Solution: Using Properties of the Gradient In Exercises 2736, find the gradient of the

In Exercises 1 - 4, use Theorem 13. 9 to find the directional derivative of the function at P in the direction of the unit vector \(u=\cos \theta i+\sin \theta j\). \(f(x, y)=x^{2}+y^{2}, \quad P(1,-2), \quad \theta=\frac{\pi}{4}\) Text Transcription: u = cos theta i +sin theta j f(x, y) = x^2 + y^2, P(1, -2), theta = pi / 4

In Exercises 1 - 4, use Theorem 13. 9 to find the directional derivative of the function at P in the direction of the unit vector \(u=\cos \theta i+\sin \theta j\). \(f(x, y)=\frac{y}{x+y}, \quad P(3,0), \quad \theta=-\frac{\pi}{6}\) Text Transcription: u = cos theta i +sin theta j f(x, y) = y / x + y, P(3,0), theta = -pi / 6

In Exercises 1 - 4, use Theorem 13. 9 to find the directional derivative of the function at P in the direction of the unit vector \(u=\cos \theta i+\sin \theta j\). \(f(x, y)=\sin (2 x+y), \quad P(0,0), \quad \theta=\frac{\pi}{3}\) Text Transcription: u = cos theta i +sin theta j f(x, y) = sin (2x + y), P(0, 0), theta = pi / 3

In Exercises 1 - 4, use Theorem 13. 9 to find the directional derivative of the function at P in the direction of the unit vector \(u=\cos \theta i+\sin \theta j\). \(g(x, y)=x e^{y}, \quad P(0,2), \quad \theta=\frac{2 \pi}{3}\) Text Transcription: u = cos theta i +sin theta j g(x, y) = xe^y, P(0, 2), theta = 2pi / 3

In Exercises 5 - 8, use Theorem 13.9 to find the directional derivative of the function at P in the direction of v. \(f(x, y)=3 x-4 x y+9 y, \quad P(1,2), \quad \mathbf{v}=\frac{3}{5} \mathbf{i}+\frac{4}{5} \mathbf{j}\) Text Transcription: f(x, y) = 3x - 4xy + 9y, P(1, 2), v = 3 / 5 i + 4 / 5 j

In Exercises 5 - 8, use Theorem 13.9 to find the directional derivative of the function at P in the direction of v. \(f(x, y)=x^{3}-y^{3}, \quad P(4,3), \quad \mathbf{v}=\frac{\sqrt{2}}{2}(\mathbf{i}+\mathbf{j})\) Text Transcription: f(x, y) = x^3 - y^3, P (4, 3), v = sqrt{2} / 2 (i + j)

In Exercises 5 - 8, use Theorem 13.9 to find the directional derivative of the function at P in the direction of v. \(g(x, y)=\sqrt{x^{2}+y^{2}}, \quad P(3,4), \quad \mathbf{v}=3 \mathbf{i}-4 \mathbf{j}\) Text Transcription: g(x, y) = sqrt{x^2 + y^2}, P(3, 4), v = 3i - 4j

In Exercises 5 - 8, use Theorem 13.9 to find the directional derivative of the function at P in the direction of v. \(h(x, y)=e^{-\left(x^{2}+y^{2}\right)}, \quad P(0,0), \quad \mathbf{v}=\mathbf{i}+\mathbf{j}\) Text Transcription: h(x, y) = e^{-(x^2 + y^2), P(0, 0), v = i + j

In Exercises 9 - 12, use Theorem 13.9 to find the directional derivative of the function at P in the direction of Q. \(f(x, y)=x^{2}+3 y^{2}, \quad P(1,1), \quad Q(4,5)\) Text Transcription: f(x, y) = x^2 + 3y^2, P(1, 1), Q(4, 5)

