Calculus I — Complete Handwritten Notes
Carefully written by Arif Hussain — covering every essential concept from limits to integration.
We are excited to share the complete Calculus I handwritten notes authored by Arif Hussain, available as a downloadable PDF on RanaMaths.com. These notes have been crafted with clarity and depth, making them ideal for university students studying calculus for the first time or anyone seeking a thorough review.
The content follows the structure of the internationally acclaimed textbook Calculus by George B. Thomas Jr. (13th Edition), which is the recommended reference for this course. The notes span all major areas — from the foundational ideas of functions and limits, through differential calculus, and into integral calculus and its real-world applications.
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Chapter 1
Functions, Domain and Range
Every calculus course begins with a solid understanding of functions — the language through which mathematics describes relationships between quantities. Arif Hussain’s notes open with a careful, well-illustrated treatment of this foundational topic.
What is a Function?
A function is a rule that assigns exactly one output to each valid input. This one-to-one (or many-to-one) correspondence is what distinguishes a function from a general relation. The notes cover different ways to represent a function — algebraically, graphically, numerically, and verbally.
Domain and Range
The domain is the complete set of valid inputs (x-values) for which a function is defined. The range is the corresponding set of outputs (y-values). Students learn to identify domain restrictions caused by:
- Division by zero (denominators equal to zero must be excluded)
- Square roots of negative numbers (radicand must be ≥ 0 over the reals)
- Logarithm of non-positive numbers
The notes also cover composite functions, piecewise functions, and the classification of functions (polynomial, rational, trigonometric, exponential, and logarithmic).
Chapter 2
Limits and Continuity
Limits are the cornerstone of calculus. They describe the behaviour of a function as its input approaches a particular value, and they underpin every concept from derivatives to integrals. Arif Hussain’s notes present limits with rigorous intuition and accessible examples.
Introduction to Limits
Understanding what it means for a function to approach a value. The notes cover one-sided limits (left and right-hand limits) and the formal definition using the ε–δ approach.
Techniques of Finding Limits
Step-by-step methods for evaluating limits:
- Direct substitution
- Factoring and cancellation
- Rationalisation of surds
- Limit laws and standard results
- Limits at infinity
Indeterminate Forms
Handling the tricky cases where direct substitution fails, yielding forms such as:
- 0/0 — most common; solved by algebra or L’Hôpital’s Rule
- ∞/∞ — divide by highest power
- 0 · ∞, ∞ − ∞, 0⁰, ∞⁰, 1^∞ — convert and apply techniques
Continuity of Functions
A function is continuous at a point a if three conditions hold:
- f(a) is defined
- limx→a f(x) exists
- limx→a f(x) = f(a)
The notes distinguish between removable, jump, and infinite discontinuities.
Chapter 3
Differential Calculus
Differentiation is the mathematical tool for measuring how a quantity changes. Arif Hussain’s notes present differentiation from its geometric motivation all the way to sophisticated techniques, ensuring a deep conceptual and computational understanding.
Concept and Idea of Differentiation
The derivative of a function f(x) at a point measures the instantaneous rate of change — the slope of the tangent line to the graph at that point. Starting from the limit definition:
The notes build intuition before introducing formal notation (Leibniz, Lagrange, and Newton notations are all covered).
Geometrical and Physical Meaning of Derivatives
- Geometrical: The derivative is the slope of the tangent line at a point on a curve.
- Physical: If s(t) is position, then s'(t) = velocity and s”(t) = acceleration. These real-world interpretations are explored with worked examples.
Rules of Differentiation
- Constant Rule: d/dx [c] = 0
- Power Rule: d/dx [xⁿ] = nxⁿ⁻¹
- Sum / Difference Rule
- Constant Multiple Rule
- Product Rule: (uv)’ = u’v + uv’
- Quotient Rule: (u/v)’ = (u’v − uv’) / v²
Techniques of Differentiation
- Derivatives of trig, inverse trig functions
- Derivatives of exponential functions (eˣ, aˣ)
- Derivatives of logarithmic functions (ln x, loga x)
- Logarithmic differentiation for complex products
Chain Rule
Used to differentiate composite functions. If y = f(g(x)), then:
Worked examples include nested compositions and trigonometric composites.
