
📲 WhatsApp Now: 9863002294

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📌 Hashtags:

#GATECalculus #Section2 #GrandMasterBikramSutradhar #BestTeacherIndia #bAstronautWay #GATE2026Prep #EngineeringMathematics #IITJAM #CalculusGATE #sirBikramSutradhar #MathsWithPurpose

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Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar

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Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar

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ICSE CLASS 10 ICSE CLASS 10 BIOLOGY BASTRONAUTWAY SirBikramSutradhar Bikram Sutradhar GrandMaster Bikram Sutradhar selina biology solutions ICSE Biology Selina Solution

📚 Class 9 Science – Complete Table of Contents (विषय-सूची)

CBSE | NCERT Curriculum | Bilingual (English + Hindi)
By Grandmaster Bikram Sutradhar | SirBikramSutradhar | BASTRONAUTWAY

🔬 PHYSICS (भौतिक विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	Motion	गति
2️⃣	Force and Laws of Motion	बल तथा गति के नियम
3️⃣	Gravitation	गुरुत्वाकर्षण
4️⃣	Work and Energy	कार्य तथा ऊर्जा
5️⃣	Sound	ध्वनि

🧪 CHEMISTRY (रसायन विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	Matter in Our Surroundings	हमारे आस-पास के पदार्थ
2️⃣	Is Matter Around Us Pure?	क्या हमारे आस-पास के पदार्थ शुद्ध हैं?
3️⃣	Atoms and Molecules	परमाणु एवं अणु
4️⃣	Structure of the Atom	परमाणु की संरचना

🧬 BIOLOGY (जीव विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	The Fundamental Unit of Life	जीवन की मौलिक इकाई
2️⃣	Tissues	ऊतक
3️⃣	Improvement in Food Resources	धनों में सुधार

📘 ADDITIONAL SECTIONS (अतिरिक्त अनुभाग)

📖 Section	Title (English + Hindi)
🔹	Glossary / पारिभाषिक शब्दावली
🔹	Answer Keys / उत्तरमाला
🔹	Summary / सारांश
🔹	Important Diagrams / महत्वपूर्ण चित्र
🔹	NCERT Exercises / एनसीईआरटी अभ्यास
🔹	Sample Papers & MCQs / मॉडल पेपर व वस्तुनिष्ठ प्रश्न

✅ All content follows the official NCERT Class 9 Science textbook structure. Each chapter will include explanations, bilingual notes, definitions, diagrams, MCQs, FAQs, and practice exercises under the BASTRONAUTWAY learning system.

class 10 Maths ncert solution.

CBSE Class 10 Mathematics (2025) NCERT Syllabus:

CBSE Class 10 Mathematics NCERT Syllabus (2025) – Chapter-Wise List

CBSE Class 10 Mathematics NCERT Syllabus (2025) – Chapter-Wise List

chapter 1. Real Numbers

chapter 2. Polynomials

chapter 3.  Linear Equations

chapter 4. Quadratic Equations

chapter 5. Arithmetic Progressions

chapter 6. Triangles

chapter 7. Coordinate Geometry

chapter 8. Introduction to Trigonometry

chapter 9. Some Applications of Trigonometry

chapter 10. Circles

chapter 11. Areas Related to Circles

chapter 12. Surface Areas and Volumes

chapter 13. Statistics

chapter 14. Probability

CBSE Class 10 Science (2025) NCERT Syllabus:

CBSE Class 10 Science NCERT Syllabus (2025) – Chapter-Wise List

CBSE Class 10 Science NCERT Syllabus (2025) – Chapter-Wise List

Chapter 1 Chemical Reactions and Equations 
Chapter 2 Acids, Bases and Salts 
Chapter 3 Metals and Non-metals 
Chapter 4 Carbon and its Compounds 
Chapter 5 Life Processes 
Chapter 6 Control and Coordination 
Chapter 7 How do Organisms Reproduce? 
Chapter 8 Heredity 
Chapter 9 Light – Reflection and Refraction 
Chapter 10 The Human Eye and the Colourful World 
Chapter 11 Electricity 
Chapter 12 Magnetic Effects of Electric Current 
Chapter 13 Our Environment

📖 Chapter 12 – Organic Chemistry (Selina Textbook)

Table of Contents:

• 12A. Organic Compounds
• 12B. Hydrocarbons: Alkanes
• 12C. Hydrocarbons: Alkenes
• 12D. Hydrocarbons: Alkynes
• 12E. Alcohols
• 12F. Carboxylic Acids
• Exercise 12 MISCELLANEOUS
• Glossary
• Model Question Paper –

📌 Exercise 12 MISCELLANEOUS – Focus Areas:

• 🧬 Consolidated practice from the entire Organic Chemistry chapter
• 🧪 Application-based questions covering alkanes, alkenes, alkynes, alcohols, and carboxylic acids
• 🧾 Writing structural formulas and chemical equations
• 💡 Distinguishing reactions, isomer identification, and conversions
• 📘 Full-syllabus revision to boost conceptual clarity and board exam readiness

Each question in this section is designed to:

• ✅ Strengthen cross-topic understanding
• 📝 Prepare students for application-based and reasoning-based questions
• 🧠 Improve problem-solving speed and exam performance

 class 5

📚 Class 9 Science – Complete Table of Contents (विषय-सूची)

CBSE | NCERT Curriculum | Bilingual (English + Hindi)
By Grandmaster Bikram Sutradhar | SirBikramSutradhar | BASTRONAUTWAY

🔬 PHYSICS (भौतिक विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	Motion	गति
2️⃣	Force and Laws of Motion	बल तथा गति के नियम
3️⃣	Gravitation	गुरुत्वाकर्षण
4️⃣	Work and Energy	कार्य तथा ऊर्जा
5️⃣	Sound	ध्वनि

🧪 CHEMISTRY (रसायन विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	Matter in Our Surroundings	हमारे आस-पास के पदार्थ
2️⃣	Is Matter Around Us Pure?	क्या हमारे आस-पास के पदार्थ शुद्ध हैं?
3️⃣	Atoms and Molecules	परमाणु एवं अणु
4️⃣	Structure of the Atom	परमाणु की संरचना

🧬 BIOLOGY (जीव विज्ञान)

🔢	Chapter Title (English)	अध्याय का नाम (हिंदी)	Page No.
1️⃣	The Fundamental Unit of Life	जीवन की मौलिक इकाई
2️⃣	Tissues	ऊतक
3️⃣	Improvement in Food Resources	धनों में सुधार

📘 ADDITIONAL SECTIONS (अतिरिक्त अनुभाग)

📖 Section	Title (English + Hindi)
🔹	Glossary / पारिभाषिक शब्दावली
🔹	Answer Keys / उत्तरमाला
🔹	Summary / सारांश
🔹	Important Diagrams / महत्वपूर्ण चित्र
🔹	NCERT Exercises / एनसीईआरटी अभ्यास
🔹	Sample Papers & MCQs / मॉडल पेपर व वस्तुनिष्ठ प्रश्न

✅ All content follows the official NCERT Class 9 Science textbook structure. Each chapter will include explanations, bilingual notes, definitions, diagrams, MCQs, FAQs, and practice exercises under the BASTRONAUTWAY learning system.