In Exercises 9 - 12, use Theorem 13.9 to find the directional derivative of the function at P in the direction of Q. \(f(x, y)=\cos (x+y), \quad P(0, \pi), \quad Q\left(\frac{\pi}{2}, 0\right)\) Text Transcription: f(x, y) = cos (x + y), P(0, pi), Q(pi / 2, 0)

In Exercises 9 - 12, use Theorem 13.9 to find the directional derivative of the function at P in the direction of Q. \(f(x, y)=e^{y} \sin x, \quad P(0,0), \quad Q(2,1)\) Text Transcription: f(x, y) = e^y sin x, P(0, 0), Q(2, 1)

In Exercises 9 - 12, use Theorem 13.9 to find the directional derivative of the function at P in the direction of Q. \(f(x, y)=\sin 2 x \cos y, \quad P(\pi, 0), \quad Q\left(\frac{\pi}{2}, \pi\right)\) Text Transcription: f(x, y) = sin 2x cos y, P(pi, 0), Q(pi / 2, pi)

In Exercises 13 - 18, find the gradient of the function at the given point. \(f(x, y)=3 x+5 y^{2}+1, \quad(2,1)\) Text Transcription: f(x, y) = 3x +5y^2 + 1, (2, 1)

In Exercises 13 - 18, find the gradient of the function at the given point. \(g(x, y)=2 x e^{y / x}, \quad(2,0)\) Text Transcription: g(x, y) = 2xe^{y / x}, (2, 0)

In Exercises 13 - 18, find the gradient of the function at the given point. \(z=\ln \left(x^{2}-y\right), \quad(2,3)\) Text Transcription: z = ln (x^2 - y), (2, 3)

In Exercises 13 - 18, find the gradient of the function at the given point. \(z=\cos \left(x^{2}+y^{2}\right), \quad(3,-4)\) Text Transcription: z = cos (x^2 + y^2), (3, - 4)

In Exercises 13 - 18, find the gradient of the function at the given point. \(w=3 x^{2}-5 y^{2}+2 z^{2}, \quad(1,1,-2)\) Text Transcription: w = 3x^2 - 5y^2 + 2z^2, (1, 1, - 2)

In Exercises 13 - 18, find the gradient of the function at the given point. w = x tan (y + z), (4, 3, -1)

In Exercises 19 - 22, use the gradient to find the directional derivative of the function at P in the direction of v. \(f(x, y)=x y, \quad P(0,-2), \quad \mathbf{v}=\frac{1}{2}(\mathbf{i}+\sqrt{3} \mathbf{j})\) Text Transcription: f(x, y) = xy, P(0, -2), v = 1 / 2 (i + sqrt{3} j)

In Exercises 19 - 22, use the gradient to find the directional derivative of the function at P in the direction of v. \(h(x, y)=e^{x} \sin y, \quad P\left(1, \frac{\pi}{2}\right), \quad \mathbf{v}=-\mathbf{i}\) Text Transcription: h(x, y) = e^x sin y, P(1, pi / 2), v = - i

In Exercises 19 - 22, use the gradient to find the directional derivative of the function at P in the direction of v. \(f(x, y, z)=x^{2}+y^{2}+z^{2}, \quad P(1,1,1), \quad \mathbf{v}=\frac{\sqrt{3}}{3}(\mathbf{i}-\mathbf{j}+\mathbf{k})\) Text Transcription: f(x, y, z) = x^2 + y^2 + z^2, P(1, 1, 1), v = sqrt{3} / 3(i - j + k)

In Exercises 19 - 22, use the gradient to find the directional derivative of the function at P in the direction of v. \(f(x, y, z)=x y+y z+x z, \quad P(1,2,-1), \quad \mathbf{v}=2 \mathbf{i}+\mathbf{j}-\mathbf{k}\) Text Transcription: f(x, y, z) = xy + yz + xz, P(1, 2, -1), v = 2 i + j - k

