121616949-math.42 - 28 Chapter 2 Instantaneous Rate Of...

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28 Chapter 2 Instantaneous Rate Of Change: The Derivative that - 19 . 6 - 4 . x > - 19 . 600001. This is something we can manipulate with a little algebra: - 19 . 6 - 4 . x > - 19 . 600001 - 4 . x > - 0 . 000001 Δ x < - 0 . 000001 / - 4 . 9 Δ x < 0 . 0000002040816327 . . . Thus, we can say with certainty that if Δ x is positive and less than 0 . 0000002, then Δ x < 0 . 0000002040816327 . . . and so - 19 . 6 - 4 . x > - 19 . 600001. We could do a similar calculation if Δ x is negative. So now we know that we can make - 19 . 6 - 4 . x within one millionth of - 19 . 6. But can we make it “as close as we want”? In this case, it is quite simple to see that the answer is yes, by modifying the calculation we’ve just done. It may be helpful to think of this as a game. I claim that I can make - 19 . 6 - 4 . x as close as you desire to - 19 . 6 by making Δ x “close enough” to zero. So the game is: you give me a number, like 10 - 6 , and I have to come up with a number representing how close Δ x must be to zero to guarantee that - 19 . 6 - 4 . x is at least as close to - 19 . 6 as you have requested.
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