Answer: In Exercises 516, find a unit normal vector to the surface at the given point

In Exercises 1–4, describe the level surface F(x, y, z) = 0. \(F(x, y, z)=3 x-5 y+3 z-15\) Text Transcription: F(x,y,z)=3x-5y+3z-15

In Exercises 1–4, describe the level surface F(x, y, z) = 0. \(F(x, y, z)=x^{2}+y^{2}+z^{2}-25\) Text Transcription: F(x,y,z)=x^2+y^2+z^2-25

In Exercises 1–4, describe the level surface F(x, y, z) = 0. \(F(x, y, z)=4 x^{2}+9 y^{2}-4 z^{2}\) Text Transcription: F(x,y,z)=4x^2+9y^2-4z^2

In Exercises 1–4, describe the level surface F(x, y, z) = 0. \(F(x, y, z)=16 x^{2}-9 y^{2}+36 z\) Text Transcription: F(x,y,z)=16x^2-9y^2+36z

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(3 x+4 y+12 z=0\) Point (0, 0, 0) Text Transcription: nablaF(x,y,z)] 3x+4y+12z=0

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(x+y+z=4\) Point (2, 0, 2) Text Transcription: nablaF(x,y,z)] x+y+z=4

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(x^{2}+y^{2}+z^{2}=6\) Point (1, 1, 2) Text Transcription: nablaF(x,y,z)] x^2+y^2+z^2=6

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(z=\sqrt{x^{2}+y^{2}}\) Point (3, 4, 5) Text Transcription: nablaF(x,y,z)] z=sqrtx^2+y^2

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(z=x^{3}\) Point (2, -1, 8) Text Transcription: nablaF(x,y,z)] z=x^3

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(x^{2} y^{4}-z=0\) Point (1, 2, 16) Text Transcription: nablaF(x,y,z)] x^2y^4-z=0

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(x^{2}+3 y+z^{3}=9\) Point (2, -1, 2) Text Transcription: nablaF(x,y,z)] x^2+3y+z^3=9

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(x^{2} y^{3}-y^{2} z+2 x z^{3}=4\) Point (-1, 1, -1) Text Transcription: nablaF(x,y,z)] x^2y^3-y^2z+2xz^3=4

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(\ln \left(\frac{x}{y-z}\right)=0\) Point (1, 4, 3) Text Transcription: nablaF(x,y,z)] ln(fracxy-z)

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(z e^{x^{2}-y^{2}}-3=0\) Point (2, 2, 3) Text Transcription: nablaF(x,y,z)] ze^x^2-y^2-3=0

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(z-x \sin y=4) Point \(\left(6, \frac{\pi}{6}, 7\right)\) Text Transcription: nablaF(x,y,z)] z-xsiny=4 (6,fracpi6,7)

In Exercises 5–16, find a unit normal vector to the surface at the given point. [Hint: Normalize the gradient vector \(\nabla F(x, y, z) ]\). Surface \(\sin (x-y)-z=2\) Point \(\left(\frac{\pi}{3}, \frac{\pi}{6},-\frac{3}{2}\right)\) Text Transcription: nablaF(x,y,z)] sin(x-y)-z=2 (fracpi3,fracpi6,-frac32)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(z=x^{2}+y^{2}+3\) (2, 1, 8) Text Transcription: z=x^2+y^2+3

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(f(x, y)=\frac{y}{x}\) (1, 2, 2) Text Transcription: f(x,y)=fracyx

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(z=\sqrt{x^{2}+y^{2}}\), (3, 4, 5) Text Transcription: z=sqrtx^2+y^2