In Exercises 23 - 26, use the gradient to find the directional derivative of the function at P in the direction of Q. \(g(x, y)=x^{2}+y^{2}+1, \quad P(1,2), \quad Q(2,3)\) Text Transcription: g(x, y) = x^2 + y^2 + 1, P(1, 2), Q(2, 3)

In Exercises 23 - 26, use the gradient to find the directional derivative of the function at P in the direction of Q. \(f(x, y)=3 x^{2}-y^{2}+4, \quad P(-1,4), \quad Q(3,6)\) Text Transcription: f(x, y) = 3x^2 - y^2 + 4, P(-1, 4), Q(3, 6)

In Exercises 23 - 26, use the gradient to find the directional derivative of the function at P in the direction of Q. \(g(x, y, z)=x y e^{z}, \quad P(2,4,0), \quad Q(0,0,0)\) Text Transcription: g(x, y, z) = xye^z, P(2, 4, 0), Q(0, 0, 0)

In Exercises 23 - 26, use the gradient to find the directional derivative of the function at P in the direction of Q. h(x, y, z) = (x + y + z), P(1, 0, 0), Q(4, 3, 1)

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point \(f(x, y)=x^{2}+2 x y\) (1, 0) Text Transcription: f(x, y) = x^2 + 2xy

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point \(f(x, y)=\frac{x+y}{y+1}\) (0, 1) Text Transcription: f(x, y) = x + y / y + 1

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point h(x, y) = x tan y \(\left(2, \frac{\pi}{4}\right)\) Text Transcription: (2, pi / 4)

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point h(x, y) = y cos (x - y) \(\left(0, \frac{\pi}{3}\right)\) Text Transcription: (0, pi / 3)

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point \(g(x, y)=y e^{-x}\) (0, 5) Text Transcription: g(x, y) = ye^- x

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point \(g(x, y)=\ln \sqrt[3]{x^{2}+y^{2}}\) (1, 2) Text Transcription: g(x, y) = ln sqrt[3] x^2 + y^2

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point \(f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}\) (1, 4, 2) Text Transcription: f(x, y, z) = sqrt{x^2 + y^2 + z^2}

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point \(w=\frac{1}{\sqrt{1-x^{2}-y^{2}-z^{2}}}\) (0, 0, 0) Text Transcription: w = 1 / sqrt{1 - x^2 - y^2 - z^2}

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point \(w=x y^{2} z^{2}\) (2, 1, 1) Text Transcription: w = xy^{2} z^{2}

In Exercises 27 - 36, find the gradient of the function and the maximum value of the directional derivative at the given point. Function Point \(f(x, y, z)=x e^{y z}\) (2, 0, -4) Text Transcription: f(x, y, z) = xe^yz

In Exercises 37 - 42, consider the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\). Sketch the graph of f in the first octant and plot the point (3, 2, 1) on the surface. Text Transcription: f(x, y) = 3 - x / 3 - y / 2

In Exercises 37 - 42, consider the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\). Find \(D_{\mathbf{u}} f(3,2)\), where \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\), using each given value of \(\theta\). (a) \(\theta=\frac{\pi}{4}\) (b) \(\theta=\frac{2 \pi}{3}\) (c) \(\theta=\frac{4 \pi}{3}\) (d) \(\theta=-\frac{\pi}{6}\) Text Transcription: f(x, y) = 3 - x / 3 - y / 2 D_u f(3, 2) u = cos theta i + sin theta j theta theta = pi / 4 theta = 2 pi / 3 theta = 4 pi / 3 theta = -pi / 6

In Exercises 37 - 42, consider the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\). Find \(D_{\mathbf{u}} f(3,2)\), where \(\mathbf{u}=\frac{\mathbf{v}}{\|\mathbf{v}\|}\), using each given vector \(\mathbf{v}\). (a) \(\mathbf{v}=\mathbf{i}+\mathbf{j}\) (b) \(\mathbf{v}=-3 \mathbf{i}-4 \mathbf{j}\) (c) \(\mathbf{v}\) is the vector from (1, 2) to (-2, 6). (d) \(\mathbf{v}\) is the vector from (3, 2) to (4, 5) Text Transcription: f(x, y) = 3 - x / 3 - y / 2 D_u f(3, 2) (\mau} = v / ||v|| v v = i + j v = - 3i - 4j

