Solved: Investigation A team of oceanographers is mapping the ocean floor to assist in

In Exercises 112, find the directional derivative of the function at in the direction of vf
x, y 3x
4xy 5y,
i 3 j P 9

In Exercises 112, find the directional derivative of the function at in the direction of v

In Exercises 112, find the directional derivative of the function at in the direction of vf
x, y xy, P
2, 3, v i j P
4,

In Exercises 112, find the directional derivative of the function at in the direction of v

In Exercises 112, find the directional derivative of the function at in the direction of vg
x, y x P
3, 4, v 3i
4j 2 y

In Exercises 112, find the directional derivative of the function at in the direction of vg
x, y arccos xy, P
1, 0, v i 5j g
x,

In Exercises 112, find the directional derivative of the function at in the direction of v

In Exercises 112, find the directional derivative of the function at in the direction of vh
x, y e P
0, 0, v i j

x2y

In Exercises 112, find the directional derivative of the function at in the direction of v

In Exercises 112, find the directional derivative of the function at in the direction of vf
x, y, z x P
1, 2,
1, v i
2j 3k 2 y

In Exercises 112, find the directional derivative of the function at in the direction of vh
x, y, z x arctan yz, P
4, 1, 1, v 1, 2,
1 f
x

In Exercises 112, find the directional derivative of the function at in the direction of vh(x, y, z xyz, P
2, 1, 1, v 2, 1, 2 h
x

In Exercises 1316, find the directional derivative of the function in the direction ofu cos  i  sin  j.
x, y x2 y 2, u

In Exercises 1316, find the directional derivative of the function in the direction ofu cos  i  sin  j.

In Exercises 1316, find the directional derivative of the function in the direction ofu cos  i  sin  j. f
x, y sin
2x
y, 

In Exercises 1316, find the directional derivative of the function in the direction ofu cos  i  sin  j. g
x, y xey , 

In Exercises 1720, find the directional derivative of the function at in the direction of Q.f
x, y x P
3, 1, Q
1,
1 2 4y

In Exercises 1720, find the directional derivative of the function at in the direction of Q.

In Exercises 1720, find the directional derivative of the function at in the direction of Q.h
x, y, z ln
x y z, P
1, 0, 0, Q
4, 3, 1 P
0, 

In Exercises 1720, find the directional derivative of the function at in the direction of Q.

In Exercises 2126, find the gradient of the function at the given point.f
x, y 3x
5y
2, 1 2

In Exercises 2126, find the gradient of the function at the given point.

In Exercises 2126, find the gradient of the function at the given point. cos
x
3,
4 2

In Exercises 2126, find the gradient of the function at the given point.

In Exercises 2126, find the gradient of the function at the given point.w 3x
1, 1,
2 2y
5yz z2

In Exercises 2126, find the gradient of the function at the given point.w x tan
y z,
4, 3,
1

In Exercises 2730, use the gradient to find the directional derivative of the function at in the direction of Q.
x, y x P
1, 2, Q
3, 6 2 y2 1, P Q.

In Exercises 2730, use the gradient to find the directional derivative of the function at in the direction of Q.

In Exercises 2730, use the gradient to find the directional derivative of the function at in the direction of Q. f
x, y e P
0, 0, Q
2, 1
x

In Exercises 2730, use the gradient to find the directional derivative of the function at in the direction of Q. P
0, 0, Q  2 ,  f
x

In Exercises 3138, find the gradient of the function and the maximum value of the directional derivative at the given point.2,  4  h

In Exercises 3138, find the gradient of the function and the maximum value of the directional derivative at the given point.

In Exercises 3138, find the gradient of the function and the maximum value of the directional derivative at the given point.g
x, y ln
1, 2  3 x2 y 2 0

In Exercises 3138, find the gradient of the function and the maximum value of the directional derivative at the given point.

In Exercises 3138, find the gradient of the function and the maximum value of the directional derivative at the given point.f
x, y, z x
1, 4, 2 2 y 2 z2 g
x,

In Exercises 3138, find the gradient of the function and the maximum value of the directional derivative at the given point.w
0, 0, 0 1 1
x2
y 2
z2 f

In Exercises 3138, find the gradient of the function and the maximum value of the directional derivative at the given point.f
x, y, z xe
2, 0,
4 yz w

In Exercises 3138, find the gradient of the function and the maximum value of the directional derivative at the given point.w xy
2, 1, 1 2z2

In Exercises 3946, use the function fx, y 3 x 3 y 2 . w Sketch the graph of in the first octant and plot the point on the surface

In Exercises 3946, use the function fx, y 3 x 3 y 2 . w

In Exercises 3946, use the function fx, y 3 x 3 y 2 . w

In Exercises 3946, use the function fx, y 3 x 3 y 2 . w

In Exercises 3946, use the function fx, y 3 x 3 y 2 . w Find where (a) is the vector from to (b) is the vector from
3, 2
4, 5. v

In Exercises 3946, use the function fx, y 3 x 3 y 2 . w

In Exercises 3946, use the function fx, y 3 x 3 y 2 . w

In Exercises 3946, use the function fx, y 3 x 3 y 2 . w Find a unit vector orthogonal to and calculate D Discuss the geometric meaning of the result.

In Exercises 4750, use the function f
x, y 9 x2 y 2 Sketch the graph of in the first octant and plot the point on the surface.

