Solved: ?Shown is a topographic map of Blue River Pine Provincial Park in British

Level curves for barometric pressure (in millibars) are shown for 6:00 am on a day in November. A deep low with pressure 972 mb is moving over northeast Iowa. The distance along the red line from \(K\) (Kearney, Nebraska) to \(S\) (Sioux City, Iowa) is 300 km. Estimate the value of the directional derivative of the pressure function at Kearney in the direction of Sioux City. What are the units of the directional derivative? Equation Transcription: Text Transcription: K S

The contour map shows the average maximum temperature for November 2004 (in \({ }^{\circ} \mathrm{C}\)). Estimate the value of the directional derivative of this temperature function at Dubbo, New South Wales, in the direction of Sydney. What are the units? Equation Transcription: Text Transcription: degree C

The wind-chill index \(W\) is the perceived temperature when the actual temperature is \(T\) and the wind speed is \(v\), so we can write \(W=f(T,\ v)\). The following table of values is an excerpt from Table 1 in Section 14.1. Use the table to estimate the value of \(D_{\mathrm{u}}\ f(-20,\ 30)\), where \(\mathbf{u}=(\mathbf{i}+\mathbf{j}) / \sqrt{2}\). Equation Transcription: Text Transcription: W T v W=f(T, v) D_u f(-20, 30) u=(i+j)/ sqrt 2

Find the directional derivative of \(f\) at the given point in the direction indicated by the angle \(\theta\). \(f(x,\ y)=xy^3-x^2,\quad\ \ \ \ (1,\ 2),\quad\ \ \ \ \theta=\pi/3\) Equation Transcription: Text Transcription: f theta f(x, y)=xy^3 - x^2, (1, 2), pi=/3

Find the directional derivative of \(f\) at the given point in the direction indicated by the angle \(\theta\). \(f(x,\ y)=y\ \cos\ (xy),\quad\ \ \ \ \ (0,1),\quad\ \ \ \ \ \theta=\pi/4\) Equation Transcription: Text Transcription: f theta f(x, y)=y cos?(xy), (0,1), theta = pi/4

Find the directional derivative of \(f\) at the given point in the direction indicated by the angle \(\theta\). \(f(x,\ y)=\sqrt{2x+3y},\ \ \ \ \ (3,1),\quad\ \ \ \ \ \theta=-\pi/6\) Equation Transcription: Text Transcription: f theta f(x, y)= sqrt 2x+3y, (3,1), theta = - pi/6

Find the directional derivative of \(f\) at the given point in the direction indicated by the angle \(\theta\). \(f(x,\ y)=\arctan(xy),\quad\ \ \ \ \ (2,-3),\quad\ \ \ \ \ \theta=3\pi/4\) Equation Transcription: Text Transcription: f theta f(x, y) = arctan(xy), (2,-3), theta = 3pi/4

(a) Find the gradient of \(f\). (b) Evaluate the gradient at the point \(P\). (c) Find the rate of change of \(f\) at \(P\) in the direction of the vector \(u\). \(f(x,\ y)=x^2e^y,\quad\ \ \ \ \ P(3,0),\quad\ \ \ \ \ \mathbf{u}=\frac{1}{5}(3\mathbf{i}-4\mathbf{j})\) Equation Transcription: Text Transcription: f P f P u f(x, y) = x^2 e^y, P(3, 0), u = 1/5(3i - 4j)

(a) Find the gradient of \(f\). (b) Evaluate the gradient at the point \(P\). (c) Find the rate of change of \(f\) at \(P\) in the direction of the vector \(u\). \(f(x,\ y)=x/y,\quad\ \ \ \ \ P(2,1),\quad\mathbf{\ \ \ \ \ u}=\frac{3}{5}\mathbf{i}+\frac{4}{5}\mathbf{j}\) Equation Transcription: Text Transcription: f P f P u f(x, y) = x/y, P(2, 1), u = 3/5 i + 4/5 j