Implicit Differentiation
When y is not isolated, differentiate both sides with respect to x treating y as a function of x, then solve for dy/dx. Essential for circles, ellipses, and more.
Rates of Change
The derivative as a rate of change is applied to physics, economics, and biology. Topics include related rates — problems where multiple quantities change with time and are connected through an equation.
Tangent and Normal Lines
Finding equations of lines tangent and normal (perpendicular) to a curve at a given point using the point-slope form, with the derivative providing the slope of the tangent.
Linear Approximation (Linearisation)
Near a point, a differentiable function is approximately linear. The linearisation of f at x = a is:
This is used in physics and engineering to approximate complicated functions with simpler linear ones.
Chapter 4
Applications of Differentiation
With differentiation mastered, the notes turn to its powerful applications — using derivatives to reveal the shape, behaviour, and optimum values of functions.
Extreme Value Functions (Extrema)
The Extreme Value Theorem guarantees that a continuous function on a closed interval attains both a maximum and a minimum. The notes distinguish between:
- Absolute (global) extrema — the highest and lowest values on the entire interval
- Local (relative) extrema — peaks and valleys in smaller neighbourhoods
Mean Value Theorems
Two key results with far-reaching implications:
- Rolle’s Theorem: If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then ∃ c ∈ (a,b) with f'(c) = 0.
- Mean Value Theorem: Guarantees a point c where the instantaneous rate equals the average rate over [a,b].
Maxima and Minima
Techniques to find and classify critical points:
- First Derivative Test — sign change of f'(x)
- Second Derivative Test — f”(c) > 0 (min) or f”(c) < 0 (max)
- Closed interval method for absolute extrema
Concavity and Inflection Points
The second derivative reveals the shape of a curve:
- f”(x) > 0 ⟹ curve is concave up (bowl-shaped)
- f”(x) < 0 ⟹ curve is concave down (dome-shaped)
- Inflection points occur where concavity changes
Chapter 5
Integral Calculus
Integration is the reverse of differentiation and also the mathematical tool for computing areas, volumes, and accumulated quantities. The notes present integration with rigorous foundations and rich applications.
Concept and Idea of Integration
Integration started as the problem of finding areas under curves. The key insight is that an integral is a limit of sums — adding up infinitely many infinitely thin rectangles. The notes connect this intuition to the formal definition.
Indefinite Integrals
The antiderivative of f(x) is a function F(x) such that F'(x) = f(x). The general indefinite integral includes the constant of integration C:
Standard integrals of polynomial, trigonometric, exponential, and logarithmic functions are all tabulated and explained.
Techniques of Integration
- Substitution (u-substitution): Reverse of the chain rule
- Integration by Parts: ∫u dv = uv − ∫v du
- Partial Fractions: For rational functions
- Trigonometric substitutions
- Trigonometric identities to simplify integrands
Riemann Sums
The definite integral is defined as the limit of Riemann sums:
Left, right, and midpoint sums are all covered, with geometric interpretations.
Definite Integrals & FTC
The Fundamental Theorem of Calculus bridges differentiation and integration:
- Part I: Differentiation of an integral
- Part II: Evaluation via antiderivatives
Improper Integrals
Integrals over infinite intervals or with unbounded integrands are evaluated using limits:
Convergence and divergence are analysed with comparison tests.
Area Under a Curve
The definite integral computes the signed area between f(x) and the x-axis. The notes cover:
- Area between two curves
- Dealing with regions below the x-axis
- Net area vs. total area
Reference Textbook
Recommended Book
Calculus — George B. Thomas Jr.
This textbook is the gold standard for undergraduate calculus worldwide. It offers exceptionally clear explanations, thousands of practice problems, and real-world applications. Arif Hussain’s handwritten notes are structured to align closely with its chapters, making it easy to cross-reference.
13th EditionAbout the Author