In Exercises 4750, use the function f
x, y 9 x2 y 2

In Exercises 4750, use the function f
x, y 9 x2 y 2

In Exercises 4750, use the function f
x, y 9 x2 y 2 Find a unit vector orthogonal to and calculate Discuss the geometric meaning of the result.

Investigation In Exercises 51 and 52, (a) use the graph to estimate the components of the vector in the direction of the maximum rate of increase in the function at the given point. (b) Find the gradient at the point and compare it with your estimate in part (a). (c) In what direction would the function be decreasing at the greatest rate? Explain.f
x, y yx, 1 10
x2
3xy y 2 , Du

Investigation In Exercises 51 and 52, (a) use the graph to estimate the components of the vector in the direction of the maximum rate of increase in the function at the given point. (b) Find the gradient at the point and compare it with your estimate in part (a). (c) In what direction would the function be decreasing at the greatest rate? Explain.f
x, y 1 2 f
x, y yx, 1 1

Investigation Consider the function at the point (a) Use a computer algebra system to graph the surface represented by the function. (b) Determine the directional derivative as a function of where Use a computer algebra system to graph the function on the interval (c) Approximate the zeros of the function in part (b) and interpret each in the context of the problem. (d) Approximate the critical numbers of the function in part (b) and interpret each in the context of the problem. (e) Find and explain its relationship to your answers in part (d). (f) Use a computer algebra system to graph the level curve of the function at the level On this curve, graph the vector in the direction of and state its relationship to the level curve.

Investigation The figure below shows the level curve of the function at the level (a) Analytically verify that the curve is a circle. (b) At the point on the level curve, sketch the vector showing the direction of the greatest rate of increase of the function. (To print an enlarged copy of the graph, go to the website www.mathgraphs.com.) (c) At the point on the level curve, sketch the vector such that the directional derivative is 0. (d) Use a computer algebra system to graph the surface to verify your answers in parts (a)(c).

In Exercises 5558, find a normal vector to the level curve fx, y c at P.

In Exercises 5558, find a normal vector to the level curve fx, y c at P.

In Exercises 5558, find a normal vector to the level curve fx, y c at P.

In Exercises 5558, find a normal vector to the level curve fx, y c at P.

In Exercises 5962, use the gradient to find a unit normal vector to the graph of the equation at the given point. Sketch your results.x
2, 10 2 1, 2
y 6, c

In Exercises 5962, use the gradient to find a unit normal vector to the graph of the equation at the given point. Sketch your results.

In Exercises 5962, use the gradient to find a unit normal vector to the graph of the equation at the given point. Sketch your results.

In Exercises 5962, use the gradient to find a unit normal vector to the graph of the equation at the given point. Sketch your results.xe
5, 0 9

Temperature Distribution The temperature at the point on a metal plate is Find the direction of greatest increase in heat from the point 3, 4.

Topography The surface of a mountain is modeled by the equation A mountain climber is at the point (500, 300, 4390). In what direction should the climber move in order to ascend at the greatest rate?

Define the derivative of the function in the direction u cos i sin j.

In your own words, give a geometric description of the directional derivative of z f
x, y. u

Write a paragraph describing the directional derivative of the function in the direction when (a) and (b) 90 .

Define the gradient of a function of two variables. State the properties of the gradient.

Sketch the graph of a surface and select a point on the surface. Sketch a vector in the plane giving the direction of steepest ascent on the surface at P.

Describe the relationship of the gradient to the level curves of a surface given by z f
x, y. P

Topography The figure shows a topographic map carried by a group of hikers. Sketch the paths of steepest descent if the hikers start at point and if they start at point (To print an enlarged copy of the graph, go to the website www.mathgraphs.com.)

Meteorology Meteorologists measure the atmospheric pressure in units called millibars. From these observations they create weather maps on which the curves of equal atmospheric pressure (isobars) are drawn (see figure). These are level curves to the function yielding the pressure at any point. Sketch the gradients to the isobars at the points and Although the magnitudes of the gradients are unknown, their lengths relative to each other can be estimated. At which of the three points is the wind speed greatest if the speed increases as the pressure gradient increases? (To print an enlarged copy of the graph, go to the website www.mathgraphs.com.)

Heat-Seeking Path In Exercises 73 and 74, find the path of a heat-seeking particle placed at point on a metal plate with a temperature field Tx, y .

Heat-Seeking Path In Exercises 73 and 74, find the path of a heat-seeking particle placed at point on a metal plate with a temperature field Tx, y .

Investigation A team of oceanographers is mapping the ocean floor to assist in the recovery of a sunken ship. Using sonar, they develop the model where is the depth in meters, and and are the distances in kilometers. (a) Use a computer algebra system to graph the surface. (b) Because the graph in part (a) is showing depth, it is not a map of the ocean floor. How could the model be changed so that the graph of the ocean floor could be obtained? (c) What is the depth of the ship if it is located at the coordinates and (d) Determine the steepness of the ocean floor in the positive direction from the position of the ship. (e) Determine the steepness of the ocean floor in the positive direction from the position of the ship. (f) Determine the direction of the greatest rate of change of depth from the position of the ship.

Temperature The temperature at the point on a metal plate is modeled by (a) Use a computer algebra system to graph the temperature distribution function. (b) Find the directions of no change in heat on the plate from the point (c) Find the direction of greatest increase in heat from the point
3, 5.

True or False? In Exercises 7780, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If then for any unit vector u

True or False? In Exercises 7780, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 7780, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

True or False? In Exercises 7780, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If for any unit vector then c =o

Find a function such that f ex cos y i
ex sin y j z k.