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Concavity definitions

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  • Concavity
    Describes the way a graph curves, indicating whether it bends upward like a smile or downward like a frown.
  • Second Derivative
    Represents the rate of change of the slope, revealing how the slope itself increases or decreases.
  • Concave Up
    A graph shape resembling a smile, where tangent lines lie below the curve and the second derivative is positive.
  • Concave Down
    A graph shape resembling a frown, where tangent lines sit above the curve and the second derivative is negative.
  • Inflection Point
    A location where the graph changes from concave up to concave down or vice versa, with the second derivative zero or undefined.
  • Tangent Line
    A straight line touching a curve at one point, used to visualize slope and its changes on a graph.
  • Slope
    A measure of steepness at a point on a graph, determined by the first derivative.
  • Sign Chart
    A tool for organizing intervals and test values to determine where the second derivative is positive or negative.
  • Interval
    A continuous range of x-values used to describe where a function is concave up or down.
  • Critical Point
    A location where the first derivative is zero or undefined, often associated with local maxima or minima.
  • Power Rule
    A differentiation shortcut for finding derivatives of terms with exponents, essential for computing higher derivatives.
  • Test Value
    A chosen x-value within an interval, plugged into the second derivative to determine its sign.
  • Ordered Pair
    A set of x and y values representing a specific point on a graph, such as an inflection point.
  • Rate of Change
    A description of how a quantity varies with respect to another, with the second derivative showing changes in slope.
  • Derivative
    A function describing the instantaneous rate of change, with the first derivative for slope and the second for concavity.