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2311_How_Derivatives_Affect_Shape_of_Graph_4.3_F10

# 2311_How_Derivatives_Affect_Shape_of_Graph_4.3_F10 - f is...

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MAC2311 How Derivatives Affect a Graph; Sections 4.3, 4.5, 4.6 Def. p.274 A critical number of a function f is a number c in the domain of f such that either ( ) 0 f c = or ( ) f c does not exist. Increasing/Decreasing Test p.287 a) If ( ) 0 f x > on an interval, then f is increasing on that interval. b) If ( ) 0 f x < on an interval, then f is decreasing on that interval. The First Derivative Test (for Local Extrema) p.288 Suppose c is a critical number of a continuous function f . a) If f changes sign from positive to negative at c , then f has a local maximum at c . b) If f changes sign from negative to positive at c , then f has a local minimum at c . c) If f does not change sign at c , then f has no a local maximum or minimum at c . Def. p.290 If the graph of f lies above all its tangent lines on an interval I , then f is concave up on I . If the graph of f lies below all its tangent lines on I , then
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Unformatted text preview: f is concave down on I . The Concavity Test p.291 a) If ( ) f x ′′ > for all x in I , then the graph of f is concave up on I . b) If ( ) f x ′′ < for all x in I , then the graph of f is concave down on I . Def. p.291 A point P on a curve ( ) y f x = is an inflection point if f is continuous there and the curve changes concavity at P . The Second Derivative Test (for Local Extrema) p.292 Suppose f ′′ is continuous near c . a) If ( ) f c ′ = and ( ) f c ′′ > , then f has a local minimum at c . b) If ( ) f c ′ = and ( ) f c ′′ < , then f has a local maximum at c . Guidelines for Curve Sketching (pp.308-309) A. Domain B. Intercepts C. Symmetry D. Asymptotes E. Intervals of Increase/Decrease F. Local Maximum or Minimum Values; Terrace Points G. Concavity and Inflection Points...
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