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03Applying Technology and Tools of Calculus

03Applying Technology and Tools of Calculus - APPLYING...

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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.
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