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(g(x, y)=\arctan \frac{y}{x}, \quad(1,0,0)\) Text Transcription: g(x,y)=arctan fracyx,(1,0,0)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(g(x, y)=x^{2}+y^{2}, \quad(1,-1,2)\) Text Transcription: g(x,y)=x^2+y^2,(1,-1,2)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(f(x, y)=x^{2}-2 x y+y^{2}, \quad(1,2,1)\) Text Transcription: f(x,y)=x^2-2xy+y^2,(1,2,1)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(z=2-\frac{2}{3} x-y, \quad(3,-1,1)\) Text Transcription: z=2-frac23x-y,(3,-1,1)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(z=e^{x}(\sin y+1), \quad\left(0, \frac{\pi}{2}, 2\right)\) Text Transcription: z=e^x(siny+1),(0,fracpi2,2)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(h(x, y)=\ln \sqrt{x^{2}+y^{2}}, \quad(3,4, \ln 5)\) Text Transcription: h(x,y)=lnsqrtx^2+y^2,(3,4,ln5)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(h(x, y)=\cos y, \quad\left(5, \frac{\pi}{4}, \frac{\sqrt{2}}{2}\right)\) Text Transcription: h(x,y)=cosy,(5,fracpi4,fracsqrt22)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(x^{2}+4 y^{2}+z^{2}=36, \quad(2,-2,4)\) Text Transcription: x^2+4y^2+z^2=36,(2,-2,4)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(x^{2}+2 z^{2}=y^{2}, \quad(1,3,-2)\) Text Transcription: x^2+2z^2=y^2,(1,3,-2)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(x y^{2}+3 x-z^{2}=8, \quad(1,-3,2)\) Text Transcription: x y^2+3x-z^2=8,(1,-3,2)

In Exercises 17–30, find an equation of the tangent plane to the surface at the given point. \(x=y(2 z-3)\), (4, 4, 2) Text Transcription: x=y(2z-3)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(x+y+z=9, \quad(3,3,3)\) Text Transcription: x+y+z=9,(3,3,3)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(x^{2}+y^{2}+z^{2}=9, \quad(1,2,2)\) Text Transcription: x^2+y^2+z^2=9,(1,2,2)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(x^{2}+y^{2}+z=9, \quad(1,2,4)\) Text Transcription: x^2+y^2+z=9,(1,2,4)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(z=16-x^{2}-y^{2}, \quad(2,2,8)\) Text Transcription: z=16-x^2-y^2,(2,2,8)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(z=x^{2}-y^{2}, \quad(3,2,5)\) Text Transcription: z=x^2-y^2,(3,2,5)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(x y-z=0, \quad(-2,-3,6)\) Text Transcription: xy-z=0,(-2,-3,6)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(x y z=10, \quad(1,2,5)\) Text Transcription: xyz=10,(1,2,5)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(z=y e^{2 x y}, \quad(0,2,2)\) Text Transcription: z=ye^2xy,(0,2,2)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(z=\arctan \frac{y}{x}, \quad\left(1,1, \frac{\pi}{4}\right)\) Text Transcription: z=arctanfracyx,(1,1,fracpi4)

In Exercises 31– 40, find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. \(y \ln x z^{2}=2, \quad(e, 2,1)\) Text Transcription: ylnxz^2=2,(e,2,1)

In Exercises 41– 46, (a) find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection. \(x^{2}+y^{2}=2, \quad z=x, \quad(1,1,1)\) Text Transcription: x^2+y^2=2,z=x,(1,1,1)

In Exercises 41– 46, (a) find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection. \(z=x^{2}+y^{2}, \quad z=4-y, \quad(2,-1,5)\) Text Transcription: z=x^2+y^2,z=4-y,(2,-1,5)

In Exercises 41– 46, (a) find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection. \(x^{2}+z^{2}=25, \quad y^{2}+z^{2}=25, \quad(3,3,4)\) Text Transcription: x^2+z^2=25,y^2+z^2=25,(3,3,4)

In Exercises 41– 46, (a) find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection. \(z=\sqrt{x^{2}+y^{2}}, \quad 5 x-2 y+3 z=22, \quad(3,4,5)\) Text Transcription: z=sqrtx^2+y^2,5x-2y+3z=22,(3,4,5)

In Exercises 41– 46, (a) find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection. \(x^{2}+y^{2}+z^{2}=14, \quad x-y-z=0, \quad(3,1,2)\) Text Transcription: x^2+y^2+z^2=14,x-y-z=0,(3,1,2)

