MAT118 Review for Midterm I

Material from Week 1

Be able to explain the Monty Hall game in your own words. See B&S, page 9 and page 23. Be able to explain the non-standard dice game (page 10 and page 24). Be able to solve a problem like 13 on page 31 and 38.

Material from Week 2

Be able to prove that $$1 + a + a^2 + a^3 + \cdots +a^n = \frac{\displaystyle 1-a^{n+1}}{\displaystyle{1-a}}$$ and understand why, if -1 < a < 1, then the term $a^{n+1} \rightarrow 0$ as $n\rightarrow \infty$.
Be able to use the formula $$1 + a + a^2 + a^3 + \cdots = \frac{\displaystyle 1}{\displaystyle 1-a}$$ when -1 < a < 1 to evaluate geometric series like $$1 + 2/3 + (2/3)^2 + (2/3)^3 + \cdots$$ or $$1 -(2/3) + (2/3)^2 - (2/3)^3 + \cdots .$$

Be able to calculate the yellow area in the "Sir Pinsky" carpet (page 15) and the variants in problems 34 and 35, page 490, or the white area in the Sierpinski triangle (page 471; see problem 47 on page 491). Understand how to play the "Chaos game" in a triangle (page 481) and be able to answer questions like 23 and 24 on page 488.

Be able to follow a "repeated replacement" rule (page 469), for example the rules 1, 2, 3, 4 on pages 472-473, to start constructing a fractal. (See the Sierpinski triangle construction on page 476).

Understand why the length of the Koch curve goes to infinity as the subdivisions get repeated (page 470; see problem 48 on page 491).

February 15, 2013