(a) Find the gradient of \(f\). (b) Evaluate the gradient at the point \(P\). (c) Find the rate of change of \(f\) at \(P\) in the direction of the vector \(u\). \(f(x,\ y)=x^2\ \ln\ y,\quad\ \ \ \ \ P(3,1),\quad\mathbf{\ \ \ \ \ u}=-\frac{5}{13}\mathbf{i}+\frac{12}{13}\mathbf{j}\) Equation Transcription: Text Transcription: f P f P u f(x, y) = x^2 ln y, P(3, 1), u = - 5/13 i + 12/13 j

(a) Find the gradient of \(f\). (b) Evaluate the gradient at the point \(P\). (c) Find the rate of change of \(f\) at \(P\) in the direction of the vector \(u\). \(f(x,\ y,\ z)=x^2yz-xyz^3,\quad\ \ \ \ \ P(2,\ -1,\ 1),\quad\mathbf{\ \ \ \ \ u}=\left\langle0,\ \frac{4}{5},\ -\frac{3}{5}\right\rangle\) Equation Transcription: ? ? Text Transcription: f P f P u f(x, y, z) = x^2 yz - xyz^3, P(2, -1, 1), u =? 0, 4/5, - 3/5?

(a) Find the gradient of \(f\). (b) Evaluate the gradient at the point \(P\). (c) Find the rate of change of \(f\) at \(P\) in the direction of the vector \(u\). \(f(x,\ y,\ z)=y^2e^{xyz},\quad\ \ \ \ \ P(0,\ 1,\ -1),\quad\mathbf{\ \ \ \ \ u}=\left\langle\frac{3}{13},\ \frac{4}{13},\ \frac{12}{13}\right\rangle\) Equation Transcription: ? ? Text Transcription: f P f P u f(x, y, z) = y^2 e^xyz, P(0, 1, -1), u = ? 3/13, 4/13, 12/13 ?

Find the directional derivative of the function at the given point in the direction of the vector \(v\). \(f(x, \ y)=e^{x} \ \sin \ y, \quad \ \ \ (0, \pi / 3), \quad \ \ \ \mathbf{v}=\langle-6, \ 8\rangle\) Equation Transcription: ? ? Text Transcription: v f(x, y) = e^x sin y, (0, pi/3), v = ? -6, 8 ?

Find the directional derivative of the function at the given point in the direction of the vector \(v\). \(f(x, \ y)=\frac{x}{x^{2}+y^{2}}, \quad \ \ \ (1, \ 2), \quad \ \ \ \mathbf{v}=\langle 3, \ 5\rangle\) Equation Transcription: ? ? Text Transcription: v f(x, y) = x / x^2 + y^2, (1, 2), v = ? 3, 5 ?

Find the directional derivative of the function at the given point in the direction of the vector \(v\). \(g(s,\ t)=s\sqrt{t},\quad\ \ \ \ \ (2,4),\quad\mathbf{\ \ \ \ \ v}=2\mathbf{i}-\mathbf{j}\) Equation Transcription: Text Transcription: v g(s, t) = s sqrt t, (2, 4), v = 2i - j

Find the directional derivative of the function at the given point in the direction of the vector \(v\). \(g(u,\ v)=u^2e^{-v},\quad\ \ \ \ \ (3,0),\quad\ \ \ \ \ \mathbf{v}=3\mathbf{i}+4\mathbf{j}\) Equation Transcription: Text Transcription: v g(u, v) = u^2 e^-v, (3, 0), v = 3i + 4j

Find the directional derivative of the function at the given point in the direction of the vector \(v\). \(f(x, \ y, \ z)=x^{2} y+y^{2} z, \quad \ \ \ (1,2,3), \quad \ \ \ \mathbf{v}=\langle 2, \ -1, \ 2\rangle\) Equation Transcription: ? ? Text Transcription: v f(x, y, z) = x^2 y + y^2 z, (1, 2, 3), v= ? 2, -1, 2 ?

