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Let where is shown at right.

  1. Evaluate and .

  2. Is ?

    Consider the symmetry of between and .

  3. Express in terms of .

    What is ? What is ?

  4. Is differentiable over the interval ? Explain.

    Points of NON-differentiability include cusps, endpoints, jumps, holes, and vertical tangents.

  5. Determine all values of in the interval where has a relative maximum.

    Recall that a local maximum exists where the derivative changes from positive to negative. This can happen where or where .

    Notice that ; after all, the derivative of an integral is the original function.

  6. Write the equation of the line tangent to at .

    What is the slope of  at (see second hint in part (e))? What is the -value?

  7. Determine all values of in the interval where has a point of inflection.

    Concavity is the slope of the slope. So inflection points are where the slope of the slope changes.

Continuous Piecewise labeled f of x, left segment from (negative 2, comma 0), to (0, comma 2), center segment from (0, comma 2) to (2, comma 0), right semicircle, vertices at the points (2, comma 0), (4, comma negative 2), & (6, comma 0).