Week 1 Exercises
1.5 Sheet 1 Limits
Exercise 1
Show that the following curves are continuous at the given \(x\) values by demonstrating that
- \(\lim\limits_{x\to1}\frac{3x+2}{2x-1} = 5\);
- \(\lim\limits_{x\to0}\frac{x+3}{x-2} = -\frac 32\);
- \(\lim\limits_{x\to3}\frac{x^3+27}{x+3} = 9\);
Exercise 2
Determine the following limits:
- \(\lim\limits_{x\to1}\frac{x^2+x-2}{x-1}\)
- \(\lim\limits_{x\to2}\frac{2x^2-x-6}{x-2}\)
- \(\lim\limits_{x\to2}\frac{x^2-4}{x-2}\)
- \(\lim\limits_{x\to0}\frac{2x^2+3x}{x}\)
- \(\lim\limits_{x\to2}\frac{x^2-x-2}{2x^2-3x-2}\)
- \(\lim\limits_{x\to\infty}\frac{x}{2x+1}\)
1.6 Sheet 2 Differentiation by first principles
For each of the following functions differentiate \(y\) with respect to \(x\) from first principles.
- \(y=4x^2\)
- \(y=6x^3\)
- \(y=3x^2+2x\)
- \(y=6x^2-4\)
- \(y=3x^3+4x-5\)
- \(y=x^4+x^2-3x\)
- \(y=\frac{1}{x^2}\)
- \(y=\frac{1}{5x+3}\)
- \(y=\frac{1}{(1+x)^2}\)
- \(y=x-\frac{2}{x}\)
[Solutions: 1. \(y=8x\); 2. \(y=18x^2\); 3. \(y=6x+2\); 4. \(y=12x\); 5. \(y=9x^2+4\); 6. \(y=4x^3+2x-3\); 7. \(y=-2/x^3\); 8. \(y=-5/(5x+3)^2\); 9. \(y=-2/(1+x)^3\); 10. \(y=1+2/x^2\)]
1.7 Sheet 3 Differentiation by rule
Differentiate each of the following functions by rule:
| 1. \(y=x^7\) | 2. \(y=6x^5\) |
| 3. \(y=4x^3\) | 4. \(s=0.5t^3\) |
| 5. \(A=\pi r^2\) | 6. \(y=x^{1/2}\) |
| 7. \(s=0.2t^2\) | 8. \(y=4x^{3/2}\) |
| 9. \(y=2\sqrt{x}\) | 10. \(y=3(\sqrt[3]{x^2})\) |
| 11. \(y=\frac1{x^2}\) | 12. \(y=\frac 1x\) |
| 13. \(y=\frac3{5x}\) | 14. \(y=\frac2{x^3}\) |
| 15. \(y=\frac1{\sqrt{x}}\) | 16. \(y=\frac2{3\sqrt{x}}\) |
| 17. \(y=\frac5{x\sqrt{x}}\) | 18. \(s=3\frac{\sqrt{t}}5\) |
| 19. \(k=\frac{0.01}{h}\) | 20. \(y=\frac5x\) |
| 21. \(y=4x^2-3x+2\) | 22. \(s=3t^3-2t^2+5t-3\) |
| 23. \(q=2u^2-u+7\) | 24. \(y=5x^4-7x^3+3x^2-2x\) |
| 25. \(s=7t^5-3t^2+7\) | 26. \(y=3.1x^{1.5}-2.4x^{0.6}\) |
| 27. \(y=\frac{x+x^3}{\sqrt{x}}\) | 28. \(y=\frac{3+x^2}x\) |
| 29. \(y=\sqrt{x}+\frac1{\sqrt{x}}\) | 30. \(y=x^3+\frac3{\sqrt{x}}\) |
| 31. \(t^{1.3}-\frac1{4t^{2.3}}\) | 32. \(y=\frac{3x^3}5-\frac{2x^2}7-\sqrt{x}\) |
| 33. \(y=0.008+\frac{0.001}{x}\) | 34. \(s=10-6t+7t^2-2t^3\) |
[Solutions: 1. \(dy/dx=7x^6\); 2. \(dy/dx=30x^4\); 3. \(dy/dx=12x^2\); 4. \(ds/dt=1.5t^2\); 5. \(dA/dr=2\pi r\); 6. \(dy/dx=1/(2x^{1/2})\); 7. \(ds/dt=0.4t\); 8. \(dy/dx=6x^{1/2}\); 9. \(dy/dx=1/\sqrt{x}\); 10. \(dy/dx=2/\sqrt[3]{x}\); 11. \(dy/dx=-2/x^3\); 12. \(dy/dx=-1/x^2\); 13. \(dy/dx=-3/(5x^2)\); 14. \(dy/dx=-6/x^4\); 15. \(dy/dx=-1/(2x^{3/2})\); 16. \(dy/dx=-1/(3x^{3/2})\); 17. \(dy/dx=-15/(2x^{5/2})\); 18. \(ds/dt=3/(10\sqrt{t})\); 19. \(dk/dh=-0.01/h^2\); 20. \(dy/dx=-5/x^2\); 21. \(dy/dx=8x-3\); 22. \(ds/dt=9t^2-4t+5\); 23. \(dq/du=4u-1\); 24. \(dy/dx=20x^3-21x^2+6x-2\); 25. \(ds/dt=35t^4-6t\); 26. \(dy/dx=4.65x^{0.5}-1.44/x^{0.4}\); 27. \(dy/dx=1/(2x^{1/2})+5x^{3/2}/2\); 28. \(dy/dx=-3/x^2+1\); 29. \(dy/dx=1/(2\sqrt{x})-1/(2x\sqrt{x})\); 30. \(dy/dx=3x^2-3/(2x^{3/2})\); 31. \(ds/dt=1.3t^{0.3}+0.575/t^{3.3}\); 32. \(dy/dx=9x^2/5-4x/7-1/(2\sqrt{x})\); 33. \(dy/dx=-0.001/x^2\); 34. \(ds/dt=-6+14t-6t^2\)]