Mathematics 201: Final Exam Objectives
The final exam will be comprehensive with an emphasis on topics
covered since the 2nd midterm exam: mathematical induction and
the theory of the integral.
This means that the objectives for the final exam include all
objectives from both the first and
the second midterm exams, and especially
those that are germane to proof by induction or the theory of
integration.
Here are the objectives related to the last 5 weeks of the course.
Axioms
State the axioms, and use them in proofs.
- Completeness property of R
- Principle of mathematical induction
- Well-ordering principle for N
Definitions
State the definitions, and use them in proofs.
- Real number system R
- Inductive set
- Recursive definition, recurrence relation
- Fibonacci sequence
- Geometric sequence, geometric series
- Binomial coefficient
- Bounded function
- Partition, refinement
- Lower sum, upper sum
- Lower integral, upper integral, integral, integrable function
- Monotone function
- Lipschitz condition, Lipschitz function
Theorems
State the theorems, and use them in proofs.
- Cardinality of a power set
- Properties of binomial coefficients
- Pascal triangle recursion
- Factorial formula
- Subset counting
- N is not bounded above in R; Archimedean property
- Partition refinement inequalities
- Integrability criterion
- Integrability of monotone and Lipschitz functions
- Linearity of the integral
- Existence of a bounded non-integrable function
