In Problems 1 through 10 we round off the indicated values to 3 decimal places.

1. 1. Approximate values 1.125 and 1.181; true value 1.213

2. 2. Approximate values 1.125 and 1.244; true value 1.359

3. 3. Approximate values 2.125 and 2.221; true value 2.297

4. 4. Approximate values 0.625 and 0.681; true value 0.713

5. 5. Approximate values 0.938 and 0.889; true value 0.851

6. 6. Approximate values 1.750 and 1.627; true value 1.558

7. 7. Approximate values 2.859 and 2.737; true value 2.647

8. 8. Approximate values 0.445 and 0.420; true value 0.405

9. 9. Approximate values 1.267 and 1.278; true value 1.287

10. 10. Approximate values 1.125 and 1.231; true value 1.333

Problems 11 through 24 call for tables of values that would occupy too much space for inclusion here. In Problems 11 through 16 we give first the final x-value, next the corresponding approximate x-values obtained with step sizes $h=0.01$ and $h=0.005$, and then the final true y-value. (All y-values rounded off accurate to 4 decimal places.)

1. 11. $1.0,-0.7048,-0.7115,-0.7183$

2. 12. 1.0, 2.9864, 2.9931, 3.0000

3. 13. 2.0, 4.8890, 4.8940, 4.8990

4. 14. 2.0, 3.2031, 3.2304, 3.2589

5. 15. 3.0, 3.4422, 3.4433, 3.4444

6. 16. 3.0, 8.8440, 8.8445, 8.8451

In Problems 17 through 24 we give first the final x-value and then the corresponding approximate y-values obtained with step sizes $h=0.1\text{,}h=0.02\text{,}h=0.004$, and $h=0.0008$ respectively. (All y-values rounded off accurate to 4 decimal places.)

1. 17. 1.0, 0.2925, 0.3379, 0.3477, 0.3497

2. 18. 2.0, 1.6680, 1.6771, 1.6790, 1.6794

3. 19. 2.0, 6.1831, 6.3653, 6.4022, 6.4096

4. 20. 2.0, $-1.3792,-1.2843,-1.2649,-1.2610$

5. 21. 2.0, 2.8508, 2.8681, 2.8716, 2.8723

6. 22. 2.0, 6.9879, 7.2601, 7.3154, 7.3264

7. 23. 1.0, 1.2262, 1.2300, 1.2306, 1.2307

8. 24. 1.0, 0.9585, 0.9918, 0.9984, 0.9997

9. 25. With both step sizes $h=0.01$ and $h=0.005$, the approximate velocity after 1 second is 16.0 ft/sec (80% of the limiting velocity of 20 ft/sec); after 2 seconds it is 19.2 ft/sec (96% of the limiting velocity).

10. 26. With both step sizes $h=1$ and $h=0.5$, the approximate population after 5 years is 49 deer (65% of the limiting population of 75 deer); after 10 years it is 66 deer (88% of the limiting population).

11. 27. With successive step sizes $h=0.1,0.01,0.001,\dots$ the first four approximations to $y(2)$ we obtain are 0.7772, 0.9777, 1.0017, and 1.0042. It therefore seems likely that $y(2)\approx 1.00$.

12. 28. With successive step sizes $h=0.1,0.01,0.001,\dots$ the first four approximations to $y(2)$ we obtain are 1.2900, 1.4435, 1.4613, and 1.4631. It therefore seems likely that $y(2)\approx 1.46$.

13. 29.

    x	$\mathit{h}\mathbf{=}0.15$ y	$\mathit{h}\mathbf{=}0.03$ y	$\mathit{h}\mathbf{=}0.006$ y
    $-1.0$	1.0000	1.0000	1.0000
    $-0.7$	1.0472	1.0512	1.0521
    $-0.4$	1.1213	1.1358	1.1390
    $-0.1$	1.2826	1.3612	1.3835
    0.2	0.8900	1.4711	0.8210
    0.5	0.7460	1.2808	0.7192

14. 30.

    x	$\mathit{h}\mathbf{=}0.1$ y	$\mathit{h}\mathbf{=}0.01$ y
    1.8	2.8200	4.3308
    1.9	3.9393	7.9425
    2.0	5.8521	28.3926

15. 31.

    x	$\mathit{h}\mathbf{=}0.1$ y	$\mathit{h}\mathbf{=}0.01$ y
    0.7	4.3460	6.4643
    0.8	5.8670	11.8425
    0.9	8.3349	39.5010