1. True or False If $F$ is differentiable at a point $P_{0}=(x_{0},y_{0},z_{0})$ on the surface $F(x,y,z)=0,$ and if ${\bf \boldsymbol \nabla }F(x_{0},y_{0},z_{0})\neq \mathbf{0}$, then the surface has a tangent plane at $P_{0}$.

2. True or False The normal line to the surface $F(x,y,z)=0$ at a point $P_{0}=(x_{0},y_{0},z_{0})$ is in the direction of the gradient ${\boldsymbol \nabla }F(x_{0},y_{0},z_{0})$, provided ${\boldsymbol \nabla }F(x_{0},y_{0},z_{0})\neq \mathbf{0}$.

3. $z=10-2x^{2}-y^{2}$ at $( 0,2,6)$

   

4. $z=4+x^{2}+y^{2}$ at $( 1,-2,9)$

   

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5. $z=4x^{2}+9y^{2}$ at $( 1,1,13)$

   

6. $z=\sqrt{x^{2}+3y^{2}}$ at $( 1,1,2)$

   

7. $x^{2}+y^{2}+z^{2}=14$ at $(1,-2,3)$

8. $2x^{2}+3y^{2}+z^{2}=12$ at $(2,1,-1)$

9. $4x^{2}+y^{2}-2z^{2}=3$ at $(1,1,-1)$

10. $z^{2}=x^{2}+3y^{2}$ at $(1,-1,-2)$

11. $z=x^{2}+y^{2}$ at $(-2,1,5)$

12. $z=2x^{2}-3y^{2}$ at $(2,1,5)$

13. $2x^{2}-y^{2}=z$ at $(1,0,2)$

14. $x^{2}+y^{2}-2z^{2}=-13$ at $(2,1,3)$

15. $x^{2}-3y^{2}-4z^{2}=2$ at $(3,1,1)$

16. $x^{2}+2y^{2}-5z^{2}=4$ at $(1,2,1)$

17. $f(x,y)=\ln ( \sec (x^{2}+y^{2}))$ at $\left( \sqrt{\dfrac{\pi }{8}},\sqrt{\dfrac{\pi }{8}},\ln \sqrt{2}\right)$

18. $f(x,y)=\dfrac{x^{2}-y^{2}}{x^{2}+y^{2}}$ at $\left(1,2, -\dfrac{3}{5}\right)$

19. $z=e^{x}\cos y$ at $\left( 0,\dfrac{\pi}{2},0\right)$

20. $z=\ln (x^{2}+y^{2})$ at $(1,-1,\ln 2)$

21. $x^{2/3}+y^{2/3}+z^{2/3}=9$ at $(1,8,-8)$

22. $x^{1/2}+y^{1/2}+z^{1/2}=6$ at $(1,4,9)$

23. Show that the same results would have been obtained in Example 1 by solving the equation $x^{2}+y^{2}-z^{2}-24=0$ for $z$ and using the function $z=f(x,y)=\sqrt{x^{2}+y^{2}-24}$ whose graph is the part of the hyperboloid lying above the $xy$-plane.

24. Show that the same results would have been obtained in Example 2 by solving the equation $x^{2}+y^{2}-z^{2}-24=0$ for $z$ and using the function $z=f(x,y)=\sqrt{x^{2}+y^{2}-24}$ whose graph is the part of the hyperboloid lying above the $xy$-plane.

25. $z=6x-4y-x^{2}-2y^{2}$

26. $z=4x-2y+x^{2}+y^{2}$

27. $z=x^{2}+2xy+y^{2}$

28. $z=x^{2}-3xy+y^{2}$

29. $z=2x^{4}-y^{2}-x^{2}-2y$

30. $z=x^{2}+y^{4}-4y^{2}-2x$

31. Solar Panel An engineer is building experimental, dome- shaped living quarters $4\,\rm{m}$ high and modeled by the function $z=4-x^{2}-y^{2}.$ She wants to bolt a flat solar panel to the dome at the point $(1,1,2)$. In what direction should the engineer drill?

