MATH 350, Review for Midterm Test 2
Test topics
- Topology of the real numbers
- Open and closed sets (3.1)
- Compactness (3.1)
- Limits and continuity
- Limit of a function (4.1)
- Definition of continuity. Continuity on a set (4.1)
- Properties of continuous functions (4.1)
- Uniform continuity (4.1)
- Differentiation
- The derivative of a function (5.1)
- Mean value theorems (5.2)
- Taylor's formula (5.2)
Important definitions/facts
- Open and closed sets; compact sets
- Limit points, boundary points, interior points, closure
- The Heine-Borel theorem
- Limits, continuity
- Global and local extrema
- Intermediate value property
- Uniformly continous functions
- Definitions of the derivative
- Generalized mean value thorem
- Taylor's formula and Lagrange's form of the remainder
Theorems that you need to know how to prove
- Basic properties of open and closed sets (Theorems 3.1-3.3)
- Properties of compact sets (Theorems 3.12 and 3.13)
- Sequential definition of limits (Theorem 4.1)
- Algebraic operations on continous functions(Theorem 4.3)
- Extreme value theorem (Corollary 4.7(b))
- Intermediate value theorem (Theorem 4.9 and Corollary 4.9)
- Differentialble function is continuous (Theorem 5.2)
- Derivatives and algebraic operations (Theorem 5.3)
- Condition on relative extremum (Theorem 5.6)
- Rolle's theorem, Mean-value theorem (Theorems 5.7, 5.8)
General rules
- All tests are closed books/notes; graphing calculators, cell
phones or laptops are not permitted. A basic scientific calculator
is OK (example: TI 30X IIS or similar) but it is not required to
answer the questions.
- The length of the test is 1 hr 15 minutes.
- Problems will come in the same format as on the quizzes:
the expectation is that you write a clear and concise proof
for the statement given in the question.