Find the directional derivative of the function at the given point in the direction of the vector \(v\). \(f(x, \ y, \ z)=x y^{2} \tan ^{-1} z, \quad \ \ \ (2, \ 1, \ 1), \quad \ \ \ \mathbf{v}=\langle 1, \ 1, \ 1\rangle\) Equation Transcription: ? ? Text Transcription: v f(x, y, z) = xy^2 tan^-1 z, (2. 1, 1), v = ? 1, 1, 1 ?

Find the directional derivative of the function at the given point in the direction of the vector \(v\). \(\begin{array}{l}h(r,\ s,\ t)=\ln(3r+6s+9t),\quad\ \ \ \ \ (1,\ 1,\ 1),\ \ \ \ \ \ v=4\mathbf{i}+12\mathbf{j}+6\mathbf{k}\end{array}\) Equation Transcription: Text Transcription: v h(r, s, t) = ln(3r + 6s + 9t), (1, 1, 1), v = 4i + 12j + 6k

Use the figure to estimate \(D_{\mathrm{u}}\ f(2,\ 2)\). Equation Transcription: Text Transcription: D_u f(2, 2)

Find the directional derivative of the function at the point \(P\) in the direction of the point \(Q\). \(f(x,\ y)=x^2y^2-y^3,\quad\ \ \ \ \ P(1,\ 2),\quad\ \ \ \ \ Q(-3,\ 5)\) Equation Transcription: Text Transcription: P Q f(x, y) = x^2 y^2 - y^3, P(1, 2), Q(-3, 5)

Find the directional derivative of the function at the point \(P\) in the direction of the point \(Q\). \(f(x,\ y)=\frac{x}{y^2},\quad\ \ \ \ \ P(3,\ -1),\quad\ \ \ \ \ Q(-2,\ 11)\) Equation Transcription: Text Transcription: P Q f(x, y) = x/y^2, P(3, -1), Q(-2, 11)

Find the directional derivative of the function at the point \(P\) in the direction of the point \(Q\). \(f(x,\ y)=\sqrt{xy},\quad\ \ \ \ \ P(2,\ 8),\quad\ \ \ \ \ Q(5,\ 4)\) Equation Transcription: Text Transcription: P Q f(x, y) = sqrt xy, P(2, 8), Q(5, 4)

Find the directional derivative of the function at the point \(P\) in the direction of the point \(Q\). \(f(x,\ y,\ z)=xy^2z^3,\quad\ \ \ \ \ P(2,\ 1,\ 1),\quad\ \ \ \ \ Q(0,\ -3,\ 5)\) Equation Transcription: Text Transcription: P Q f(x, y, z) = xy^2 z^3 , P(2, 1, 1), Q(0, -3, 5)

Find the directional derivative of the function at the point \(P\) in the direction of the point \(Q\). \(f(x,\ y,\ z)=xy-xy^2z^2,\quad\ \ \ \ \ P(2,\ -1,\ 1),\quad\ \ \ \ \ Q(5,\ 1,\ 7)\) Equation Transcription: Text Transcription: P Q f(x, y, z) = xy - xy^2 z^2 , P(2, -1, 1), Q(5, 1, 7)

The contour map of a function \(f\) is shown. At points \(P\), \(Q\), and \(R\), draw an arrow to indicate the direction of the gradient vector. Equation Transcription: Text Transcription: f P Q R

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. \(f(x,\ y)=5xy^2,\quad\ \ \ (3,-2)\) Equation Transcription: Text Transcription: f f(x, y) = 5xy^2, (3, -2)

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. \(f(s,\ t)=\frac{s}{s^2+t^2},\quad\ \ \ (-1,\ 1)\) Equation Transcription: Text Transcription: f f(s, t) = s / s^2 + t^2, (-1, 1)