In Exercises 41– 46, (a) find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point, and (b) find the cosine of the angle between the gradient vectors at this point. State whether or not the surfaces are orthogonal at the point of intersection. \(z=x^{2}+y^{2}, \quad x+y+6 z=33, \quad(1,2,5)\) Text Transcription: z=x^2+y^2,x+y+6z=33,(1,2,5)

In Exercises 47–50, find the angle of inclination of the tangent plane to the surface at the given point. \(3 x^{2}+2 y^{2}-z=15, \quad(2,2,5)\) Text Transcription: 3x^2+2y^2-z=15,(2,2,5)

In Exercises 47–50, find the angle of inclination of the tangent plane to the surface at the given point. \(2 x y-z^{3}=0, \quad(2,2,2)\) Text Transcription: 2xy-z^3=0,(2,2,2)

In Exercises 47–50, find the angle of inclination of the tangent plane to the surface at the given point. \(x^{2}-y^{2}+z=0, \quad(1,2,3)\) Text Transcription: x^2-y^2+z=0,(1,2,3)

In Exercises 47–50, find the angle of inclination of the tangent plane to the surface at the given point. \(x^{2}+y^{2}=5, \quad(2,1,3)\) Text Transcription: x^2+y^2=5,(2,1,3)

In Exercises 51–56, find the point(s) on the surface at which the tangent plane is horizontal. \(z=3-x^{2}-y^{2}+6 y\) Text Transcription: z=3-x^2-y^2+6y

In Exercises 51–56, find the point(s) on the surface at which the tangent plane is horizontal. \(z=3 x^{2}+2 y^{2}-3 x+4 y-5\) Text Transcription: z=3x^2+2y^2-3x+4y-5

In Exercises 51–56, find the point(s) on the surface at which the tangent plane is horizontal. \(z=x^{2}-x y+y^{2}-2 x-2 y\) Text Transcription: z=x^2-xy+y^2-2x-2y

In Exercises 51–56, find the point(s) on the surface at which the tangent plane is horizontal. \(z=4 x^{2}+4 x y-2 y^{2}+8 x-5 y-4\) Text Transcription: z=4x^2+4xy-2y^2+8x-5y-4

In Exercises 51–56, find the point(s) on the surface at which the tangent plane is horizontal. \(z=5 x y\) Text Transcription: z=5xy

In Exercises 51–56, find the point(s) on the surface at which the tangent plane is horizontal. \(z=x y+\frac{1}{x}+\frac{1}{y}\) Text Transcription: z=xy+frac1x+frac1y

In Exercises 57 and 58, show that the surfaces are tangent to each other at the given point by showing that the surfaces have the same tangent plane at this point. \(\begin{array}{l} x^{2}+2 y^{2}+3 z^{2}=3, x^{2}+y^{2}+z^{2}+6 x-10 y+14=0 \\ (-1,1,0)\end{array}\) Text Transcription: x^2+2y^2+3z^2=3,x^2+y^2+z^2+6x-10y+14=0 (-1,1,0)

In Exercises 57 and 58, show that the surfaces are tangent to each other at the given point by showing that the surfaces have the same tangent plane at this point. \(\begin{array}{l} x^{2}+y^{2}+z^{2}-8 x-12 y+4 z+42=0, x^{2}+y^{2}+2 z=7 \\ (2,3,-3)\end{array}\) Text Transcription: x^2+y^2+z^2-8x-12y+4z+42=0,x^2+y^2+2z=7 (2,3,-3)

In Exercises 59 and 60, (a) show that the surfaces intersect at the given point, and (b) show that the surfaces have perpendicular tangent planes at this point. \(z=2 x y^{2}, 8 x^{2}-5 y^{2}-8 z=-13, \quad(1,1,2)\) Text Transcription: z=2xy^2,8x^2-5y^2-8z=-13,(1,1,2)

