### Home > CALC > Chapter 7 > Lesson 7.1.5 > Problem 7-55

Let

*f*′(*x*) =and *f*(−3) =. Homework Help ✎ Estimate

*f*(−3.1).Find

*f*″(*x*).Use concavity to determine if your answer to part (a) is an under or over estimate of the actual value of

*f*(*x*). Justify your answer.

Do NOT find an *f*(*x*) function. That would be tedious and unnecessary work.

This is another way of wording a very familiar problem:

'Given *f* '(*x*) and a point on *f*(*x*), find the tangent line to *f*(*x*) at *x* = −3 and use it to estimate the value of *f*(−3.1).'

Split the absolute value into a piecewise function and differentiate.

Or, you could recall a special note about the antiderivative of *x*^{−1}: While you can generally leave it as ln(*x*), it actually equals ln|*x*|.

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.