Calculus Problems PDF

## Calculus Problems ### Part 1 1. If $u = log(x^2 + y^2 + z^2 - 3xyz)$, show that $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = \frac{-3}{(x + y + z)^2}$ 2. State and prove the Euler's theorem on homogeneous function of two va...

## Calculus Problems ### Part 1 1. If $u = log(x^2 + y^2 + z^2 - 3xyz)$, show that $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} = \frac{-3}{(x + y + z)^2}$ 2. State and prove the Euler's theorem on homogeneous function of two variables. 3. State and prove the Ertet's theorem on hom**...** 4. If $u = tan^{-1}(\frac{x^2 y}{x^4 + y^2})$, $x + xy$ show that $x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = -(1 + sin^2u)sin 2u$. 5. If $u = log(\frac{x + y}{x - y})$, prove that $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 1$. 6. If $u = sin(\frac{x + y}{x - y})$, show that $\frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = tan u$. #### Part 2 7. Find $\frac{dz}{dt}$ when $z = x^2y + xy^2$, $x = at^2$, $y = 2t$. Verify by direct substitution. 8. State and prove the Euler's theorem on homogeneous functions of three variables. ### Part 3 9. For the cycloid $x = a(t + sin t)$, $y = a(1 - cos t)$, prove that $\kappa = 4acos(\frac{t}{2})$. 10. Find the radius of curvature for the curve $r = a(1 - cos \theta)$. 11. Prove that the radius of curvature at the point $(-a, a)$ on the curve $x^2 y = a(x^2 + y^2)$ is $-2a$. 12. Find $\kappa$ at the origin of the curves $x^2y^2 + x^3z + x^2y + xy = 0$. 13. Find the coordinates of the center of curvature at the point $(x, y)$ of the parabola $y^2 = 4ax$. 14. Obtain the evolute of the parabola $y^2 = 4ax$. 15. Find the length of the parabola arc of the evolute of the parabola $y = ax^2$ which is intercepted between the parabola. 16. Find the length of the arc $y = cosh(\frac{x}{2})$ from the vertex $(0, 1)$ to any point $(x, y)$. 17. Find the whole length of the astroid $x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}$. 18. Rectify the curve $x = a(t + sin t)$, $y = a(1 - cos t)$. 19. Discuss the maximum or minimum values of $u$ when $u = x^2y - 3axy$. 20. Discuss the maximum or minimum values of $u$ given by $u = x^3(1 - x - y)$. 21. Show that the minimum value of $u = xy + \frac{3}{x} + \frac{a}{y}$ is $3\sqrt[3]{a}$.