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The First Derivative Test definitions

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  • Derivative
    A measure indicating how a function's output changes as its input changes, often interpreted as the slope of the tangent line.
  • Critical Point
    A value where the derivative is zero or undefined, serving as a boundary for intervals of increasing or decreasing behavior.
  • Sign Chart
    A number line divided by critical points, used to test the sign of the derivative in each interval for function analysis.
  • Interval
    A continuous range of x-values, often separated by critical points, over which function behavior is examined.
  • Local Maximum
    A point where a function changes from increasing to decreasing, creating a peak relative to nearby values.
  • Local Minimum
    A point where a function changes from decreasing to increasing, forming a valley compared to surrounding values.
  • First Derivative Test
    A method using sign changes of the derivative around critical points to identify local maxima and minima.
  • Increasing Interval
    A range where the derivative is positive, indicating the function's output rises as input increases.
  • Decreasing Interval
    A range where the derivative is negative, showing the function's output falls as input increases.
  • Test Value
    A chosen x-value within an interval, plugged into the derivative to determine the sign and thus the function's behavior.
  • Tangent Line
    A straight line touching a curve at a single point, with a slope equal to the function's derivative at that point.
  • Slope
    A numerical value representing the steepness of a tangent line, directly linked to the derivative.
  • Polynomial Function
    A function composed of terms with non-negative integer exponents, often used in derivative and critical point examples.
  • Open Interval
    A range of values that does not include its endpoints, typically used when analyzing intervals between critical points.
  • Local Extrema
    Collective term for local maxima and minima, representing peaks and valleys in a function's graph.