📘 Calculus: GATE Engineering Mathematics – Section 2
📘 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
🔍 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.
📐 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!
📈 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)
🧮 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.
📉 4. Maxima and Minima
To find extrema:
- Compute f′(x) and solve f′(x) = 0
- Use f″(x):
- If f″(x) > 0 → local minima
- If f″(x) < 0 → local maxima
✨ Trick: Maxima/minima without boundary check = dangerous in GATE!
∫ 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!)
📊 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
🎯 50 Most Important Questions – GATE Level
Here are sample practice questions curated with GATE trends in mind:
- Evaluate lim x→0 (sin x) / x
- Prove continuity of f(x) = |x| at x = 0
- Use MVT for f(x) = x² + 3x in [1,3]
- Expand eˣ using Maclaurin series up to x⁴
- Find local maxima of f(x) = x³ − 6x² + 9x
- Evaluate ∫₀^π sin²x dx
- Area under f(x) = |x| from −1 to 1
- ∫ x log x dx using integration by parts
- Show f(x) = x³ is continuous and differentiable
- 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
💫 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
📌 Hashtags:
#GATECalculus #Section2 #GrandMasterBikramSutradhar #BestTeacherIndia #bAstronautWay #GATE2026Prep #EngineeringMathematics #IITJAM #CalculusGATE #sirBikramSutradhar #MathsWithPurpose
Written By
Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar
bAstronautWay : A Government-Approved Trademark Brand
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Written By
Full Stack Developer and 5-Time World Record Holder, Grandmaster Bikram Sutradhar
bAstronautWay
SirBikramSutradhar on YouTube
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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 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 11. Areas Related to Circles
chapter 12. Surface Areas and Volumes
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.
