1. $f(x,y) =e^{x}\ln y$

   1. (a) $f( 1,1)$
   2. (b) $f(x+\Delta x, y)$
   3. (c) $f(x, y+\Delta y)$

2. $f(x,y) =2x^{2}+6xy-y^{3}$

   1. (a) $f ( 1,1 )$
   2. (b) $f(x+\Delta x, y)$
   3. (c) $f(x, y+\Delta y)$

3. $f(x,y,z) =e^{x}\sin ^{-1}(y+2z)$

   1. (a) $f\left( \ln 3,\dfrac{1}{2},\dfrac{1}{4}\right)$
   2. (b) $f\left( 1,0,\dfrac{1}{4}\right)$
   3. (c) $f( 0,0,0)$

4. $z=f(x,y)=\ln ( x^{2}-3y)$

5. $z=f(x,y)=\sqrt{9-x^{2}-4y^{2}}$

{(x, y) | x² + 4y² ≤ 9}

6. $z=f(x,y) =\dfrac{25xy}{\sqrt{5-y^{2}}}$

7. $z=f(x,y) = \dfrac{(y-3x)^2}{x+2y}$

8. Find the domain of $w=f(x,y,z)=\dfrac{y^{2}+z^{2}}{x^{2}}.$

9. Find the domain of $w=f(x,y,z)=e^{x+y}\ln z$.

10. $z=f(x,y)=x-y+5$

11. $z=f(x, y)=\sin x$

12. $z=f(x,y)=\ln x$

13. $z=f(x,y)=e^{y}$

14. For $z=f(x, y)=x^{2}-2y,$ graph the level curves corresponding to $c=-4,$ $-1,$ $0,$ $1,$ $4.$

15. For $z=f(x, y)=\sqrt{x^{2}+y^{2}},$ graph the level curves corresponding to $c=0, 1, 4, 9.$

16. For $z=f(x,y) =e^{4x^{2}+y^{2}},$ graph the level curves corresponding to $c=1,$ $e,$ $e^{4},$ $e^{16}$.

17. Describe the level surfaces associated with the function $w=f(x, y, z)=4x^{2}+y^{2}+z^{2}$.

18. Describe the level surfaces associated with the function $w=f(x, y, z)=x+y+2z$.

19. Find $\lim\limits_{(x,y) \rightarrow \left( {\pi }/{2},0\right) }\dfrac{\sin x\cos y}{x}$.

20. Find $\lim\limits_{(x, y)\rightarrow (1, 2)}\dfrac{4x^{2}+y^{2} }{2x+y}$.

21. Let $f(x, y)=\dfrac{3xy^{2}}{x^{2}+y^{4}}$.

    1. (a) Show that $\lim\limits_{(x, y)\rightarrow (0, 0)}f(x, y)=0$ along the lines $y=mx$.
    2. (b) Find $\lim\limits_{(x, y)\rightarrow (0, 0)}f(x, y)$ along the parabola $x=y^{2}$.
    3. (c) What can you conclude?

860

22. Show that $\lim\limits_{(x, y, z)\rightarrow (0, 0, 0)}\dfrac{ 4xy}{x^{2}+y^{2}+z^{2}}$ does not exist.

23. Determine where the function $f(x,y) =2x^{2}y-y^{2}+3$ is continuous.

24. Determine where the function $R(x,y) =\dfrac{xy}{ x^{2}-y^{2}}$ is continuous.

25. 1. (a) Determine where the function $f(x,y) =\tan ^{-1}\dfrac{1}{x^{2}+y^{2}}$ is continuous.
    2. (b) Find $\lim\limits_{(x,y) \rightarrow ( 0,1) }\tan ^{-1}\dfrac{1}{x^{2}+y^{2}}.$

26. Find $f_{x}$ and $f_{y}$ for $z=f(x, y)=e^{x^{2}+y^{2}}\sin (xy)$.

27. Find $f_{x}$ and $f_{y}$ for $z=f(x, y)=\dfrac{x+y}{y}$.

28. Find $\dfrac{\partial z}{\partial x}$ and $\dfrac{\partial z}{ \partial y}$ for $z=f(x, y)=\sqrt{x-2y^{2}}$.

29. Find $\dfrac{\partial z}{\partial x}$ and $\dfrac{\partial z}{ \partial y}$ for $z=f(x, y)=e^{x}\ln ( 5x+2y)$.

30. For $f(x, y)=\sqrt{x^{2}-y^{2}}$, find $f_{x}(2,1)$ and $f_{y}(2,-1)$.

31. For $F(x, y)=e^{x}\sin y$, find $F_{x}\!\left( 0,\dfrac{\pi }{6} \right)$ and $F_{y}\!\left( 0,\dfrac{\pi }{6}\right)$.

32. Find symmetric equations for the tangent line to the curve of intersection of the ellipsoid $\dfrac{x^{2}}{24}+\dfrac{y^{2}}{12}+\dfrac{ z^{2}}{6}=1$, and the plane $y=1$, where $x=4$ and $z$ is positive.

