?Orthogonal Surfaces Two surfaces are called orthogonal at a point of intersection if

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 direc- tional derivative of the pressure function at Kearney in the direction of Sioux City. What are the units of the directional derivative?

The contour map shows the average maximum temperature for November 2004 (in °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?

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,). The following table of values is an excerpt from Table 1 in Section 14.1. Use the table to estimate the value of , where

Find the directional derivative of at the given point in the direction indicated by the angle .

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

Find the directional derivative of at the given point in the direction indicated by the angle .

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

(a) Find the gradient of . (b) Evaluate the gradient at the point . (c) Find the rate of change of at in the direction of the vector .

(a) Find the gradient of . (b) Evaluate the gradient at the point . (c) Find the rate of change of at in the direction of the vector .

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

(a) Find the gradient of . (b) Evaluate the gradient at the point . (c) Find the rate of change of at in the direction of the vector .

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

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

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

Find the directional derivative of the function at the given point in the direction of the vector .

Find the directional derivative of the function at the given point in the direction of the vector .

Find the directional derivative of the function at the given point in the direction of the vector \(v\). \(f(x, y, z)=x y^{2} \tan ^{-1} z,(2,1,1), v=\langle 1,1,1\rangle\) 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 \(v\). \(h(r, s, t)=\ln (3 r+6 s+9 t),(1,1,1), v=4 i+12 j+6 k\) 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 Du f (2, 2).

Find the directional derivative of the function at the point in the direction of the point .

Find the directional derivative of the function at the point in the direction of the point .

Find the directional derivative of the function at the point in the direction of the point .

Find the directional derivative of the function at the point in the direction of the point .

Find the directional derivative of the function at the point in the direction of the point .

The contour map of a function is shown. At points . and , draw an arrow to indicate the direction of the gradient vector.

Find the maximum rate of change of at the given point and the direction in which it occurs.

Find the maximum rate of change of at the given point and the direction in which it occurs.

Find the maximum rate of change of \(f\) at the given point and the direction in which it occurs. \(f(x, y)=\sin (x y),(1,0)\) 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. \(f(x, y, z)=x \ln (y z),\left(1,2, \frac{1}{2}\right)\) Equation Transcription: Text Transcription: f f(x,y,z)=x ln? (yz), (1,2,½)

Find the maximum rate of change of at the given point and the direction in which it occurs.

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

Direction of Most Rapid Decrease (a) Show that a differentiable function decreases most rapidly at in the direction opposite the gradient vector, that is, in the direction of , and that the maximum rate of decrease is . (b) Use the result of part (a) to find the direction in which the function decreases fastest at the point . What is the rate of decrease?

Find the directions in which the directional derivative of at the point has the value 2 .

Find all points at which the direction of greatest rate of change of the function is .

Near a buoy, the depth of a lake at the point with coordinates is , where , and are measured in meters. A fisherman in a small boat starts at the point and moves toward the buoy, which is located at . Is the water under the boat getting deeper or shallower when he departs? Explain.

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.

The temperature at a point is given by where is measured in and in meters. (a) Find the rate of change of temperature at the point in the direction toward the point . (b) In which direction does the temperature increase fastest at (c) Find the maximum rate of increase at .

Suppose that over a certain region of space the electrical potential is given by (a) Find the rate of change of the potential at in the direction of the vector (b) In which direction does change most rapidly at ? (c) What is the maximum rate of change at ?

Suppose you are climbing a hill whose shape is given by the equation , where , and are measured in meters, and you are standing at a point with coordinates . The positive -axis points east and the positive -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 horizontal does the path in that direction begin?

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 \(\overrightarrow{AB}\) is 3 , and the directional derivative at A in the direction of \(\overrightarrow{AC}\) is 26 . Find the directional derivative of f at A in the direction of the vector \(\overrightarrow{AD}\).

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.

Show that the operation of taking the gradient of a function has the given property. Assume that and are differentiable functions of and and that are constants. (a) (b) (c) (d)

Sketch the gradient vector for the function whose level curves are shown. Explain how you chose the direction and length of this vector.

Second Directional Derivatives The second directional derivative of is If and , calculate

Second Directional Derivatives The second directional derivative of is (a) If is a unit vector and has continuous second partial derivatives, show tha (b) Find the second directional derivative of in the direction of .

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

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.

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.

If , find the gradient vector and use it to find the tangent line to the level curve at the point . Sketch the level curve, the tangent line, and the gradient vector.

If , find the gradient vector and use it to find the tangent line to the level curve at the point . Sketch the level curve, the tangent line, and the gradient vector.

Show that the equation of the tangent plane to the ellipsoid at the point can be written as

Find the equation of the tangent plane to the hyperboloid at and express it in a form similar to the one in Exercise 57 .

Show that the equation of the tangent plane to the elliptic paraboloid at the point can be written as

At what point on the ellipsoid is the tangent plane parallel to the plane ?

Are there any points on the hyperboloid where the tangent plane is parallel to the plane ?

Show that the ellipsoid and the sphere are tangent to each other at the point . (This means that they have a common tangent plane at the point.)

Show that every plane that is tangent to the cone passes through the origin.

Show that every normal line to the sphere passes through the center of the sphere.

Where does the normal line to the paraboloid z = x2 + y2 at the point (1, 1, 2) intersect the paraboloid a second time?

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 \(x^{2}+y^{2}+z^{2}=102\)?

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

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.

Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 + y2 and the ellipsoid 4x2 + y2 + z2 = 9 at the point (?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.

Where does the helix r(t) = ?cos ????t, sin ????t, t? intersect the paraboloid z = x2 + y2? 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.)

The helix r(t) = ?cos(????t/2), sin(????t/2), t? intersects the sphere x2 + y2 + z2 = 2 in two points. Find the angle of intersection at each point.

Orthogonal Surfaces Two surfaces are called orthogonal 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 ?F ? 0 and ?G ? 0 if and only if FxGx + FyGy + FzGz = 0 at P

Orthogonal Surfaces Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Use Exercise 73 to show that the surfaces z2 = x2 + y2 and x2 + y2 + z2 = r2 are orthogonal at every point of intersection. Can you see why this is true without using calculus?

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 ????f at this point? If so, how would you do it?

(a) Show that the function f(x, y) = is continuous and the partial derivatives fx and fy 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).

Show that if z = f(x, y) is differentiable at x0 = ?x0, y0?, then [Hint: Use Definition 14.4.7 directly.]