?A long piece of galvanized sheet metal with width is to be bent into a symmetric form

(a) What is a function of two variables? (b) Describe three methods for visualizing a function of two variables.

What is a function of three variables? How can you visualize such a function?

What does \(\lim \limits _{(x,\ y)\rightarrow(a,\ b)}\ f(x,\ y)=L\) mean? How can you show that such a limit does not exist? Equation Transcription: Text Transcription: Lim _(x, y) rightarrow (a, b) f(x, y) = L

(a) What does it mean to say that \(f\) is continuous at \((a, \ b)\)? (b) If \(f\) is continuous on \(\mathbb{R}^{2}\), what can you say about its graph? Equation Transcription: ?2 Text Transcription: f (a, b) f ?^2

(a) Write expressions for the partial derivatives \(f_x(a,\ b)\) and \(f_y(a,\ b)\) as limits. (b) How do you interpret \(f_x(a,\ b)\) and \(f_y(a,\ b)\) geometrically? How do you interpret them as rates of change? (c) If \(f(x,\ y)\) is given by a formula, how do you calculate \(f_{x}\) and \(f_{y}\)? Equation Transcription: Text Transcription: f_x(a, b) f_y(a, b) f_x(a, b) f_y(a, b) f(x, y) f_x f_y

What does Clairaut's Theorem say?

How do you find a tangent plane to each of the following types of surfaces? (a) A graph of a function of two variables, \(z=f(x, \ y)\) (b) A level surface of a function of three variables, \(F(x, \ y, \ z)=k\) Equation Transcription: Text Transcription: z = f(x, y) F(x, y, z) = k

Define the linearization of \(f\) at \((a, \ b)\). What is the corresponding linear approximation? What is the geometric interpretation of the linear approximation?

(a) What does it mean to say that \(f\) is differentiable at \(a, \ b)\) (b) How do you usually verify that \(f\) is differentiable? Equation Transcription: Text Transcription: f (a, b) f

If \(z=f(x, \ y)\), what are the differentials \(dx\), \(dy\), and \(dz\)? Equation Transcription: Text Transcription: z = f(x, y) dx dy dz

State the Chain Rule for the case where \(z=f(x, \ y)\) and \(x\) and \(y\) are functions of one variable. What if \(x\) and \(y\) are functions of two variables? Equation Transcription: Text Transcription: z = f(x, y) x y x y

If \(z\) is defined implicitly as a function of \(x\) and \(y\) by an equation of the form \(F(x, \ y, \ z)=0\), how do you find \(\partial z / \partial x\) and \(\partial z / \partial y\)? Equation Transcription: Text Transcription: z x y F(x, y, z)=0 partial z / partial x partial z / partial y

(a) Write an expression as a limit for the directional derivative of \(f\) at \(\left(x_0,\ y_0\right)\) in the direction of a unit vector \(\mathbf{u}=\langle a, \ b\rangle\). How do you interpret it as a rate? How do you interpret it geometrically? (b) If \(f\) is differentiable, write an expression for \(D_{\mathrm{u}}\ f\left(x_0,\ y_0\right)\) in terms of \(f_{x}\) and \(f_{y}\). Equation Transcription: ? ? Text Transcription: f (x_0, y_0) u = ? a, b ? f D_u f(x_0, y_0) f_x f_y

(a) Define the gradient vector \(\nabla f\) for a function \(f\) of two or three variables. (b) Express \(D_{\mathrm{u}}\ f\) in terms of \(\nabla f\). (c) Explain the geometric significance of the gradient. Equation Transcription: Text Transcription: nabla f f D_u f nabla f

What do the following statements mean? (a) \(f\) has a local maximum at \((a, b)\). (b) \(f\) has an absolute maximum at \((a, b)\). (c) \(f\) has a local minimum at \((a, b)\). (d) \(f\) has an absolute minimum at \((a, b)\). (e) \(f\) has a saddle point at \((a, b)\). Equation Transcription: Text Transcription: f (a, b)

(a) If \(f\) has a local maximum at \((a, \ b)\), what can you say about its partial derivatives at \((a, \ b)\)? (b) What is a critical point of \(f\)? Equation Transcription: Text Transcription: f (a, b)

State the Second Derivatives Test.

(a) What is a closed set in \(\mathbb{R}^{2}\)? What is a bounded set? (b) State the Extreme Value Theorem for functions of two variables. (c) How do you find the values that the Extreme Value Theorem guarantees? Equation Transcription: ?2 Text Transcription: ?^2

Explain how the method of Lagrange multipliers works in finding the extreme values of f(x, y, z) subject to the constraint g(x, y, z)=k. What if there is a second constraint h(x, y, z)=c?

