CHAPTER 12
Mathematical Analysis
Deepen calculus by studying convergence, continuity, the central theorems of differential calculus, Taylor approximation and the rigorous foundations of integration.
MATHEMATICAL ANALYSIS
Why calculus works
Bounds, Intervals and Absolute Value
Build the language used to state precise results in analysis.
Learn →02Sequences and Convergence
Understand what it means for infinitely many terms to approach a limit.
Learn →03Limits More Rigorously
Move from intuitive limits toward the epsilon-delta definition.
Learn →04Continuity
Study continuity as preservation of limiting behaviour.
Learn →05Intermediate Value Theorem
Show why a continuous graph cannot jump over intermediate values.
Learn →06Extreme Value Theorem
Understand why continuous functions on closed bounded intervals attain maxima and minima.
Learn →07Rolle's Theorem
Relate equal endpoint values to a horizontal tangent somewhere inside.
Learn →08Mean Value Theorem
Connect an average rate of change with an instantaneous rate of change.
Learn →09Consequences of the Mean Value Theorem
Derive monotonicity, uniqueness and constancy results from the MVT.
Learn →10Taylor's Theorem
Approximate smooth functions with polynomials and understand the remainder.
Learn →11Fundamental Theorem of Calculus
Study the theorem that rigorously connects accumulation and differentiation.
Learn →12Infinite Series and Convergence
Understand when an infinite sum has a finite value.
Learn →13Analysis View of Calculus
Bring limits, continuity, derivatives, integrals and the main theorems into one logical structure.
Review →The central idea
Analysis explains exactly which assumptions make the familiar rules of calculus valid.