#### Answer

$-0.763$ and $0.6072$

#### Work Step by Step

$e^xcosx \approx 1 +x$
$f(x) = e^xcosx$
$f(0) = e^0 cos(0)$
$f(0) = 1$
$f'(x) = e^xcosx + e^x(-sinx)$
$f'(x) = e^xcosx - e^xsinx$
$f'(0) = e^0cos(0) - e^0sin(0)$
$f'(0) = 1$
$L(x) = f(0) + x(f'(0))$
$L(x) = 1 + x(1)$
$L(x) = 1 + x$
$L(x) = 1 + x(1)$
This means that the original answer given by the book is correct. Now we have to find the values of x for which the linear approximation is accurate to within 0.1.
$f(x) - L(x) < 0,1$
$e^xcosx -x -1 < 0.1$
$-0.1 < e^xcosx -x -1 < 0.1$
$-0.1 + 1 < e^xcosx -x < 0.1 + 1$
$0.9 < e^xcosx -x < 1.1$
Using the graph of $e^xcosx -x$ and the values of $y$ we can see the approximations to be $-0.763$ and $0.6072$