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Calculus 3: test 2

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1. Draw the contour map (level curves) of the function $f(x,y)=e^{y/x}$. Explain what the level curves are.

2. Find the equation of the tangent plane to the surface $z=\sqrt {4-x^{2}-2y^{2}}$ at the point $(1,-1,1).$

3. Find $\frac{dw}{dt},$ where $w=xy+yz^{2},$ $x=e^{t},$ $y=e^{t}\sin t,$ $z=e^{t}\cos t$.

4. Find the directional derivative of the function $f(x,y)=x/(y+z)$ at the point $(4,1,1)$ in the direction of the vector $v=<0,2,-1>$.

5. Use a Riemann sum with $m=n=2$ to estimate the value of the integral $\int\int_{D}\sin(x+y)dA,$ where $R=[0,\pi]\times\lbrack0,\pi].$ Choose your own sample points.

6. Find the mass of the lamina that occupies the region $D$ bounded by $y=e^{x},$ $y=0,$ $x=0,$ $x=1,$ and its density function is $\rho(x,y)=y.$

7. Evaluate the iterated integral $% %TCIMACRO{\dint \limits_{0}^{1}}% %BeginExpansion {\displaystyle\int\limits_{0}^{1}} %EndExpansion% %TCIMACRO{\dint \limits_{0}^{z}}% %BeginExpansion {\displaystyle\int\limits_{0}^{z}} %EndExpansion% %TCIMACRO{\dint \limits_{0}^{y}}% %BeginExpansion {\displaystyle\int\limits_{0}^{y}} %EndExpansion ze^{-y^{2}}dxdydz.$


  1. Find the partial derivatives of the function $f(x,y)=\dfrac{x}{y}+\sin(xy)+xe^{2y}$.
  2. Find the equation of the tangent plane to the surface $z=y\cos(x-y)$ at the point $(2,2,2).$
  3. Set up as a max/min problem, but do not solve, the following: "Find the dimensions of a rectangular box of maximal volume such that the sum of lengths of its edges is equal to 10".
  4. Find the volume of the solid under the plane $x+2y-z=0$ and above the region bounded by $y=x$ and $y=x^{4}$.
  5. Find the area of the part of the surface $z=1+3x+2y^{2}$ that lies above the triangle with vertices $(0,0),(0,1),$ and $(2,1).$
  6. Find the maximum rate of change of the function $f(x,y)=\sin(xy)$ at the point $(1,0)$ and the direction in which it occurs.
  7. Evaluate $\int\int\int_{D}2xdV,$ where

\[ D=\{(x,y,z):0\leq y\leq2,0\leq x\leq\sqrt{4-y^{2}},0\leq z\leq y\}. \]