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(f_y(a,\ b)=\lim \limits _{y\rightarrow b} \ \frac{f(a,y)-f(a,b)}{y-b}\) Equation Transcription: Text Transcription: f_y (a, b) = lim _y rightarrow b f(a, y) - f(a, b) / y - b

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. There exists a function \(f\) with continuous second-order partial derivatives such that \(f_x(x,\ y)=x+y^2\) and \(f_y(x,\ y)=x-y^2\). Equation Transcription: Text Transcription: f f_x(x, y) = x + y^2 f_y(x, y) = x - y^2

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(f_{xy}=\frac{\partial^2f}{\partial x\ \partial y}\) Equation Transcription: Text Transcription: f_xy = partial^2 f / partial x partial y

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. \(D_{\mathrm{k}}\ f(x,\ y,\ z)=f_z(x,\ y,\ z)\) Equation Transcription: Text Transcription: D_k f(x, y, z) = f_z(x, y, z)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f(x,\ y)\rightarrow L\) as \((x,\ y)\rightarrow(a,\ b)\) along every straight line through \((a,\ b)\), then \(\lim_{(x,\ y)\rightarrow(a,\ b)}\ f(x,\ y)=L\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f_x(a,\ b)\) and \(f_y(a,\ b)\) both exist, then \(f\) is differentiable at \((a, \ b)\) Equation Transcription: Text Transcription: f_x(a, b) f_y(a, b) f (a, b)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f\) has a local minimum at \((a, \ b)\) and \(f\) is differentiable at \((a, \ b)\), then \(\nabla f(a,\ b)=\mathbf{0}\). Equation Transcription: Text Transcription: f (a, b) f (a, b) nabla f(a, b)=0

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f\) is a function, then \(\lim \limits _{(x,\ y)\rightarrow(2,\ 5)}\ f(x,\ y)=f(2,\ 5)\) Equation Transcription: Text Transcription: f lim_(x, y) rightarrow (2, 5) f(x, y) = f(2, 5)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f(x, \ y)=ln \ y\), then \(\nabla \ f(x, \ y)=1/y\) Equation Transcription: Text Transcription: f(x, y)=ln y nabla f(x, y)=1/y

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \((2, \ 1)\) is a critical point of \(f\) and \(f_{xx}(2,\ 1)\ f_{yy}(2,\ 1)<\left[f_{xy}(2,1)\right]^2\) then \(f\) has a saddle point at \((2, \ 1)\). Equation Transcription: Text Transcription: (2, 1) f f_xx(2,1)f_yy(2,1)<[f_xy(2,1)]^2 f (2, 1)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f(x, \ y)=\sin \ x+\sin \ y\), then \(-\sqrt{2} \leqslant D_{u} \ f(x, \ y) \leqslant \sqrt{2}\). Equation Transcription: ? ? Text Transcription: f(x,y) = sin ?x + sin? y - sqrt 2 leqslant D_u f(x,y) leqslant sqrt 2

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(f(x, \ y)\) has two local maxima, then \(f\) must have a local minimum. Equation Transcription: Text Transcription: f(x, y) f

Find and sketch the domain of the function. f(x, y) = ln?(x + y + 1)

Find and sketch the domain of the function. \(f(x,\ y)=\sqrt{4-x^2-y^2}+\sqrt{1-x^2}\) Equation Transcription: Text Transcription: f(x, y) = sqrt 4 - x^2 - y^2 + sqrt 1 - x^2

Sketch the graph of the function. \(f(x,\ y)=1-y^2\) Equation Transcription: Text Transcription: f(x, y) = 1 - y^2

Sketch the graph of the function. \(f(x,\ y)=x^2+(y-2)^2\) Equation Transcription: Text Transcription: f(x, y) = x^2 + (y - 2)^2

Sketch several level curves of the function \(f(x,\ y)=\sqrt{4x^2+y^2}\) Equation Transcription: Text Transcription: f(x, y) = sqrt 4x^2 + y^2

Sketch several level curves of the function. \(f(x,\ y)=e^x+y\) Equation Transcription: Text Transcription: f(x, y) = e^x + y

Make a rough sketch of a contour map for the function whose graph is shown.

The contour map of a function \(f\) is shown. (a) Estimate the value of \(f(3, \ 2)\). (b) Is \(f_x(3,\ 2)\) positive or negative? Explain. (c) Which is greater, \(f_y(2,\ 1)\) or \(f_y(2,\ 2)\)? Explain. Equation Transcription: Text Transcription: f f(3, 2) f_x(3, 2) f_y(2, 1) f_y(2, 2)

Evaluate the limit or show that it does not exist. \(\lim \limits _{(x, \ y) \rightarrow(1, \ 1)} \frac{2 x y}{x^{2}+2 y^{2}}\)

Evaluate the limit or show that it does not exist. \(\lim \limits _{(x, y) \rightarrow(0,0)} \frac{2 x y}{x^{2}+2 y^{2}}\) Equation Transcription: Text Transcription: lim_(x,y) right arrow (0, 0) 2xy/x^2 +2y^2

