Elementary Functions and Calculus I Math 131 (Sec 42), Autumn 2004 Practice Mid-term 1

1. Express the repeating decimal 0.69\overline{69} as a fraction. Show all of your working.

2. Sketch the points (x, y) which satisfy the equation 3x^{2}+4y^{2} = 1. What is the name of the shape it describes?

3. Use only the basic properties of the real numbers (commutivity, associativity, distributivity, identity elements and inverses) and of the order relation < (trichotomy, transitivity, addition and multiplication) to prove that if a < b then a < \frac{a+b}{2} < b.

4. State the ε − δ definition of a limit of a function f at a point c.

5. Prove that

lim_{x→−2} \frac{x^{2} − 9}{3 − x} = −1

using the ε − δ definition.

6. Prove that

lim_{x→3}(5x + 9) = 24

using the ε − δ definition.

7. Prove or give a counter-example to the following statement: If two real numbers x and y satisfy x ≤ y then ax ≤ ay for any real number a.