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

$$C(x) = \frac {(\sin (2x)+2)^{|x|}}{5(x^2+1)}$$

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 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.