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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 \(P_{0}\).

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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 \({\boldsymbol \nabla }F(x_{0},y_{0},z_{0})\neq \mathbf{0}\).

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

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

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

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

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

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

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

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

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

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

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\(2x^{2}-y^{2}=z\) at \((1,0,2)\)

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

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

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

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

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

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

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

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

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

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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.

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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.

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\(z=6x-4y-x^{2}-2y^{2}\)

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\(z=4x-2y+x^{2}+y^{2}\)

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\(z=x^{2}+2xy+y^{2}\)

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\(z=x^{2}-3xy+y^{2}\)

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\(z=2x^{4}-y^{2}-x^{2}-2y\)

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\(z=x^{2}+y^{4}-4y^{2}-2x\)

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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?

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1. 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. Graph each surface.

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1. 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. Graph each surface.

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1. 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. Graph each surface.

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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.

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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 \((-x_{0},-y_{0},-z_{0})\).

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

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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 \(2x-3y+z=4\).

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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 \(x_{0}x+y_{0}y=a^{2}\).

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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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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.

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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 \(\left( 1,1,\dfrac{\pi }{2} \right) \).

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

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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.