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. \(f(x,\ y)=\sin(xy),\quad\ \ \ (1,\ 0)\) Equation Transcription: Text Transcription: f f(x, y) = sin(xy), (1, 0)

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. \(f(x,\ y,\ z)=x\ \ln(yz),\quad\ \ \ \left(1,\ 2,\ \frac{1}{2}\right)\) Equation Transcription: Text Transcription: f f(x, y, z) = x ln(yz), (1, 2, 1/2)

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. \(f(x,\ y,\ z)=x/(y+z),\quad\ \ \ (8,1,3)\) Equation Transcription: Text Transcription: f f(x, y, z) = x/(y + z), (8, 1, 3)

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. \(f(p,\ q,\ r)=\arctan(pqr),\quad\ \ \ (1,\ 2,\ 1)\) Equation Transcription: Text Transcription: f f(p, q, r) = arctan(pqr), (1, 2, 1)

Direction of Most Rapid Decrease (a) Show that a differentiable function \(f\) decreases most rapidly at \(x\) in the direction opposite the gradient vector, that is, in the direction of \(-\nabla f(\mathbf{x})\), and that the maximum rate of decrease is \(-|\nabla f(\mathbf{x})|\). (b) Use the result of part (a) to find the direction in which the function \(f(x,\ y)=x^4y-x^2y^3\) decreases fastest at the point \((2,\ -3)\). What is the rate of decrease? Equation Transcription: ???? ???? Text Transcription: f x - nabla f(x) -|nabla f(x)| f(x, y)=x^4 y - x^2 y^3 (2, -3)

Find the directions in which the directional derivative of \(f(x,\ y)=x^2+xy^3\) at the point \((2,\ 1)\) has the value \(2\). Equation Transcription: Text Transcription: f(x, y)=x^2 + xy^3 (2, 1) 2

Find all points at which the direction of greatest rate of change of the function \(f(x,\ y)=x^2+y^2-2x-4y\) is \(\mathbf{i}+\mathbf{j}\). Equation Transcription: Text Transcription: f(x, y)=x^2 + y^2 - 2x - 4y i+j

Near a buoy, the depth of a lake at the point with coordinates \((x,\ y)\) is \(z=200+0.02 x^{2}-0.001 y^{3}\), where \(x\), \(y\), and \(z\) are measured in meters. A fisherman in a small boat starts at the point \((80,\ 60)\) and moves toward the buoy, which is located at \((0,\ 0)\). Is the water under the boat getting deeper or shallower when he departs? Explain. Equation Transcription: Text Transcription: (x, y) z=200 + 0.02x^2 - 0.001y^3 x y z (80, 60) (0, 0)

The temperature \(T\) in a metal ball is inversely proportional to the distance from the center of the ball, which we take to be the origin. The temperature at the point \((1,\ 2,\ 2)\) is \(120^{\circ}\). (a) Find the rate of change of \(T\) at \((1,\ 2,\ 2)\) in the direction toward the point \((2,\ 1,\ 3)\). (b) Show that at any point in the ball the direction of greatest increase in temperature is given by a vector that points toward the origin. Equation Transcription: Text Transcription: T (1, 2, 2) 120 degree T (1, 2, 2) (2, 1, 3)

The temperature at a point \((x,\ y,\ z)\) is given by \(T(x,\ y,\ z)=200e^{-x^2-3y^2-9z^2}\) where \(T\) is measured in \({ }^{\circ} \mathrm{C}\) and \(x,\ y,\ z\) in meters. (a) Find the rate of change of temperature at the point \(P(2,\ -1,\ 2)\) in the direction toward the point \((3,\ -3,\ 3)\). (b) In which direction does the temperature increase fastest at \(P\)? (c) Find the maximum rate of increase at \(P\). Equation Transcription: Text Transcription: (x, y, z) T(x, y, z) = 200e^-x^2 - 3^y2 - 9z^2 T C x, y, z P(2, -1,2) (3, -3, 3) P P