In Exercises 59 and 60, (a) show that the surfaces intersect at the given point, and (b) show that the surfaces have perpendicular tangent planes at this point. \(\begin{array}{l} x^{2}+y^{2}+z^{2}+2 x-4 y-4 z-12=0 \\ 4 x^{2}+y^{2}+16 z^{2}=24, \quad(1,-2,1) \end{array}\) Text Transcription: x^2+y^2+z^2+2x-4y-4z-12=0 4x^2+y^2+16z^2=24,(1,-2,1)

Find a point on the ellipsoid \(x^{2}+4 y^{2}+z^{2}=9\) where the tangent plane is perpendicular to the line with parametric equations \(x=2-4 t, y=1+8 t \text {, and } z=3-2 t \text {. }\) Text Transcription: x^2+4y^2+z^2=9 x=2-4t,y=1+8tandz=3-2t

Find a point on the hyperboloid \(x^{2}+4 y^{2}-z^{2}=1\) where the tangent plane is parallel to the plane \(x+4 y-z=0\). Text Transcription: x^2+4y^2-z^2=1 x+4y-z=0

Give the standard form of the equation of the tangent plane to a surface given by \(F(x, y, z)=0 \text { at }\left(x_{0}, y_{0}, z_{0}\right)\) Text Transcription: F(x,y,z)=0at(x_0,y_0,z_0)

For some surfaces, the normal lines at any point pass through the same geometric object. What is the common geometric object for a sphere? What is the common geometric object for a right circular cylinder? Explain.

Discuss the relationship between the tangent plane to a surface and approximation by differentials.

Consider the elliptic cone given by \(x^{2}-y^{2}+z^{2}=0\). (a) Find an equation of the tangent plane at the point (5, 13, -12). (b) Find symmetric equations of the normal line at the point (5, 13, -12).

Consider the function \(f(x, y)=\frac{4 x y}{\left(x^{2}+1\right)\left(y^{2}+1\right)}\) on the intervals \(-2 \leq x \leq 2\) and \(0 \leq y \leq 3\). (a) Find a set of parametric equations of the normal line and an equation of the tangent plane to the surface at the point (1, 1, 1). (b) Repeat part (a) for the point \(\left(-1,2,-\frac{4}{5}\right)\). (c) Use a computer algebra system to graph the surface, the normal lines, and the tangent planes found in parts (a) and (b). Text Transcription: f(x,y)=frac4xy(x^2+1)(y^2+1) -2leqxleq2 0leqyleq3 (-1,2,-frac45)

Consider the function \(f(x, y)=\frac{\sin y}{x}\) on the intervals \(-3 \leq x \leq 3\) and \(0 \leq y \leq 2 \pi\). (a) Find a set of parametric equations of the normal line and an equation of the tangent plane to the surface at the point \(\left(2, \frac{\pi}{2}, \frac{1}{2}\right)\). (b) Repeat part (a) for the point \(\left(-\frac{2}{3}, \frac{3 \pi}{2}, \frac{3}{2}\right)\). (c) Use a computer algebra system to graph the surface, the normal lines, and the tangent planes found in parts (a) and (b). Text Transcription: f(x,y)=fracsinyx -3leqxleq3 (2,fracpi2,frac12) (-frac23,frac3pi2,frac32)

Consider the functions \(f(x, y)=6-x^{2}-y^{2} / 4\) and \(g(x, y)=2 x+y\). (a) Find a set of parametric equations of the tangent line to the curve of intersection of the surfaces at the point (1, 2, 4), and find the angle between the gradient vectors. (b) Use a computer algebra system to graph the surfaces. Graph the tangent line found in part (a). Text Transcription: f(x,y)=6-x^2-y^2/4 g(x,y)=2x+y

Consider the functions \(f(x, y)=\sqrt{16-x^{2}-y^{2}+2 x-4 y}\) and \(g(x, y)=\frac{\sqrt{2}}{2} \sqrt{1-3 x^{2}+y^{2}+6 x+4 y}\). (a) Use a computer algebra system to graph the first-octant portion of the surfaces represented by f and g. (b) Find one first-octant point on the curve of intersection and show that the surfaces are orthogonal at this point. (c) These surfaces are orthogonal along the curve of intersection. Does part (b) prove this fact? Explain. Text Transcription: f(x,y)=sqrt16-x^2-y^2+2x-4y g(x,y)=fracsqrt22sqrt1-3x^2+y^2+6x+4y

