Exam theorems¶
The list of theoretical results of the course that you must be able to state and prove. Next to each one, the chapter of the notes where the proof is given.
How to study them
For each theorem: read the statement and try to rebuild the proof on your own; only then open the «Proof» box in the chapter and compare.
Numbers¶
- Formula for the sum of the first \(n\) terms of a geometric progression — Summations and geometric progressions
- Bernoulli's inequality — Principle of induction
- Newton's binomial formula — Factorials, binomial coefficients and triangle inequality
- Triangle inequality for real and complex numbers — Factorials, binomial coefficients and triangle inequality, Complex numbers
- \(\R\) is not countable — Cardinality of infinite sets
- Strictly monotone functions are invertible — Inverse functions
Sequences¶
- Uniqueness of the limit — Sequences and limits of sequences
- Monotone convergence theorem — Sequences and limits of sequences
- Algebra of limits (finite limits) — Computing limits of sequences
- Sign-preservation theorems — Sequences and limits of sequences
- Comparison theorems — Computing limits of sequences
- Convergence of Euler's sequence — Euler's number
- Asymptotic behavior theorem — Comparisons and asymptotic estimates
- Ratio test and applications to the hierarchy of infinities — Hierarchies of infinities and ratio test
- Convergence of Heron's sequence — Recursively defined sequences
- Error estimate for Heron's algorithm — Recursively defined sequences
Limits of functions¶
- Algebra of limits (finite limits) — Computing limits of functions
- Sign preservation — Limits of functions, asymptotes and continuity
- Comparison theorems — Computing limits of functions
- Algebra of continuous functions — Continuous functions
- Standard limits — Standard limits and asymptotic estimates
- Zeros theorem (Bolzano's theorem) — Zeros theorem and bisection method
- Errors of the bisection method — Zeros theorem and bisection method
- Weierstrass theorem — Weierstrass theorem and intermediate value theorem
- Intermediate value theorem — Weierstrass theorem and intermediate value theorem
- Monotonicity of functions — Monotone functions on an interval and invertibility
Derivatives¶
- Derivatives of elementary functions — Derivatives of elementary functions
- A differentiable function is continuous — The derivative
- Algebra of derivatives — Rules of differentiation
- Derivative of a composite function — Rules of differentiation
- Derivative of the inverse function — Rules of differentiation
- Fermat's theorem — Mean value theorem, maxima and minima
- Lagrange's (mean value) theorem — Mean value theorem, maxima and minima
- Monotonicity test — Mean value theorem, maxima and minima
- L'Hôpital's rule — L'Hôpital's rule and differentiability
- Limit of the derivative theorem — L'Hôpital's rule and differentiability
- Maclaurin formula with Peano remainder — Differential calculus and approximations
- Error estimate for Newton's method — Newton's method
Numerical series¶
- If a series converges, its sequence of remainders tends to \(0\) — Numerical series
- Comparison test — Series with non-negative terms
- Limit comparison test — Series with non-negative terms
- Ratio test — Series with non-negative terms
- Root test — Series with non-negative terms
Integrals and differential equations¶
In preparation
The chapters on integrals and differential equations are not yet part of the online notes. The exam results for these parts are: mean value theorem for integrals; fundamental theorem of calculus and its corollary; formulas for computing indefinite integrals by substitution and by parts; general solution of a first-order linear homogeneous differential equation; structure of the general solution of a first-order linear non-homogeneous differential equation; formula for a solution of a first-order linear non-homogeneous differential equation.