03Applying Technology and Tools of Calculus

03Applying Technology and Tools of Calculus - x-axis- not...

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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
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Unformatted text preview: 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. Ex. 2 : Rational Function ) 4 )( 4 ( ) 2 )( 2 ( 2 ) 4 )( 4 ( ) 4 ( 2 16 8 2 2 2 2-+-+ =-+-=--= x x x x x x x x x y-symmetry: -x-intercepts: -y-intercepts: -vertical asymptotes: -horizontal asymptote: -derivatives: Ex. 3: Radical Function 3 2 ) 4 (-= x y-symmetry: -x-intercepts: -y-intercepts: -vertical asymptotes: -horizontal asymptotes: -derivatives: vertical tangent lines: Practice Question : 3 4 3 1 3 6 x x y + = Assign: p.299 #1-39odds, 62; MM p.387 #8, 9...
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This note was uploaded on 07/12/2011 for the course MATH 241 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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03Applying Technology and Tools of Calculus - x-axis- not...

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