32. 1. (a) Two surfaces are said to be tangent at a common point $\boldsymbol P_{0}$ if each has the same tangent plane at $\boldsymbol P_{0}$. Show that the surfaces $x^{2}+z^{2}+4y=0$ and $x^{2}+y^{2}+z^{2}-6z+7=0$ are tangent at the point $(0,-1,2)$.
    2. (b) Graph each surface.

33. 1. (a) Show that the surfaces $x^{2}+y^{2}+z^{2}=1$ and $x^{2}+y^{2}-z^{2}=1$ are tangent at every point of intersection.
    2. (b) Graph each surface.

34. 1. (a) Two surfaces are orthogonal at a common point $\boldsymbol P_{0}$ if their normal lines at $\boldsymbol P_{0}$ are orthogonal. Show that the surfaces $x^{2}+y^{2}+z^{2}=4$ and $x^{2}+y^{2}-z^{2}=0$ are orthogonal at $(0,\sqrt{2} ,\sqrt{2})$. In fact, show that the surfaces are orthogonal at every point of intersection.
    2. (b) Graph each surface.

35. Normal Lines of a Sphere Show that the normal lines of a sphere $x^{2}+y^{2}+z^{2}=R^{2}$ pass through the center of the sphere.

36. Normal Line of a Sphere Let $P_{0}=(x_{0},y_{0},z_{0})$ be a point on the sphere $x^{2}+y^{2}+z^{2}=R^{2}$. Show that the normal line to the sphere at $P_{0}$ passes through the point $(-x_{0},-y_{0},-z_{0})$.

37. Tangent Plane Find equations of the tangent plane and normal line to $\sin (x+y)+\sin (y+z)=1$ at the point $\left( 0,\dfrac{\pi }{ 2},\dfrac{\pi }{2}\right)$.

38. Tangent Plane to an Ellipsoid Find the points on the surface $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}+\dfrac{z^{2}}{36}=1$, where the tangent plane is parallel to the plane $2x-3y+z=4$.

39. Tangent Plane to a Cylinder Show that the tangent plane to the cylinder $x^{2}+y^{2}=a^{2}$ at the point $(x_{0},y_{0},z_{0})$ is given by the equation $x_{0}x+y_{0}y=a^{2}$.

40. Tangent Plane to a Cone Show that the tangent plane to the cone $\dfrac{z^{2}}{c^{2}}=\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}$ at the point $(x_{0},y_{0},z_{0})\neq (0,0,0)$ is given by the equation $\dfrac{ z_{0}}{c^{2}}z=\dfrac{x_{0}}{a^{2}}x+\dfrac{y_{0}}{b^{2}}y$. Why must the point of tangency be different from the origin?

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41. Show that at any point, the sum of the intercepts on the coordinate axes of the tangent plane to the surface $x^{1/2}+y^{1/2}+z^{1/2}=a^{1/2}$, $a>0$, is a constant.

42. Tangent Line Find an equation of the tangent line to the curve of intersection of the surfaces $x\sin (yz) =1$ and $ze^{y^{2}-x^{2}}=\dfrac{\pi }{2}$ at the point $\left( 1,1,\dfrac{\pi }{2} \right)$.

43. Find equations of the tangent plane and normal line to $(yz)^{xz}=16$ at the point $(2,1,2).$

44. Consider the tangent plane to the graph of a differentiable function $z=f(x,y)$ at a point $(x_{0},y_{0})$. Suppose $\mathbf{v}_{1}$ is a tangent vector on the plane whose $\mathbf{j}$ component is $0$ and $\mathbf{v}_{2}$ is a tangent vector on the plane whose $\mathbf{i}$ component is $0$. Must $\mathbf{ v}_{1}$ and $\mathbf{v}_{2}$ be at right angles? See the figure.