Single-Variable Calculus

Limits, derivatives, integrals and series — built in the right order

About this course

Calculus of one variable developed from the limit: differentiation and its uses, integration and the fundamental theorem, and infinite series.

What you will be able to do

  • State and use the limit definition, and compute limits rigorously
  • Differentiate fluently and apply derivatives to optimisation and rates
  • Explain and use the Fundamental Theorem of Calculus
  • Evaluate integrals and apply them to areas, volumes and arc length
  • Determine convergence of sequences and series, and build Taylor series

What this course covers

  • Series as limit of partial sums
  • Geometric series sum formula
  • Divergence test and the harmonic series
  • Grandi's series and the danger of assuming convergence
  • Comparison test and p-series convergence boundary
  • Telescoping series worked example
  • Kempner series and tail behaviour
  • Extreme values: candidates and existence
  • The Mean Value Theorem: bridge from local to global
  • Corollaries of the MVT: constant, increasing, decreasing functions
  • First and second derivative tests for shape
  • Inflection points: necessary vs. sufficient condition
  • Curve sketching as a systematic checklist
  • Capstone: the MVT as the one bridge — lecture synthesis
  • The slice-approximate-sum method
  • Area between two curves
  • Volumes by disk and washer slicing
  • Volumes by cylindrical shells
  • Rohan's staircase and why length needs Pythagoras
  • Arc length formula
  • Worked arc length example and closing synthesis
  • The two founding questions of calculus
  • The derivative as a limit of a difference quotient
  • The definite integral as a limit of Riemann sums
  • Precise meaning of a limit
  • Local linearity of smooth curves
  • The Fundamental Theorem of Calculus
  • End-to-end worked example and course map
  • Riemann Sums and the Squeeze Definition
  • The Definite Integral and Signed Area
  • Average Value of a Function
  • Why the Sample Point Does Not Matter
  • Preview of the Fundamental Theorem of Calculus
  • Computing an Integral from the Definition
  • The Epsilon-Delta Definition of a Limit
  • One-Sided Limits and When Two-Sided Limits Fail
  • Continuity and Its Three Failure Modes
  • Intermediate Value Theorem
  • Extreme Value Theorem
  • Worked Limit Proof: Two Methods
  • Differential equations as function-finding problems
  • Solving dy/dt = ky by separation of variables and uniqueness
  • Half-life, doubling time, and the rule of seventy
  • Newton's law of cooling via variable shift
  • Why e is the natural base for exponential functions
  • Separable equations as a general technique
  • Limits of exponential models and the fitting-vs-validation distinction
  • Optimization: reducing to one variable
  • Fermat's principle as an optimization
  • Linearization and the tangent-line approximation
  • Differentials and error propagation
  • L'Hôpital's rule
  • Tin-can optimization: optimal shape is a ratio
  • Capstone: derivative as decision and estimate
  • The Fundamental Theorem: Core Idea and Motivation
  • First Fundamental Theorem: Differentiating an Accumulated Area
  • Second Fundamental Theorem: Evaluation via Antiderivatives
  • Definite vs. Indefinite Integrals and the Constant of Integration
  • Net Change Theorem: Integral of a Rate
  • Hypothesis Failure: Integrating Across a Discontinuity
  • Chain Rule with Accumulation Functions and Synthesis
  • Power series as functions of X
  • Radius of convergence and interval of convergence
  • Taylor (Maclaurin) coefficients from derivatives
  • The big three: exponential, sine, and cosine series
  • Two ways Taylor series fail
  • Capstone: local derivatives determine global behaviour
  • Worked example: series for ln(1+x) by integration
  • The Derivative: Definition and Notation
  • Three Ways a Derivative Can Fail
  • Differentiable Implies Continuous (One-Way Arrow)
  • Derivative as a Function; Zeros and Shape Reading
  • The Weierstrass Monster Function
  • First-Principles Worked Example: Derivative of 1/x
  • Three Power Series and Their Coefficient Patterns
  • Euler's Formula Derived by Substituting i into the Exponential Series
  • Geometric Meaning: Exponential as Unit-Circle Rotation
  • What Euler's Formula Explains: Derivatives, Pythagoras, Angle Addition
  • Extending Definitions Beyond Their Original Domain
  • The Limit as the Single Idea Underlying All of Calculus
  • Capstone Verification and Closing Summary
  • Integration as reverse differentiation
  • Substitution (u-substitution)
  • Substitution in definite integrals
  • Integration by parts
  • Strategy: choosing the right technique
  • Worked example: integrating ln x by parts
  • Gabriel's Horn: finite volume, infinite surface
  • Trigonometric integrals: lowering the power
  • Trigonometric substitution: a root as a triangle
  • Partial fractions: adding fractions backwards
  • Necessary vs. sufficient conditions for convergence
  • Improper integrals: definition as a limit
  • Comparison test and the 1/xᵖ benchmark
  • The Power Rule from the Binomial Theorem
  • The Product Rule and the Vanishing Corner
  • The Quotient Rule
  • The Chain Rule: Rates Multiply
  • Derivatives of Sine, Cosine, and the Exponential
  • The Unified Toolkit and Clause Four
  • Explicit vs. implicit relations
  • Implicit differentiation
  • Derivatives of inverse functions
  • Implicit function theorem — local existence of Y(x)
  • Related rates — ladder problem
  • Related rates — balloon and four-step procedure
  • Folium of Descartes — capstone worked example

Syllabus

  1. Limits and the Derivative

    1. L1 · What Calculus Is For
    2. L2 · What a Limit Actually Is
    3. L3 · The Derivative, Properly
    4. L4 · The Rules, Earned
    5. L5 · Tangled Together

  2. Using the Derivative

    1. L6 · What the Derivative Tells You
    2. L7 · The Best You Can Do

  3. The Integral and the Fundamental Theorem

    1. L8 · Infinitely Many Nothings
    2. L9 · Why It Is One Subject
    3. L10 · Reading the Rules Backwards
    4. L11 · Integrals That Run to Infinity

  4. Applications

    1. L12 · Areas, Volumes and Arc Length
    2. L13 · Growth, Decay and a First Differential Equation

  5. Sequences and Series

    1. L14 · Sequences and Infinite Series
    2. L15 · Power Series and Taylor Series
    3. L16 · The Infinite Made Finite