Week 5 Exercises

5.4 Sheet 12 - Partial Differentiation

For each of the functions below, obtain where possible \[ \frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}, \frac{\partial^2 z}{\partial x^2}, \frac{\partial^2 z}{\partial y^2}, \frac{\partial^2 z}{\partial y\partial x}, \frac{\partial^2 z}{\partial x\partial y} \]

  1. \(z=3x^2+4xy-5y^2\).
  2. \(z=xy^2+1/y-y/x^2\).
  3. \(z=\sin(2x)\cos(2y)\).
  4. \(z=x\cos(y)-y\cos(x)\).
  5. \(z=\sin(3x+2y)\).

{Solutions: 1. \(\frac{\partial z}{\partial x}=6x+4y\), \(\frac{\partial z}{\partial y}=4 x - 10 y\), \(\frac{\partial^2 z}{\partial x^2}=6\), \(\frac{\partial^2 z}{\partial y^2}=-10\), \(\frac{\partial^2 z}{\partial y\partial x}=4\), \(\frac{\partial^2 z}{\partial x\partial y}=4\); 2. \(y^2+{\frac{2y}{x^3}}\), \(2xy-{\frac{1}{y^2}}-{\frac{1}{x^2}}\), \(-{\frac{6y}{x^4}}\), \(2x+{\frac{2}{y^3}}\), \(2y+{\frac{2}{x^3}}\), \(2y+{\frac{2}{x^3}}\); 3. \(2\cos \left(2x\right)\cos \left(2y\right)\), \(-2\sin \left(2x\right)\sin \left(2y\right)\), \(-4\sin \left(2x\right)\cos \left(2y\right)\), \(-4\sin \left(2x\right)\cos \left(2y\right)\), \(-4\cos \left(2x\right)\sin \left(2y\right)\), \(-4\cos \left(2x\right)\sin \left(2y\right)\); 4. \(\cos y+y\sin x\), \(-x\sin y-\cos x\), \(y\cos x\), \(-x\cos y\), \(\sin x-\sin y\), \(\sin x-\sin y\); 5. \(3\cos \left(3x+2y\right)\), \(2\cos \left(3x+2y\right)\), \(-9\sin \left(3x+2y\right)\), \(-4\sin \left(3x+2y\right)\), \(-6\sin \left(3x+2y\right)\), \(-6\sin \left(3x+2y\right)\); }

5.5 Sheet 12A - Indefinite Integration

Integrate the following functions with respect to the variable

  1. \(x^2+3-1/x\)
  2. \(x+1/\sqrt{x}\)
  3. \((4x^3-3x)/\sqrt{x}\)
  4. \(1/x^3+x^3\)
  5. \((x+1)^2\)
  6. \((1+\sqrt{x})^2\)
  7. \(5x^5-\sqrt(x)\)
  8. \(e^{0.2x}+x\)
  9. \(\cos(5x)\)
  10. \((t+1)(3+t)\)
  11. \(\sin(ax)+\cos(bx)\)
  12. \(\sqrt{r}(1+r)\)
  13. \((2+\sqrt{t})^2\)
  14. \(\cos(3x)-\sin(2x)\)
  15. \(\sin(100\pi t)-\cos(100\pi t)\)
  16. \(\sec^2(x)-\sin(x)\)
  17. \(\sec^2(\theta)+\cos(3\theta)\)
  18. \(4\sec^2(x)-\text{cosec}^2(x)\)
  19. \(4/x-1/x^2\)
  20. \(e^{2x}-1/x\)
  21. \(\sin(x/2)\)
  22. \(e^x+2/x-3/e^x\)
  23. \((t+1)/t\)
  24. \((x+1/x)^2\)
  25. \(1/(2x)\)
  26. \((e^{3x}-e^{-3x})/2\)
  27. \(\text{cosec}^2(x)-\sin(x)\)”
  28. \(v^{-1.2}-v^{-1}\)”

{Solutions: 1. \(-\ln(|x|)+{\frac{x^3}{3}}+3x+C\); 2. \({\frac{x^2}{2}}+2\sqrt{x}+C\); 3. \(\frac{8}{7}x^{7/2} - 2x^{3/2}+C\); 4. \({\frac{x^4}{4}}-{\frac{1}{2x^2}}+C\); 5. \(\frac{1}{3}(x+1)^3+C\); 6. \({\frac{x^2}{2}}+\frac{4x^{\frac{3}{2}}}{3}+x+C\); 7. \({\frac{5x^6}{6}}-\frac{2x^{\frac{3}{2}}}{3}+C\); 8. \(5.0e^{0.2x}+\frac{x^2}{2}+C\); 9. \(\frac{1}{5}sin(5x)+C\); 10. \((t^3+6t^2+9t)/3+C\); 11. \(\frac{\sin(bx)}{b}-\frac{\cos(ax)}{a}+C\); 12. \((6r^{5/2}+10r^{3/2})/15+C\); 13. \(t^2/2 + 8t^{3/2}/3+4t+C\); 14. \(\frac{\sin(3x)}{3}+\frac{\cos(2x)}{2}+C\); 15. \(-\frac{1}{100\pi}\cos(100\pi t) - \frac{1}{100\pi}\sin(100\pi t) + C\); 16. \(\tan(x)+\cos(x)+C\); 17. \(\tan(\theta) + \frac{1}{3}\sin(3\theta)+C\); 18. \(4\tan(x) + \cot(x) +C\); 19. \(4\ln(|x|) + \frac{1}{x} +C\); 20. \(\frac{1}{2}\exp(2x)-\ln(|x|) + C\); 21. \(-2\cos(x/2)+C\); 22. \(\exp(x) + 2\ln(|x|) +3\exp(-x)+C\); 23. \(\ln(|t|) + t + C\); 24. \(2\ln(|x|) + x - \frac{1}{x} + C\); 25. \(\frac{1}{2}\ln(|x|)+C\); 26. \(\frac{1}{6}(\exp(3x)+\exp(-3x)) + C\); 27. \(\cot(x)+\cos(x)+C\); 28. \(-\ln(|v|)-5 v^{-1/5}+C\); }