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  1. Graph on graph paper. Then without a calculator, sketch on the same set of axes.

    Black continuous curve, labeled y = x raised to the 2 thirds power, coming from upper left, concave down, turning at the origin, continuing up & right, still concave down, & gray curves, labeled d y divided by d x, left curve coming from left below x axis, continuing to negative infinity left of y axis, right curve coming from infinity right of y axis, continuing to right above x axis.

  2. Describe the graphs of and at .

  3. Use your graphing calculator to determine the slope of at the vertex.

    Though the derivative does not exist, some graphing calculators (inaccurately) say that the derivative is at a cusp.

  4. Calculators are not always accurate. Some graphing calculators incorrectly determine slopes at a vertex, as well as other cusps, because they use the symmetric difference quotient to calculate the slope of a tangent line.

    For , use , to calculate for and . What do you notice? What leads a calculator to give a false derivative of  at ?

Compute without a calculator