M1410202 - 2.10 SECTION 2.2: POLYNOMIAL FUNCTIONS OF HIGHER...

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2.10 SECTION 2.2: POLYNOMIAL FUNCTIONS OF HIGHER DEGREE PART A: INFINITY The Harper Collins Dictionary of Mathematics defines infinity , denoted by , as “a value greater than any computable value.” The term “value” may be questionable! Likewise, negative infinity, denoted by −∞ , is a value lesser than any computable value. Warning: and −∞ are not numbers. They are more conceptual. We sometimes use the idea of a “point at infinity” in graphical settings. PART B: LIMITS The concept of a limit is arguably the key foundation of calculus. (It is the key topic of Chapter 2 in the Calculus I: Math 150 textbook at Mesa .) Example lim x →∞ f x ( ) = −∞ ” is read “the limit of f x ( ) as x approaches infinity is negative infinity.” It can be rewritten as: f x ( ) → −∞ as x →∞ ,” which is read “ f x ( ) approaches negative infinity as x approaches infinity.” The Examples in Part C will help us understand these ideas!
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2.11 PART C: BOWLS AND SNAKES Let a represent a nonzero real number. Recall that the graphs for ax 2 , ax 4 , ax 6 , ax 8 , etc. are “bowls.” If a > 0 , then the bowls open upward . If a < 0 , then the bowls open downward . Examples The graph of f x ( ) = x 2 is on the left, and the graph of g x ( ) = x 2 is on the right. lim x →∞ f x ( ) = lim x →−∞ f x ( ) = lim x →∞ g x ( ) = −∞ lim x →−∞ g x ( ) = −∞
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2.12 Let a represent a nonzero real number. Recall that the graphs for ax 3 , ax 5 , ax 7 , ax 9 , etc. are “snakes.” If a > 0 , then the snakes rise from left to right. If a < 0 , then the snakes fall from left to right. Examples The graph of f x ( ) = x 3 is on the left, and the graph of g x ( ) = x 3 is on the right. lim x →∞ f x ( ) = lim x →−∞ f x ( ) = −∞ lim x →∞ g x ( ) = −∞ lim x →−∞ g x ( ) =
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2.13 PART D: THE “ZOOM OUT” DOMINANCE PROPERTY Example The graph of f x ( ) = 4 x 3 5 x 2 7 x + 2 is below. Observe that 4 x 3 is the leading (i.e., highest-degree) term of f x ( ) . If we “zoom out,” we see that the graph looks similar to the graph for 4 x 3 (in blue below).
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2.14 To determine the “long run” behavior of the graph of f x ( ) as x →∞ and as x → −∞ , it is sufficient to consider the graph for the leading term. (See p.123 .) Even if we don’t know the graph of f x ( ) , we do know that the graph for 4 x 3 is a rising snake (in particular, a “stretched” version of the graph for x 3 ). We can conclude that: lim x →∞ f x ( ) = lim x →−∞ f x ( ) = −∞ “Zoom Out” Dominance Property of Leading Terms The leading term of a polynomial
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M1410202 - 2.10 SECTION 2.2: POLYNOMIAL FUNCTIONS OF HIGHER...

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