In Exercises 37 - 42, consider the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\). Find \(\nabla f(x, y)\). Text Transcription: f(x, y) = 3 - x / 3 - y / 2 nabla f(x, y)

In Exercises 37 - 42, consider the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\). Find the maximum value of the directional derivative at (3, 2). Text Transcription: f(x, y) = 3 - x / 3 - y / 2

In Exercises 37 - 42, consider the function \(f(x, y)=3-\frac{x}{3}-\frac{y}{2}\). Find a unit vector \(\mathbf{u}\) orthogonal to \(\nabla f(3,2)\) and calculate \(D_{\mathbf{u}} f(3,2)\). Discuss the geometric meaning of the result. Text Transcription: f(x, y) = 3 - x / 3 - y / 2 u nabla f(3, 2) D_u f(3, 2)

In Exercises 43 and 44, (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. \(f(x, y)=\frac{1}{10}\left(x^{2}-3 x y+y^{2}\right)\) (1, 2) Text Transcription: f(x, y) = 1 / 10 (x^2 - 3xy + y^2)

In Exercises 43 and 44, (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. \(f(x, y)=\frac{1}{2} y \sqrt{x}\) (1, 2) Text Transcription: f(x, y) = 1 / 2 y sqrt{x}

Consider the function \(f(x, y)=x^{2}-y^{2}\) at the point (4, -3, 7). (a) Use a computer algebra system to graph the surface represented by the function. (b) Determine the directional derivative \(D_{\mathbf{u}} f(4,-3)\) as a function of \(\theta\), where \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). Use a computer algebra system to graph the function on the interval \([0,2 \pi)\). (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 \(\|\nabla f(4,-3)\|\) and explain its relationship to your answers in part (d). (f) Use a computer algebra system to graph the level curve of the function f at the level c = 7. On this curve, graph the vector in the direction of \(\nabla f(4,-3)\), and state its relationship to the level curve. Text Transcription: f(x, y) = x^2 - y^2 D_u f(4, -3) theta u = cos theta i + sin theta j [0, 2 pi) ||nabla f(4, -3)|| nabla f(4, -3)

Consider the function \(f(x, y)=\frac{8 y}{1+x^{2}+y^{2}}\) . (a) Analytically verify that the level curve of f(x, y) at the level c = 2 is a circle. (b) At the point \((\sqrt{3}, 2)\) on the level curve for which c = 2, 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 MathGraphs.com.) (c) At the point \((\sqrt{3}, 2)\) on the level curve, sketch a 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). Text Transcription: f(x, y) = 8y / 1 + x^2 + y^2 (sqrt{3}, 2)

In Exercises 47 - 50, find a normal vector to the level curve f(x, y) = c at P. f(x, y) = 6 - 2x - 3y c = 6, P(0, 0)

In Exercises 47 - 50, find a normal vector to the level curve f(x, y) = c at P. \(f(x, y)=x^{2}+y^{2}\) c = 25, P(3, 4) Text Transcription: f(x, y) = x^2 + y^2

In Exercises 47 - 50, find a normal vector to the level curve f(x, y) = c at P. f(x, y) = xy c = - 3, P(- 1, 3)

In Exercises 47 - 50, find a normal vector to the level curve f(x, y) = c at P. \(f(x, y)=\frac{x}{x^{2}+y^{2}}\) \(c=\frac{1}{2}, \quad P(1,1)\) Text Transcription: f(x, y) = x / x^2 + y^2 c = 1 / 2, P(1, 1)

In Exercises 51 - 54, (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f(x, y) = c at P, (c) find the tangent line to the level curve f(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. \(f(x, y)=4 x^{2}-y\) c = 6, P(2, 10) Text Transcription: f(x, y) = 4x^2 - y

