## Test topics

1. Topology of the real numbers
1. Open and closed sets (3.1)
2. Compactness (3.1)

2. Limits and continuity
1. Limit of a function (4.1)
2. Definition of continuity. Continuity on a set (4.1)
3. Properties of continuous functions (4.1)
4. Uniform continuity (4.1)

3. Differentiation
1. The derivative of a function (5.1)
2. Mean value theorems (5.2)
3. 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.