Suppose that over a certain region of space the electrical potential \(V\) is given by \(V(x,\ y,\ z)=5x^2-3xy+xyz\). (a) Find the rate of change of the potential at \(P(3,\ 4,\ 5)\) in the direction of the vector \(\mathbf{v}=\mathbf{i}+\mathbf{j}-\mathbf{k}\). (b) In which direction does \(V\) change most rapidly at \(P\)? (c) What is the maximum rate of change at \(P\)? Equation Transcription: Text Transcription: V V(x, y, z) = 5x^2 - 3xy + xyz P(3, 4, 5) v=i+j-k V P P

Suppose you are climbing a hill whose shape is given by the equation \(z=1000-0.005 x^{2}-0.01 y^{2}\), where \(x\), \(y\), and \(z\) are measured in meters, and you are standing at a point with coordinates \((60,\ 40,\ 966)\). The positive \(x\)-axis points east and the positive \(y\)-axis points north. (a) If you walk due south, will you start to ascend or descend? At what rate? (b) If you walk northwest, will you start to ascend or descend? At what rate? (c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizon-tal does the path in that direction begin? Equation Transcription: Text Transcription: z = 1000 - 0.005x^2 - 0.01y^2 x y z (60, 40, 966) y

Let \(f\) be a function of two variables that has continuous partial derivatives and consider the points \(A(1,\ 3)\), \(B(3,\ 3)\), \(C(1,\ 7)\), and \(D(6,\ 15)\). The directional derivative of \(f\) at \(A\) in the direction of the vector \(\vec{A B}\) is \(3\), and the directional derivative at \(A\) in the direction of \(\vec{A C}\) is \(26\). Find the directional derivative of \(f\) at \(A\) in the direction of the vector \(\vec{A D}\). Equation Transcription: Text Transcription: f A(1, 3) B(3, 3) C(1, 7) D(6, 15) f A vector AB 3 A vector AC 26 f A vector AD

Shown is a topographic map of Blue River Pine Provincial Park in British Columbia. Draw curves of steepest descent from point \(A\) (descending to Mud Lake) and from point \(B\). Equation Transcription: Text Transcription: A B

Show that the operation of taking the gradient of a function has the given property. Assume that \(u\) and \(v\) are differentiable functions of \(x\) and \(y\) and that \(a,\ b\) are constants. (a) \(\nabla(au+bv)=a\ \nabla u+b\ \nabla v\) (b) \(\nabla(uv)=u\ \nabla v+v\ \nabla u\) (c) \(\nabla\left(\frac{u}{v}\right)=\frac{v\ \nabla u-u\ \nabla v}{v^2}\) (d) \(\nabla u^n=nu^{n-1}\ \nabla u\) Equation Transcription: Text Transcription: u v x y a, b nabla (au+bv)=a nabla u + b nabla v nabla (uv)=u nabla v + v nabla u nabla (u/v)= frac v nabla u - u nabla v / v^2 nabla u^n = nu^n-1 nabla u

Sketch the gradient vector \(\nabla f(4,\ 6)\) for the function \(f\) whose level curves are shown. Explain how you chose the direction and length of this vector. Equation Transcription: Text Transcription: nabla f(4, 6) f

Second Directional Derivatives The second directional derivative of \(f(x,\ y)\) is \(D_{\mathrm{u}}^2\ f(x,y)=D_{\mathrm{u}}\left[D_{\mathrm{u}}\ f(x,\ y)\right]\) If \(f(x,\ y)=x^3+5x^2y+y^3\) and \(\mathbf{u}=\left\langle\frac{3}{5},\ \frac{4}{5}\right\rangle\), calculate \(D_{\mathrm{u}}^2\ f(2,\ 1)\). Equation Transcription: ? ? Text Transcription: f(x, y) D_u ^2 f(x, y) = D_u [D_u f(x, y)] f(x, y) = x^3 + 5x^2 y + y^3 u= ? 3/5, 4/5 ? D_u ^2 f(2, 1)

Second Directional Derivatives The second directional derivative of \(f(x,\ y)\) is \(D_{\mathrm{u}}^2\ f(x,y)=D_{\mathrm{u}}\left[D_{\mathrm{u}}\ f(x,\ y)\right]\) (a) If \(\mathbf{u}=\langle a, \ b\rangle\) is a unit vector and \(f\) has continuous second partial derivatives, show that \(D_{u}^{2} \ f=f_{x x} a^{2}+2 f_{x y} a b+f_{y y} b^{2}\) (b) Find the second directional derivative of \(f(x,\ y)=xe^{2y}\) in the direction of \(\mathbf{v}=\langle 4, \ 6\rangle\) Equation Transcription: ? ? ? ? Text Transcription: f(x, y) D_u ^2 f(x, y) = D_u [D_u f(x, y)] u= ? a, b ? f D_u ^2 f = f_xx a^2 + 2f_xy ab + f_yy b^2 f(x, y)=xe^2y u= ? 4, 6 ?

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. \(2(x-2)^2+(y-1)^2+(z-3)^2=10,\ \ \ \left(3,\ 3,\ 5\right)\) Equation Transcription: Text Transcription: 2(x - 2)^2 + (y - 1)^2 + (z - 3)^2 = 10, (3, 3, 5)

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. \(x=y^2+z^2+1,\quad\ \ \ (3,\ 1,\ -1)\) Equation Transcription: Text Transcription: x = y^2 + z^2 + 1, (3, 1, - 1)

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. \(xy^2z^3=8,\quad(2,\ 2,\ 1)\) Equation Transcription: Text Transcription: xy^2 z^3 = 8, (2, 2, 1)

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. \(xy+yz+zx=5,\quad\ \ \ (1,\ 2,\ 1)\) Equation Transcription: Text Transcription: xy + yz + zx = 5, (1, 2, 1)

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. \(x+y+z=e^{xyz},\quad\ \ \ (0,\ 0,\ 1)\) Equation Transcription: Text Transcription: x + y + z = e^xyz, (0, 0,1)

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point. \(x^{4}+y^{4}+z^{4}=3 x^{2} y^{2} z^{2},(1,1,1)\) Equation Transcription: Text Transcription: x^4 + y^4 + z^4 = 3x^2 y^2 z^2, (1,1,1)

Graph the surface, the tangent plane, and the normal line at the given point on the same screen. Choose a viewpoint so that you get a good view of all three objects. \(x y+y z+z x=3,(1,1,1)\) Equation Transcription: Text Transcription: xy + yz + zx = 3, (1,1,1)

Graph the surface, the tangent plane, and the normal line at the given point on the same screen. Choose a viewpoint so that you get a good view of all three objects. \(x y z=6,(1,2,3)\) Equation Transcription: Text Transcription: xyz = 6, (1,2,3)

If \(f(x, y)=x y\), find the gradient vector \(\nabla f(3,2)\) and use it to find the tangent line to the level curve \(f(x, y)=6\) at the point \((3, 2)\). Sketch the level curve, the tangent line, and the gradient vector. Equation Transcription: Text Transcription: f(x,y) = xy nabla f(3,2) f(x.y)=6 (3,2)

If \(g(x, y)=x^{2}+y^{2}-4 x\), find the gradient vector \(\nabla g(1,2)\) and use it to find the tangent line to the level curve \(g(x, y)=1\) at the point \((1, 2)\). Sketch the level curve, the tangent line, and the gradient vector. Equation Transcription: Text Transcription: g(x,y) = x^2 + y^2 - 4x nabla g(1,2) g(x,y)=1 (1,2)

