B find the dimensions of the box of maximum volume

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Unformatted text preview: d11 (written x → ∞). For example, given f (x) = x2 , as x → −∞, we imagine substituting x = −100, x = −1000, etc., into f to get f (−100) = 10000, f (−1000) = 1000000, and so on. Thus the function values are becoming larger and larger positive numbers (without bound). To describe this behavior, we write: as x → −∞, f (x) → ∞. If we study the behavior of f as x → ∞, we see that in this case, too, f (x) → ∞. The same can be said for any function of the form f (x) = xn where n is an even natural number. If we generalize just a bit to include vertical scalings and reflections across the x-axis,12 we have 7 Make sure you choose some x-values between −1 and 1. Herein lies one of the possible origins of the term ‘even’ when applied to functions. 9 Of course, there are no ends to the x-axis. 10 We think of x as becoming a very large negative number far to the left of zero. 11 We think of x as moving far to the right of zero and becoming a very large positive number. 12 See Theorems 1.4 and 1.5 in Section 1...
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