change in $y$ using a first order approximation, this gives us
    the differential notation  $$dy = f'(c)dx.$$  A second order
    approximation is  $$dy = f'(c)dx + \frac{f''(c)}{2!}dx^2.$$  A
    third order approximation is $$dy = f'(c)dx +
    \frac{f''(c)}{2!}dx^2+ \frac{f'''(c)}{3!}dx^3.$$
  </div>

  <div class="example">
<p>    This example refers back to <a href="#diff-sphere">this problem above</a>. We
    wanted a spherical tank of radius 3m, but due to manufacturing
    error the radius was sli