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Consider the equation .

  1. What is ?

    Implicitly differentiate.

  2. Write the equation of the line tangent to the curve at .

    Notice that the derivative you found in part (a) has both - and -values.

  3. If , estimate using the tangent line.

    Since , it is easier to evaluate in a linear equation than in the curve shown above. Use the tangent line to estimate the -value at .

  4. Using , determine if the tangent line approximation is an overestimate or an underestimate. Justify your answer in words.

    The sign of the 2nd derivative at will determine if the tangent line is above or below the curve.

    When a function is concave up, the tangent line will be below the curve.
    When it is concave down, the tangent line will be above the curve.
    Verify these statements by sketching a few examples.