Stitz-Zeager_College_Algebra_e-book

6 making and selling 71 portaboys yields a maximized

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online