(a) Find the tangent line to the function at the specified value of x.
(b) Use the tangent line from (a) to estimate the value of the function at (x + Δ x).
(a) The slope of the tangent line at is
The value of the function at is 1, so we get the following tangent line:
(b) We plug
into the tangent line and get
It turns out that when you know a value of f at a particular x and you're asked to approximate the value of f somewhere nearby, you don't need to go to the trouble of finding the tangent line equation. As we saw earlier (link back up to where we reviewed slope), it's enough to know the slope of the tangent line, yold, and Δ x. That's enough information to find ynew.