In Exercises 71 and 72, show that the tangent plane to the quadric surface at the point can be written in the given form. \(\text { Ellipsoid: } \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\) \(\text { Plane: } \frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}+\frac{z_{0} z}{c^{2}}=1\) Text Transcription: Ellipsoid:fracx^2a^2+fracy^2b^2+fracz^2c^2=1 Plane:fracx_0xa^2+fracy_0yb^2+fracz_0zc^2=1

In Exercises 71 and 72, show that the tangent plane to the quadric surface at the point can be written in the given form. \(\text { Hyperboloid: } \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\) \(\text { Plane: } \frac{x_{0} x}{a^{2}}+\frac{y_{0} y}{b^{2}}-\frac{z_{0} z}{c^{2}}=1\) Text Transcription: Hyperboloid:fracx^2a^2+fracy^2b^2-fracz^2c^2=1 Plane:fracx_0xa^2+fracy_0yb^2-fracz_0zc^2=1

Show that any tangent plane to the cone \(z^{2}=a^{2} x^{2}+b^{2} y^{2}\) passes through the origin. Text Transcription: z^2=a^2x^2+b^2y^2

Let f be a differentiable function and consider the surface \(z=x f(y / x)\). Show that the tangent plane at any point \(P\left(x_{0}, y_{0}, z_{0}\right)\) on the surface passes through the origin. Text Transcription: z=xf(y/x) P(x_0,y_0,z_0)

Consider the following approximations for a function f(x, y) centered at (0, 0). Linear approximation: \(P_{1}(x, y)=f(0,0)+f_{x}(0,0) x+f_{y}(0,0) y\) Quadratic approximation: \(\begin{aligned} P_{2}(x, y)=& f(0,0)+f_{x}(0,0) x+f_{y}(0,0) y+\\ & \frac{1}{2} f_{x x}(0,0) x^{2}+f_{x y}(0,0) x y+\frac{1}{2} f_{y y}(0,0) y^{2} \end{aligned}\) [Note that the linear approximation is the tangent plane to the surface at (0, 0, f(0, 0)).] (a) Find the linear approximation of \(f(x, y)=e^{(x-y)}\) centered at (0, 0). (b) Find the quadratic approximation of \(f(x, y)=e^{(x-y)}\) centered at (0, 0). (c) If x = 0 in the quadratic approximation, you obtain the second-degree Taylor polynomial for what function? Answer the same question for y = 0. (d) Complete the table. (e) Use a computer algebra system to graph the surfaces \(z=f(x, y), z=P_{1}(x, y), \text { and } z=P_{2}(x, y)\). Text Transcription: P_1(x,y)=f(0,0)+f_x(0,0)x+f_y(0,0)y P_2(x,y)=&f(0,0)+f_x(0,0)x+f_y(0,0)y+ frac12f_xx(0,0)x^2+f_xy(0,0)xy+frac12f_yy(0,0)y^2 f(x,y)=e^(x-y) z=f(x,y) z=P_1(x,y) z=P_2(x,y)

Repeat Exercise 75 for the function \(f(x, y)=\cos (x+y)\). Text Transcription: f(x,y)=cos(x+y)

Prove that the angle of inclination \(\boldsymbol{\theta}\) of the tangent plane to the surface z = f(x,y) at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is given by \(\cos \theta=\frac{1}{\sqrt{\left[f_{x}\left(x_{0}, y_{0}\right)\right]^{2}+\left[f_{y}\left(x_{0}, y_{0}\right)\right]^{2}+1}}\). Text Transcription: theta (x_0,y_0,z_0) costheta=frac1sqrt[f_x (x_0,y_0)]^2+[f_y(x_0,y_0)]^2+1

Prove Theorem 13.14.