33. Find symmetric equations for the tangent line to the curve of intersection of the surface $z=4x^{2}-y^{2}+7$:

    1. (a) With the plane $y=-2$ at the point $(1,-2,7)$.
    2. (b) With the plane $x=1$ at the point $(1,-2,7)$.

34. Boyle’s Law The volume $V$ of a gas varies directly with the temperature $T$ and inversely with the pressure $P$.

    1. (a) Find the rate of change of the volume $V$ with respect to the temperature $T.$
    2. (b) Find the rate of change of the volume $V$ with respect to the pressure $P.$
    3. (c) If $T$ and $P$ are functions of $t$, what is $\dfrac{\partial V}{\partial t}$?

35. $z=f(x,y) =( x+y^{2}) e^{3x}$

36. $z=f(x,y) =\sec (xy)$

37. $w=f(x,y,z)=e^{xyz}$

38. $w=f(x,y,z)=ze^{xy}$

39. $f(x,y,z)=e^{x}\sin y+e^{y}\sin z$

40. $w=f(x,y,z)=z\tan ^{-1}\dfrac{y}{x}$

41. For the function $z=x^{3}y^{2}-2xy^{4}+3x^{2}y^{3},$ show that $x\dfrac{\partial z}{\partial x}+y\dfrac{\partial z}{\partial y}=5z$.

42. Find the change $\Delta z$ in $z=f(x,y)=xy^{2}+2$ from $(x_{0},y_{0})$ to $(x_{0}+\Delta x, y_{0}+\Delta y)$. Use the answer to calculate the change in $z$ from $(1,0)$ to $(0.9,0.2)$.

43. Show that the function $z=f(x,y)=xy-5y^{2}$ is differentiable at any point $(x,y)$ in its domain by:

    1. (a) Finding $\Delta z$.
    2. (b) Finding $\eta _{1}$ and $\eta _{2}$ so that $\Delta z=f_{x}(x_{0},y_{0})\Delta x+ f_{y}(x_{0},y_{0})\Delta y+\eta _{1}\Delta x+\eta _{2}\Delta y$.
    3. (c) Show that $\lim\limits_{(\Delta x, \Delta y)\rightarrow (0, 0)}\eta_{1}=0$ and $\lim\limits_{(\Delta x, \Delta y)\rightarrow (0, 0)}\eta _{2}=0$.

44. $z=x\sqrt{1+y^{2}}$

45. $z=\sin ^{-1}\dfrac{x}{y}$, $y>0$

46. $w=ze^{xy}$

47. $w=\ln (xyz)$

48. Use differentials to estimate the change in $z=x\sqrt{1+y^{2}}$ from $( 4,0)$ to $( 4.1,0.1) .$

49. Use differentials to estimate the change in $z=\sin ^{-1}\dfrac{x}{y}$ from $( 0,1)$ to $( 0.1,1.1) .$

50. Use the differential of $f(x,y)=y^{2}\cos x$ to find an approximate value of $f(0.05,1.98)$. (Compare your answer with a calculator result.)

51. Electricity The electrical resistance $R$ of a wire is $R=k \dfrac{L}{D^{2}}$ where $L$ is the length of the wire, $D$ is the diameter of the wire, and $k$ is a constant. If $L$ has a $1\%$ error and $D$ has a $2\%$ error, what is the approximate maximum percentage error in the computation of $R$?

52. Find $\dfrac{dz}{dt}$ if $z=\sin (xy)-x\sin y,$ where $x=e^{t}$ and $y=te^{t}$.

53. Find $\dfrac{dw}{dt}$ if $w=\dfrac{x}{y}+\dfrac{y}{z}+\dfrac{z}{ x},$ where $x=\dfrac{1}{t}$, $y=\dfrac{1}{t^{2}}$, and $z=\dfrac{1}{t^{3}}$.

54. Find $\dfrac{\partial w}{\partial u}$ and $\dfrac{\partial w}{ \partial v}$ if $w=xy+yz-xz,$ where $x=u+v$, $y=uv$, and $z=v$. Express each answer in terms of $u$ and $v.$

55. Find $\dfrac{\partial u}{\partial r}$ and $\dfrac{\partial u}{ \partial s}$ if $u=\sqrt{x^{2}+y^{2}+z^{2}},$ where $x=r\cos s$, $y=r\sin s$ , and $z=\sqrt{r^{2}+s^{2}}$. Express each answer in terms of $r$ and $s.$

56. Find $\dfrac{\partial z}{\partial x}$ and $\dfrac{\partial z}{ \partial y}$ if $z=f(x,y)$ is defined implicitly by $F(x,y,z)=x^{2}+y^{2}-2xyz=0.$

57. Find $\dfrac{\partial z}{\partial x}$ and $\dfrac{\partial z}{ \partial y}$ if $z=f(x,y)$ is defined implicitly by $F(x,y,z)=2x\sin y+2y\sin x+2xyz=0.$

58. If $z=uf( u^{2}+v^{2}) ,$ show that $2v\dfrac{ \partial z}{\partial u}-2u\dfrac{\partial z}{\partial v}=\dfrac{2vz}{u}.$