For the function,
(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 function value at is
The derivative function is
f '(x) = 2sin xcos x
so the slope of our tangent line is
Putting this together, the tangent line is
For a sanity check, we can put this into our calculator and make sure it looks like a tangent line, which it does.
(b) To estimate , we plug this value into the tangent line: