What are we supposed to do with a function like this?

The algebra is beyond us.

The formula is too difficult to manipulate with algebra. So, we concentrate on approximations.

The function is too complicated to consider as a whole. So, we think of it in pieces - small pieces.

If we are only interested in approximating small pieces of the function, then we can use a replacement function that does a pretty good job of approximating this function over a small interval - a replacement that is easier to work with.

And, our favorite replacement functions are linear functions.

\(C(x)\) appears to be linear-ish on the interval \((6.6, 7.0)\). Our plan is to create a linear function that does a pretty good job of approximating \(C(x)\) on the interval. Graphically, that means a tangent line around \(6.8\). We need two data for this. We need \(C(6.8)\) and \(C'(6.8)\).

Our linear approximation will be \(approxC(x) = C'(6.8)(x-6.8) + C(6.8)\).

Of course, it will only be useful on \((6.6, 7.0)\), if that.

Learning Outcomes

In this section, students will

• create linear approximations.
• accumulation via linear approximation.