Sketch $f\left(x\right)=x+\cos\left(x\right)$.

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 $f\left(x\right)$ 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 $f\left(c\right) = 0$.

2. Let $x_{1} = –1$. Use Newton’s Method to calculate $x_{3}$.

   Answer (b):

   Check your work: $x_{3} = −0.739$

3. How close is $x_{3}$ to the actual root of $f$ ? Calculate the error.

   Hint (c):

   $x_{3}$ and the actual root should be very close.