6 making and selling 71 portaboys yields a maximized

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Unformatted text preview: nd closer to 5 4x3 0.00000025 −0.00000000125 0.000025 −0.00000125 0.0025 −0.00125 0.0025 0.00125 0.000025 0.00000125 0.00000025 0.00000000125 In other words, as x → ±∞, f (x) ≈ 4x3 (1 − 0 + 0) = 4x3 , which is the leading term of f . The formal proof of Theorem 3.2 works in much the same way. Factoring out the leading term leaves f (x) = an xn 1 + an−1 a2 a1 a0 + ... + + + n−2 n−1 an x an x an x an xn As x → ±∞, any term with an x in the denominator becomes closer and closer to 0, and we have f (x) ≈ an xn . Geometrically, Theorem 3.2 says that if we graph y = f (x), say, using a graphing calculator, and continue to ‘zoom out,’ the graph of it and its leading term become indistinguishable. Below are the graphs of y = 4x3 − x + 5 (the thicker line) and y = 4x3 (the thinner line) in two different windows. A view ‘close’ to the origin. A ‘zoomed out’ view. Let’s return to the function in Example 3.1.5, f (x) = x3 (x − 3)2 (x +2) x2 + 1 , whose sign diagram and graph are reproduced below for reference. Theorem 3.2 tells us that the end behavior is the same as that of its leading term, x8 . This tells us that the gr...
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