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Let and .

  1. Estimate f(−3.1).

    Do NOT find an function. That would be tedious and unnecessary work.

    This is another way of wording a very familiar problem:
    'Given and a point on , find the tangent line to at and use it to estimate the value of .'

  2. Find .

    Split the absolute value into a piecewise function and differentiate.
    Or, you could recall a special note about the antiderivative of : While you can generally leave it as , it actually equals .

  3. Use concavity to determine if your answer to part (a) is an under or over estimate of the actual value of . Justify your answer.

    If a function is concave down, all tangent lines will be above the curve.
    If a function is concave up, all tangent lines will be below the curve. Sketch this to verify.