In Exercises 51 - 54, (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f(x, y) = c at P, (c) find the tangent line to the level curve f(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. \(f(x, y)=x-y^{2}\) c = 3, P(4, -1) Text Transcription: f(x, y) = x - y^2

In Exercises 51 - 54, (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f(x, y) = c at P, (c) find the tangent line to the level curve f(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. \(f(x, y)=3 x^{2}-2 y^{2}\) c = 1, P(1, 1) Text Transcription: f(x, y) = 3x^2 - 2y^2

In Exercises 51 - 54, (a) find the gradient of the function at P, (b) find a unit normal vector to the level curve f(x, y) = c at P, (c) find the tangent line to the level curve f(x, y) = c at P, and (d) sketch the level curve, the unit normal vector, and the tangent line in the xy-plane. \(f(x, y)=9 x^{2}+4 y^{2}\) c = 40, P(2, -1) Text Transcription: f(x, y) = 9x^2 + 4y^2

Define the derivative of the function z =f(x, y) in the direction \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). Text Transcription: u = cos theta i + sin theta j

Write a paragraph describing the directional derivative of the function f in the direction \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) when (a) \(\theta=0^{\circ}\) and (b) \(\theta=90^{\circ}\). Text Transcription: u = cos theta i + sin theta j theta = 0^circ theta = 90^circ

Gradient 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 P on the surface. Sketch a vector in the x y-plane giving the direction of the 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).

Consider the function \(f(x, y)=9-x^{2}-y^{2}\) . (a) Sketch the graph of f in the first octant and plot the point (1, 2, 4) on the surface. (b) Find \(D_{\mathbf{u}} f(1,2)\), where \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\), for \(\theta=-\pi / 4\). (c) Repeat part (b) for \(\theta=\pi / 3\). (d) Find \(\nabla f(1,2)\) and \(\|\nabla f(1,2)\|\). (e) Find a unit vector \(\mathbf{u}\) orthogonal to \(\nabla f(1,2)\) and calculate \(D_{\mathbf{u}} f(1,2)\. Discuss the geometric meaning of the result. Text Transcription: f(x, y) = 9 - x^2 - y^2 D_u f(1,2)\) u = cos theta i + sin theta j theta = -pi / 4 theta = pi / 3 nabla f(1,2) ||nabla f(1,2)|| u

The surface of a mountain is modeled by the equation \(h(x, y)=5000-0.001 x^{2}-0.004 y^{2}\) . 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? Text Transcription: h(x, y) = 5000 - 0.001x^2 - 0. 004 y^2

The figure shows a topographic map carried by a group of hikers. Sketch the paths of steepest descent when the hikers start at point A and when they start at point B. (To print an enlarged copy of the graph, go to MathGraphs.com.)

The temperature at the point (x, y) on a metal plate is \(T=\frac{x}{x^{2}+y^{2}}\) . Find the direction of the greatest increase in heat from the point (3,4). Text Transcription: T = x / x^2 + y^2

The temperature at the point (x, y) on a metal plate is modeled by \(T(x, y)=400 e^{-\left(x^{2}+y\right) / 2}, x \geq 0, y \geq 0\). (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 (3, 5). (c) Find the direction of the greatest increase in heat from the point (3, 5). Text Transcription: T(x, y) = 400e^-(x^2 + y) / 2, x geq 0, y geq 0

In Exercises 65 and 66, the temperature in degrees Celsius on the surface of a metal plate is given by T(x, y), where x and y are measured in centimeters. Find the direction from point P where the temperature increases most rapidly and the rate of increase. \(T(x, y)=80-3 x^{2}-y^{2}, \quad P(-1,5)\) Text Transcription: T(x, y) = 80 - 3x^2 - y^2, P(-1, 5)

