$y = f(x)$ satisfies $\frac { d y } { d x }= 2x$ and its graph contains the point $(−1, 5)$.

1. Use Euler' s Method to sketch $y$ for $−1 ≤ x ≤ 1$. Use $Δx = 0.5$.

   Hint (a):

   Show all steps and carefully record the coordinate points at $x =-0.5$, $x = 0$, $x = 0.5$ and $x = 1$.
   

2. Use implicit integration to find $y$.

   Hint (b):

   Use $(−1, 5)$ to help you find the value of $C$ after you integrate.

3. Compare your answers to parts (a) and (b). Find the largest difference between the actual $y$ value and the Euler's Method $y$ value for $x = − 1$, $−0.5$, $0$, $0.5$, and $1$. This largest difference estimates the error of Euler's Method. If $Δx = 0.1$ instead, do you think this error would be larger or smaller?