Generating...                               quiz04a_n21

  1. Let $z = f(x,y)$ where $x = r\,\cos s$ and $y = r\,\sin s$ . Express $\dfrac{\partial z}{\partial r}$ in terms of $f_x$, $f_y$, $r$ and $s$.

    $f_x ( -r\,\sin s) + f_y ( r\,\cos s)$ $f_x ( \sin s) + f_y ( \cos s)$ $f_x ( \cos s) + f_y ( \sin s)$ $f_x ( r\,\cos s) + f_y ( -r\,\sin s)$

  2. Let $f(x,y) = -\frac{y}{x+y}$. Find the tangent plane at $(x,y) = \left[ 1 , 2 \right] $.

    $\displaystyle z=\frac{6-2\,x+y}{9}$ $\displaystyle z=-\frac{6-2\,x+y}{9}$ $\displaystyle z=-\frac{-9-x+2\,y}{9}$ $\displaystyle z=\frac{-9-x+2\,y}{9}$

  3. Let $f(x,y) = x\,e^{y}$. Find the first partial derivative $f_{ y}(x,y)$ .

    $\displaystyle x\,e^{y}$ $\displaystyle x^{y}\,\ln x$ $\displaystyle x^{-1+y}\,y$ $\displaystyle e^{y}$

  4. Let $z = f(x,y)$ where $x = e^{s}\,\cos t$ and $y = e^{s}\,\sin t$ . Express $\dfrac{\partial z}{\partial s}$ in terms of $f_x$, $f_y$, $s$ and $t$.

    $f_x ( e^{s}\,\cos t) + f_y ( e^{s}\,\sin t)$ $f_x ( -e^{s}\,\sin t) + f_y ( e^{s}\,\cos t)$ $f_x ( e^{s}\,\cos t) + f_y ( -e^{s}\,\sin t)$ $f_x ( e^{s}\,\sin t) + f_y ( e^{s}\,\cos t)$

  5. Let $f(x,y) = \sqrt{x}\,y^{\frac{3}{2}}$. Find the tangent plane at $(x,y) = \left[ 4 , 1 \right] $.

    $\displaystyle z=\frac{-8+x+12\,y}{4}$ $\displaystyle z=\frac{-41+12\,x+y}{4}$ z = −11 + 4x + 3y z = −8 + 3x + 4y

  6. Let $f(x,y) = \frac{x}{x-y}$. Find all the second partial derivatives.

    $f_{xx}(x,y) = \frac{2\,y}{\left(-x+y\right)^3}$; $f_{xy}(x,y) = -\frac{x+y}{\left(-x+y\right)^3}$; $f_{yy}(x,y) = \frac{2\,x}{\left(-x+y\right)^3}$ $f_{xx}(x,y) = -\frac{2\,y}{\left(-x+y\right)^3}$; $f_{xy}(x,y) = \frac{x+y}{\left(-x+y\right)^3}$; $f_{yy}(x,y) = -\frac{2\,x}{\left(-x+y\right)^3}$ $f_{xx}(x,y) = \frac{2\,x}{\left(-x+y\right)^3}$; $f_{xy}(x,y) = \frac{y}{\left(-x+y\right)^2}$; $f_{yy}(x,y) = \frac{2\,y}{\left(-x+y\right)^3}$ $f_{xx}(x,y) = -\frac{2\,x}{\left(-x+y\right)^3}$; $f_{xy}(x,y) = -\frac{y}{\left(-x+y\right)^2}$; $f_{yy}(x,y) = -\frac{2\,y}{\left(-x+y\right)^3}$

  7. Let $z = \frac{x}{x+y}$. Find the total differential $dz$.

    $\displaystyle \frac{-y\,dx+x\,dy}{\left(x+y\right)^2}$ $\displaystyle \frac{y^2\,dx+x^2\,dy}{\left(x+y\right)^2}$ $\displaystyle -\frac{-y\,dx+x\,dy}{\left(x+y\right)^2}$ $\displaystyle -\frac{dx+dy}{\left(x+y\right)^2}$

  8. Let $z = f(x,y)$ where $x = t^2$ and $y = t$ . Suppose that $f_x(x,y) = \frac{x\,y^2\,\left(x+2\,y\right)}{\left(x+y\right)^2}$ and $f_y(x,y) = \frac{x^2\,y\,\left(2\,x+y\right)}{\left(x+y\right)^2}$ . Find $\dfrac{dz}{dt}$.

    $\displaystyle \frac{t^2\,\left(3+2\,t\right)}{\left(1+t\right)^2}$ $\displaystyle \frac{t\,\left(2+t\right)}{\left(1+t\right)^2}$ $\displaystyle \frac{t^4\,\left(5+4\,t\right)}{\left(1+t\right)^2}$ $\displaystyle \frac{t^3\,\left(4+3\,t\right)}{\left(1+t\right)^2}$

  9. Let $f(x,y) = \sin \left(2\,x+3\,y\right)$. Find all the second partial derivatives.

    $f_{xx}(x,y) = -9\,\cos \left(2\,x+3\,y\right)$; $f_{xy}(x,y) = -2\,\sin \left(2\,x+3\,y\right)$; $f_{yy}(x,y) = -4\,\cos \left(2\,x+3\,y\right)$ $f_{xx}(x,y) = -4\,\sin \left(2\,x+3\,y\right)$; $f_{xy}(x,y) = -6\,\sin \left(2\,x+3\,y\right)$; $f_{yy}(x,y) = -9\,\sin \left(2\,x+3\,y\right)$ $f_{xx}(x,y) = -4\,\cos \left(2\,x+3\,y\right)$; $f_{xy}(x,y) = -6\,\cos \left(2\,x+3\,y\right)$; $f_{yy}(x,y) = -9\,\cos \left(2\,x+3\,y\right)$ $f_{xx}(x,y) = -9\,\sin \left(2\,x+3\,y\right)$; $f_{xy}(x,y) = 2\,\cos \left(2\,x+3\,y\right)$; $f_{yy}(x,y) = -4\,\sin \left(2\,x+3\,y\right)$

  10. Let $z = f(x,y)$ where $x = t^2$ and $y = t$ . Suppose that $f_x(x,y) = 2\,x\,y\,\cos \left(x^2\,y\right)$ and $f_y(x,y) = x^2\,\cos \left(x^2\,y\right)$ . Find $\dfrac{dz}{dt}$.

    $\displaystyle -5\,t^4\,\sin t^5$ $\displaystyle 4\,t^3\,\cos t^4$ $\displaystyle -4\,t^3\,\sin t^4$ $\displaystyle 5\,t^4\,\cos t^5$


Department of Mathematics
Last modified: 2025-10-30