Sketch $f(x) = x +\operatorname{cos }x$.

1. Use the Intermediate Value Theorem to show there is a root between $-1$ and $0$.

   Step 1(a):

   State the conditions of the Intermediate Value Theorem: 'Since $f(x)$ is continuous on the closed domain $[-1, 0]$...

   Step 2(a):

   Prove the thesis of the IVT: ... and $f(-1) < 0$ and $f(0) > 0$, there must be a value $c$, on $-1 < c < 0$, such that $f(c) = 0$.

2. Let $x_1 =-1$. Use Newton's Method to find $x_3$.

   Answer (b):

   Check your work: $x_3 = −0.739$

3. How close is $x_3$ to the actual root of $f(x)$? Calculate the error.

   Hint (c):

   $x_3$ and the actual root should be very close.