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**Unformatted text preview: **MATH 8, SECTION 1, WEEK 5 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Wednesday, Oct. 27th’s lecture, where we studied how differentiability and extrema are related; also C 1 functions! 1. Random Question Question 1.1. Suppose you have a N × N grid of 1 × 1 squares. Consider the following game you can play on this board: • Starting configuration: put one coin on the square in the bottom-right-hand corner of our board. • Moves: If we ever have a coin such that the square immediately above this coin and the square immediately to the right of this coin are empty, we can remove this coin from the board, and put one new coin to the north and one new coin to the east of this square. Given this setup, the question is this: Is there some finite sequence of moves that can empty the highlighted green region of all its coins? 2. Interpretations of the Derivative Last class, we defined the derivative as follows: Definition 2.1. For a function f differentiable at a point a , we say that f ( a ) = lim h → f ( a + h )- f ( a ) ( a + h )- a . There are a pair of useful ways to interpret this limit: • If we think of t as a measurement of time and f ( t ) as a function that outputs a measurement of distance, f ( t ) is measuring the change in distance over the change in time – in other words, the velocity of the distance function f . Similarly, f 00 ( t ) is measuring the change in velocity over the change in distance: i.e. the acceleration . 1 2 TA: PADRAIC BARTLETT • If we think of f ( x ) as a function that outputs the height at location x , f ( t ) is measuring the change in height over the change in x-position – in other...

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