Show that the equation of the tangent plane to the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1\) at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) can be written as \(\frac{x x_{0}}{a^{2}}+\frac{y y_{0}}{b^{2}}+\frac{z z_{0}}{c^{2}}=1\) Equation Transcription: Text Transcription: x^2/ a^2 + y^2/ b^2 + z^2/ c^2 = 1 (x_0, y_0, z_0) Xx_0 /a^2 + yy_0 /b^2 + zz_0 /c^2 = 1

Find the equation of the tangent plane to the hyperboloid \(x^{2} / a^{2}+y^{2} / b^{2}-z^{2} / c^{2}=1\) at \(\left(x_{0}, y_{0}, z_{0}\right)\) and express it in a form similar to the one in Exercise \(57\). Equation Transcription: Text Transcription: x^2/ a^2 + y^2/ b^2 - z^2/ c^2 = 1 (x_0, y_0, z_0) 57

Show that the equation of the tangent plane to the elliptic paraboloid \(z / c=x^{2} / a^{2}+y^{2} / b^{2}\) at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) can be written as \(\frac{2 x x_{0}}{a^{2}}+\frac{2 y y_{0}}{b^{2}}=\frac{z+z_{0}}{c}\) Equation Transcription: Text Transcription: z/c=x^2/ a^2 + y^2/ b^2 (x_0, y_0, z_0) 2xx_0 /a^2 + 2yy_0 /b^2 = z + z_0 /c

At what point on the ellipsoid \(x^{2}+y^{2}+2 z^{2}=1\) is the tangent plane parallel to the plane \(x+2 y+z=1\)? Equation Transcription: Text Transcription: x^2 + y^2 + 2z^2 = 1 x + 2y + z = 1

Are there any points on the hyperboloid \(x^{2}-y^{2}-z^{2}=1\) where the tangent plane is parallel to the plane \(z=x+y\)? Equation Transcription: Text Transcription: x^2 - y^2 - z^2 = 1 z = x + y

Show that the ellipsoid \(3 x^{2}+2 y^{2}+z^{2}=9\) and the sphere \(x^{2}+y^{2}+z^{2}-8 x-6 y-8 z+24=0\) are tangent to each other at the point \((1, 1, 2)\). (This means that they have a com-mon tangent plane at the point.) Equation Transcription: Text Transcription: 3x^2 + 2y^2 + z^2 = 9 x^2 + y^2 + z^2 - 8x - 6y - 8z + 24 = 0 (1,1,2)

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

Show that every normal line to the sphere \(x^{2}+y^{2}+z^{2}=r^{2}\) passes through the center of the sphere. Equation Transcription: Text Transcription: x^2 + y^2 + z^2 = r^2

Where does the normal line to the paraboloid \(z=x^{2}+y^{2}\) at the point \((1, 1, 2)\) intersect the paraboloid a second time? Equation Transcription: Text Transcription: z = x^2 + y^2 (1,1,2)

At what points does the normal line through the point \((1, 2, 1)\) on the ellipsoid \(4 x^{2}+y^{2}+4 z^{2}=12\) intersect the sphere \(x^{2}+y^{2}+z^{2}=102\)? Equation Transcription: Text Transcription: (1,2,1) 4x^2 + y^2 + 4z^2 = 12 x^2 + y^2 + z^2 = 102

Show that the sum of the \(x-, y-, \text { and } z-\) intercepts of any tangent plane to the surface \(\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{c}\) is a constant. Equation Transcription: Text Transcription: x-, y-, and z- sqrt x + sqrt y + sqrt z = sqrt c

Show that the pyramids cut off from the first octant by any tangent planes to the surface \(xyz = 1\) at points in the first octant must all have the same volume. Equation Transcription: Text Transcription: xyz=1

Find parametric equations for the tangent line to the curve of intersection of the paraboloid \(z=x^{2}+y^{2}\) and the ellipsoid \(4 x^{2}+y^{2}+z^{2}=9\) at the point \((-1, 1, 2)\). Equation Transcription: Text Transcription: z=x2+y2 4x2+y2+z2=9 (-1, 1, 2)

