1. 1.

   x	Improved Euler y	Actual y
   0.1	1.8100	1.8097
   0.2	1.6381	1.6375
   0.3	1.4824	1.4816
   0.4	1.3416	1.3406
   0.5	1.2142	1.2131

Note: In Problems 2 through 10, we give the value of x, the corresponding improved Euler value of y, and the true value of y.

1. 2. 0.5, 1.3514, 1.3191

2. 3. 0.5, 2.2949, 2.2974

3. 4. 0.5, 0.7142, 0.7131

4. 5. 0.5, 0.8526, 0.8513

5. 6. 0.5, 1.5575, 1.5576

6. 7. 0.5, 2.6405, 2.6475

7. 8. 0.5, 0.4053, 0.4055

8. 9. 0.5, 1.2873, 1.2874

9. 10. 0.5, 1.3309, 1.3333

In Problems 11 through 16 we give the final value of x, the corresponding values of y with $h=0.01$ and with $h=0.005$, and the true value of y

1. 11. 1.0, $-0.71824,-0.71827,-0.71828$

2. 12. 1.0, 2.99995, 2.99999, 3.00000

3. 13. 2.0, 4.89901, 4.89899, 4.89898

4. 14. 2.0, 3.25847, 3.25878, 3.25889

5. 15. 3.0, 3.44445, 3.44445, 3.44444

6. 16. 3.0, 8.84511, 8.84509, 8.84509

In Problems 17 through 24 we give the final value of x and the corresponding values of y for $h=0.1$, 0.02, 0.004, and 0:0008.

1. 17. 1.0, 0.35183, 0.35030, 0.35023, 0.35023

2. 18. 2.0, 1.68043, 1.67949, 1.67946, 1.67946

3. 19. 2.0, 6.40834, 6.41134, 6.41147, 6.41147

4. 20. 2.0, $-1.26092,-1.26003,-1.25999,-1.25999$

5. 21. 2.0, 2.87204, 2.87245, 2.87247, 2.87247

6. 22. 2.0, 7.31578, 7.32841, 7.32916, 7.32920

7. 23. 1.0, 1.22967, 1.23069, 1.23073, 1.23073

8. 24. 1.0, 1.00006, 1.00000, 1.00000, 1.00000

9. 25. With both step sizes $h=0.01$ and $h=0.005$ the approximate velocity after 1 second is 15.962 ft/sec (80% of the limiting velocity of 20 ft/sec); after 2 seconds it is 19.185 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 49.391 deer (65% of the limiting population of 75 deer); after 10 years it is 66.113 deer (88% of the limiting population).

11. 27. With successive step sizes $h=0.1,0.01,0.001,\dots$ the first three approximations to $y(2)$ we obtain are 1.0109, 1.0045, and 1.0045. It therefore seems likely that $y(2)\approx 1.0045$.

12. 28. With successive step sizes $h=0.1,0.01,0.001,\dots$ the first four approximations to $y(2)$ we obtain are 1.4662, 1.4634, 1.4633, and 1.4633. It therefore seems likely that $y(2)\approx 1.4633$.

13. 29. Impact speed approximately 43.22 m/s

14. 30. Impact speed approximately 43.48 m/s