Chapter 9 - Section 9.2 - Direction Fields and Euler''s Method - 9.2 Exercises - Page 598: 19

See explanation

$y'=y$ and $y(0)=1$ --- a) i) $h=0.4$ $y_1=1+0.4(1)=1.4$ $x_1=0.4$ ii) $h=0.2$ $y_1=1+0.2(1)=1.2$ $x_1=0.2$ $y_2=1.2+0.2(1.2)=1.44$ $x_2=0.4$ iii) $h=0.1$ $y_1=1+0.1(1)=1.1$ $x_1=0.1$ $y_2=1.1+0.1(1.1)=1.21$ $x_2=0.2$ $y_3=1.21+0.1(1.21)=1.331$ $x_3=0.3$ $y_4=1.331+0.1(1.331)=1.4641$ $x_4=0.4$ b) Attached as image. The value approaches $e^x$ as $h \to 0$. The values are underestimates since $e^x$ is increasing between $0 \le x \le 0.4$. c) i) $Err \approx 0.0918$ ii) $Err \approx 0.0518$ iii) $Err \approx 0.02772$ It appears as if the error halves as you half the step size.