In Exercises 65 and 66, the temperature in degrees Celsius on the surface of a metal plate is given by T(x, y), where x and y are measured in centimeters. Find the direction from point P where the temperature increases most rapidly and the rate of increase. \(T(x, y)=50-x^{2}-4 y^{2}, \quad P(2,-1)\) Text Transcription: T(x, y) = 50 - x^2 - 4y^2, P(2, -1)

In Exercises 67 and 68, find the path of a heat-seeking particle placed at point P on a metal plate with a temperature field T(x, y). \(T(x, y)=400-2 x^{2}-y^{2}, \quad P(10,10)\) Text Transcription: T(x, y) = 400 - 2x^2 - y^2, P(10, 10)

In Exercises 67 and 68, find the path of a heat-seeking particle placed at point P on a metal plate with a temperature field T(x, y). \(T(x, y)=100-x^{2}-2 y^{2}, \quad P(4,3)\) Text Transcription: T(x, y) = 100 - x^2 - 2y^2, P(4, 3)

In Exercises 69-72, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x, y)=\sqrt{1-x^{2}-y^{2}}\), then \(D_{\mathbf{u}} f(0,0)=0\) for any unit vector u. Text Transcription: f(x, y) = sqrt{1 - x^2 - y^2} D_u f(0, 0) = 0

In Exercises 69-72, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f(x, y) = x + y, then \(-1 \leq D_{\mathbf{u}} f(x, y) \leq 1\). Text Transcription: -1 leq D_u f(x, y) leq 1

In Exercises 69-72, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(D_{\mathbf{u}} f(x, y)\) exists, then \(D_{\mathbf{u}} f(x, y)=-D_{-\mathbf{u}} f(x, y)\). Text Transcription: D_u f(x, y)\) D_u f(x, y) = -D_-u f(x, y)\)

In Exercises 69-72, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(D_{\mathbf{u}} f\left(x_{0}, y_{0}\right)=c\) for any unit vector \(\mathbf{u}\), then c = 0. Text Transcription: D_u f(x_0, y_0) = c u

Find a function f such that \(\nabla f=e^{x} \cos y \mathbf{i}-e^{x} \sin y \mathbf{j}+z \mathbf{k}\) Text Transcription: nabla f = e^x cos y i - e^x sin y j + z k

A team of oceanographers is mapping the ocean floor to assist in the recovery of a sunken ship. Using sonar, they develop the model \(D=250+30 x^{2}+50 \sin \frac{\pi y}{2}, \quad 0 \leq x \leq 2,0 \leq y \leq 2\) where D is the depth in meters, and x and y 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 x = 1 and y = 0. 5? (d) Determine the steepness of the ocean floor in the positive x-direction from the position of the ship. (e) Determine the steepness of the ocean floor in the positive y-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. Text Transcription: D = 250 + 30x^2 + 50 sin pi y / 2, 0 leq x leq 2, 0 leq y leq 2

Consider the function \(f(x, y)=\sqrt[3]{x y}\) . (a) Show that f is continuous at the origin. (b) Show that \(f_{x}\) and \(f_{y}\) exists at the origin, but that the directional derivatives at the origin in all other directions do not exist. (c) Use a computer algebra system to graph f near the origin to verify your answers in parts (a) and (b). Explain. Text Transcription: f(x, y) = sqrt[3]{xy} f_x f_y

Consider the function \(f(x, y)= \begin{cases}\frac{4 x y}{x^{2}+y^{2}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0)\end{cases}\) and the unit vector \(\mathbf{u}=\frac{1}{\sqrt{2}}(\mathbf{i}+\mathbf{j})\) . Does the directional derivative of f at P(0, 0) in the direction of \(\mathbf{u}\) exist? If f(0, 0) were defined as 2 instead of 0, would the directional derivative exist? Text Transcription: f(x, y) = {4 xy / x^2 + y^2 , (x, y) neq(0, 0) \\ 0, (x, y) = (0, 0) u = 1 / sqrt{2} (i + j) u