Let  $y=f\left(x\right)$  be the solution to the differential equation  $\frac{dy}{dx}=y-10x^2$ with the initial condition  $f\left(0\right)=3$  .  What is the approximation for  $f\left(0.4\right)$  
 if Euler’s method is used, starting at  $x=0$   with steps of size 0.2 ?

Let $y=g\left(x\right)$  be the solution to the differential equation  $\frac{dy}{dx}=xy-2$   with the initial condition  $g\left(2\right)=-1$ . What is the approximation for g(1)  if Euler’s method is used, starting at x=2  with two steps of equal size?

 The table above gives values of $f'$ , the derivative of the function  $f$ , for selected values of x.  If  $f\left(2\right)=6$  , what is the approximation for  $f\left(1.4\right)$  obtained by using Euler's method, starting at  $x=2$   with two steps of equal size?