(a)The plane \(y+z=3\) intersects the cylinder \(x^{2}+y^{2}=5\) in an ellipse. Find parametric equations for the tangent line to this ellipse at the point \((1, 2, 1)\). (b)Graph the cylinder, the plane, and the tangent line on the same screen. Equation Transcription: Text Transcription: y + z = 3 x^2 + y^2 = 5 (1,2,1)

Where does the helix \(r(t)=\langle\cos \pi t, \sin \pi t, t\rangle\) intersect the paraboloid \(z=x^{2}+y^{2}\)? What is the angle of intersection between the helix and the paraboloid? (This is the angle between the tangent vector to the curve and the tangent plane to the paraboloid.) Equation Transcription: Text Transcription: r(t)=< cos pi t, sin pi t, t> z=x^2+y^2

The helix \(r(t)=\langle\cos (\pi t / 2), \sin (\pi t / 2), t\rangle\) intersects the sphere \(x^{2}+y^{2}+z^{2}=2\) in two points. Find the angle of intersection at each point. Equation Transcription: Text Transcription: r(t)=cos(pi t/2), sin(pi t/2), t x^2+y^2+z^2=2

Orthogonal Surfaces Two surfaces are called orthogo-nal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations \(F(x, y, z)=0\) and \(G(x, y, z) = 0\) are orthogonal at a point \(P\) where \(\nabla F \neq 0\) and \(\nabla \mathrm{G} \neq 0\) if and only if \(\mathrm{F}_{\mathrm{x}}\mathrm{G}_{\mathrm{x}}+\mathrm{F}_{\mathrm{y}}\mathrm{G}_{\mathrm{y}}+\mathrm{F}_{z} \mathrm{G}_{z}=0\) at \(P\) Equation Transcription: Text Transcription: F(x,y,z)=0 G(x,y,z)=0 P nabla F neq 0 nabla G neq 0 F_x G_x + F_y G_y + F_z G_z = 0

Orthogonal Surfaces Two surfaces are called orthogo-nal at a point of intersection if their normal lines are perpendicular at that point. Use Exercise \(73\) to show that the surfaces \(z^{2}=x^{2}+y^{2}\) and \(x^{2}+y^{2}+z^{2}+r^{2}\) are orthogonal at every point of intersection. Can you see why this is true without using calculus? Equation Transcription: Text Transcription: 73 z^2 = x^2 + y^2 x^2 + y^2 + z^2 + r^2

Suppose that the directional derivatives of \(f(x, y)\) are known at a given point in two nonparallel directions given by unit vectors \(u\) and \(v\). Is it possible to find \(\nabla f\) at this point? If so, how would you do it? Equation Transcription: Text Transcription: f(x,y) u v nabla f

(a)Show that the function \(f(x, y)=\sqrt[3]{x y}\) is continuous and the partial derivatives \(\mathrm{f}_{\mathrm{x}}\) and \(\mathrm{f}_{\mathrm{y}}\) exist at the origin, but the directional derivatives in all other directions do not exist. (b)Graph f near the origin and comment on how the graph confirms part (a). Equation Transcription: Text Transcription: f(x,y)= cube root xy f_x f_y

Show that if \(z = f(x, y)\) is differentiable at \(x_{0}=\left\langle x_{0}, y_{0}\right\rangle\), then \(\lim \limits_{x \rightarrow x_{0}} \frac{f(x)-\left[f\left(x_{0}\right)+\nabla f\left(x_{0}\right) \cdot\left(x-x_{0}\right)\right]}{\left|x-x_{0}\right|}\) [Hint: Use Definition \(14.4.7\) directly.] Equation Transcription: Text Transcription: z=f(x,y) z_0 = <x_0, y_0> lim_x right arrow x_0 f(x) - [f(x_0) + nabla f(x_0) cdot (x - x_0)] / |x - x_0| 14.4.7