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Unformatted text preview: A PPLYING T ECHNOLOGY AND T OOLS OF C ALCULUS Properties of Graphs we are Interested In • X-intercepts • Y-intercepts • Relative extrema • Inflection points • Symmetries • Periodicity • Intervals of increase and decrease • Concavity • Asymptotes • Behaviour as ± ∞ → x Polynomials : Domain is R Continuous everywhere Differentiable everywhere no sharp corners or vertical tangents As ± ∞ → x , ± ∞ → ) ( x f At most: n x-intercepts; n-1 relative extrema; n-2 inflections points Multiplicity: A root, r x = , of a polynomial ) ( x p has multiplicity m if m r x ) (- divides ) ( x p but 1 ) ( +- m r x does not. A root of multiplicity 1 is a simple root. Even multiplicity Roots of odd multiplicity Simple Roots- tangent to x-axis- inflection points- no inflection point- no inflection point- tangent to x-axis- not tangent- does not cross x-axis- crosses x-axis- crosses x-axis Ex. 1 : 2 3 ) 1 )( 2 ( 2 3- + = +- = x x x x y We want exact locations of intercepts, relative extrema, inflection points.We want exact locations of intercepts, relative extrema, inflection points....
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