Key takeaways
- Nine short units, from limits to the three vector integral theorems.
- Formula tables for standard limits, derivatives and integrals, plus Wallis' formula.
- Solved examples on maxima-minima, integration by parts and change of order.
- Ten practice questions at the end, in the style of university and GATE papers.
- Original Uncodemy revision notes, not official university notes; match them with your syllabus.
About these handwritten notes
These are original Uncodemy revision notes for Calculus, written in a clean handwritten style so they feel like a topper's notebook rather than a textbook. They are short on purpose: each unit gives the key definitions, formulas, a solved example or two and the points that are easy to forget.
The notes follow the common syllabus that most Indian B.Tech universities use for this subject, including universities that follow the AICTE model curriculum. They are not official notes from any university. Unit names, order and depth differ from one university to another, so check your own university syllabus and mark any topic that is missing before you rely on the notes for an exam. The PDF is written for quick revision, so read your textbook first if a topic is completely new to you.
What's inside the calculus notes
The notes are split into nine units. Here is what each one covers, in one line.
Unit 1: Limits and continuity
What a limit means, left and right limits, continuity, a table of standard limits and L'Hospital's rule for 0/0 and ∞/∞ forms.
Unit 2: Differentiation
Standard derivatives, product, quotient and chain rules, Rolle's, Lagrange's and Cauchy's mean value theorems, and Taylor and Maclaurin series.
Unit 3: Partial derivatives
Partial and total derivatives, Euler's theorem on homogeneous functions and the Jacobian with its key properties.
Unit 4: Maxima and minima
Tests for one variable and two variables (the rt - s² test), a full solved example and the Lagrange multiplier method for constraint problems.
Unit 5: Integration
Standard integrals, properties of definite integrals, Wallis' formula for 0 to π/2, beta and gamma functions and a solved integration-by-parts example.
Unit 6: Applications of integration
Area between curves, volume of revolution about the x-axis and arc length, with a short worked example.
Unit 7: Multiple integrals
Double and triple integrals, change of order of integration, change to polar coordinates and a worked example.
Unit 8: Vector calculus basics
Gradient, divergence, curl, directional derivative and the three integral theorems: Green's, Stokes' and Gauss divergence.
Unit 9: Most asked questions (practice)
Ten practice questions that mix every unit, for a final self-test.
How to revise calculus with these notes
Calculus is a practice subject. Reading the notes once will not get you marks; solving with them will.
- Learn the tables first. Standard limits, derivatives and integrals are the base of every later unit. Write them out from memory until you get them all right.
- One unit per sitting. Read the unit, close the PDF and write the key results on a blank page. Then check what you missed.
- Redo every solved example. Cover the solution, solve it yourself, then compare step by step. In maxima and minima, check that you found all the stationary points.
- Draw before you integrate. For change of order and area problems, sketch the region and mark the strip. Most mistakes come from wrong limits, not from the integration.
- Use symmetry in Lagrange problems. The notes point out that when f and the constraint are symmetric, the answer is often at x = y = z. Check that first, then confirm.
- Check numericals with a tool. After solving a definite integral by hand, you can verify the value with the Simpson's rule calculator.
Exam focus: high-weightage calculus topics
The notes end with a tip based on how papers are usually set. In GATE engineering mathematics, short questions on limits, maxima and minima, definite integrals and gradient or directional derivative come up often. In university exams, change of order of integration and the three vector theorems are common long questions.
Topics worth extra time:
- Mean value theorems: state and verify Rolle's or Lagrange's theorem for a given function on an interval.
- Series expansion: Maclaurin expansion of products such as e^x sin x up to a few terms.
- Euler's theorem: verify it for a homogeneous function, or use it to find x ∂u/∂x + y ∂u/∂y.
- Maxima and minima: functions such as x³ + y³ - 3xy, plus a Lagrange multiplier problem.
- Beta and gamma functions: evaluate integrals of sin^m x cos^n x from 0 to π/2.
- Multiple integrals: change the order of integration and evaluate, usually with an integral that cannot be done in the original order.
- Vector calculus: find div and curl of a field and verify Green's theorem over a simple region.
Typical questions
The practice set in the PDF includes questions like "Evaluate lim (x→0) (1 - cos x)/x² using L'Hospital's rule", "Find the extreme value of x² + y² + z² subject to x + y + z = 3" and "Change the order of integration in ∫₀¹ ∫ₓ¹ e^(y²) dy dx". Treat them as practice, not as a prediction of your paper.
Who these notes are for
- First-year B.Tech students of any branch taking Engineering Mathematics 1 or a calculus paper.
- GATE aspirants revising the calculus part of engineering mathematics, which appears in most GATE papers.
- Data science and machine learning learners who want a quick refresher on derivatives, partial derivatives and optimisation before studying gradient descent.
If you want unit-wise practice questions with answer outlines, use the matching Engineering Mathematics 1 important questions PDF alongside these notes.
Common mistakes to avoid in calculus
- Using L'Hospital's rule on the wrong form. It works only for 0/0 or ∞/∞. Convert forms like 0 x ∞ or 1^∞ first.
- Skipping the conditions of a theorem. For Rolle's or Lagrange's theorem, first show the function is continuous on the closed interval and differentiable on the open interval. Many students lose marks by jumping straight to the formula.
- Missing a stationary point. In two-variable maxima and minima, solve f_x = 0 and f_y = 0 completely and test every point with rt - s².
- Wrong limits after change of order. Always draw the region again and read the new limits from the sketch.
- Mixing up beta and gamma forms. Write the formula for ∫ sin^m x cos^n x dx in terms of the beta function on your formula sheet and copy it exactly.
- Sign errors in curl. Expand the curl determinant slowly, one component at a time.
A simple 5-day revision plan
- Day 1: Units 1 and 2. Limits, L'Hospital's rule, mean value theorems and Taylor series.
- Day 2: Units 3 and 4. Euler's theorem, Jacobians, maxima-minima and Lagrange multipliers.
- Day 3: Units 5 and 6. Integration tables, Wallis' formula, beta-gamma and applications.
- Day 4: Units 7 and 8. Change of order, polar coordinates and the vector theorems.
- Day 5: Solve the ten practice questions in Unit 9 with a timer, then revise only what you got wrong.
Download Calculus Handwritten Notes PDF
Calculus Handwritten Notes PDF
The PDF carries the same content as this page in a clean, print-ready layout, so you can read it offline and keep it for revision.
7 pages · 294 KB · PDF · Updated 5 Nov 2026