A metal plate is situated in the \(xy\)-plane and occupies the rectangle \(0 \leqslant x \leqslant 10\), \(0 \leqslant y \leqslant 8\) where \(x\) and \(y\) are measured in meters. The temperature at the point \((x, \ y)\) in the plate is \(T(x, \ y)\), where \(T\) is measured in degrees Celsius. Temperatures at equally spaced points were measured and recorded in the table. (a) Estimate the values of the partial derivatives \(T_x(6,\ 4)\) and \(T_y(6,\ 4)\). What are the units? (b) Estimate the value of \(D_{\mathrm{u}}T(6,\ 4)\), where \(\mathbf{u}=(\mathbf{i}+\mathbf{j}) / \sqrt{2}\). Interpret your result. (c) Estimate the value of \(T_{xy}(6,\ 4)\) Equation Transcription: ? ? ? ? Text Transcription: xy 0 leqslant x leqslant 10 0 leqslant y leqslant 8 x y (x, y) T(x, y) T T_x (6, 4) T_y (6, 4) D_u T(6, 4) u = (i+j)/ sqrt 2 T_xy (6, 4)

Find a linear approximation to the temperature function \(T(x, \ y)\) in Exercise 11 near the point \((6, \ 4)\). Then use it to estimate the temperature at the point \((5, \ 3.8)\). Equation Transcription: Text Transcription: T(x, y) (6, 4) (5, 3.8)

Find the first partial derivatives. \(f(x,\ y)=\left(5y^3+2x^2y\right)^8\) Equation Transcription: Text Transcription: f(x, y) = (5y^3 + 2x^2 y)^8

Find the first partial derivatives \(g(u,\ v)=\frac{u+2v}{u^2+v^2}\) Equation Transcription: Text Transcription: g(u,v) = u+2v / u^2 + v^2

Find the first partial derivatives. \(F(\alpha,\ \beta)=\alpha^2\ \ln\left(\alpha^2+\beta^2\right)\) Equation Transcription: Text Transcription: F(alpha,beta) = alpha^2 ln(alpha^?2 + beta^2

Find the first partial derivatives. \(G(x,\ y,\ z)=e^{xz}\ \sin(y/z)\) Equation Transcription: Text Transcription: G(x,y,z) = e^xz sin?(y/z)

Find the first partial derivatives. \(S(u,\ v,\ w)=u\ \arctan(v\sqrt{w})\) Equation Transcription: Text Transcription: S(u, v, w) = u arctan?(v sqrt w)

The speed of sound traveling through ocean water is a function of temperature, salinity, and pressure. It has been modeled by the function \(C=1449.2+4.6 T-0.055 T^{2}+0.00029 T^{3}+(1.34-0.01 T)(S-35)+0.016 D\) where \(C\) is the speed of sound (in meters per second), \(T\) is the temperature (in degrees Celsius), \(S\) is the salinity (the concentration of salts in parts per thousand, which means the number of grams of dissolved solids per \(1000 \ g\) of water), and \(D\) is the depth below the ocean surface (in meters). Compute \(\partial C / \partial T\), \(\partial C / \partial S\) and \(\partial C / \partial D\) when \(T=10^{\circ} \mathrm{C}\), \(S=35\) parts per thousand, and \(D=100 \ m\). Explain the physical significance of these partial derivatives. Equation Transcription: Text Transcription: C = 1449.2 + 4.6T - 0.055T^2 + 0.00029T^3 + (1.34 - 0.01T)(S - 35) + 0.016D C T S 1000 g D partial C / partial T partial C / partial S partial C / partial D T = 10 degree C S = 35 D = 100 m

Find all second partial derivatives of \(f\). \(f(x,\ y)=4x^3-xy^2\) Equation Transcription: Text Transcription: f f(x, y) = 4x^3 - xy^2

Find all second partial derivatives of \(f\). \(z=x e^{-2 y}\) Equation Transcription: Text Transcription: f z = xe^-2y

Find all second partial derivatives of \(f\). \(f(x,\ y,\ z)=x^{k}y^{l}z^{m}\)

Find all second partial derivatives of \(f\). \(v=r \cos (s+2 t)\)

If , show that

If \(z=\sin (x+\sin t)\), show that \(\frac{\partial z}{\partial x} \frac{\partial^{2} z}{\partial x \partial t}=\frac{\partial z}{\partial t} \frac{\partial^{2} z}{\partial x^{2}}\) Equation Transcription: Text Transcription: z=sin?(x+sin? t) partial z/partial x partial^2 z/partial /x partial t= partial z/partial t partial^2 z/partial x^2

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. \(z=e^{x} \cos y,(0,0,1)\) Equation Transcription: Text Transcription: z=e^x cos ?y, (0,0,1)

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

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. sin (x y z) = x + 2y + 3z, (2,-1,0)

Use a computer to graph the surface and its tangent plane and normal line at on the same screen. Choose the domain and viewpoint so that you get a good view of all three objects.

Find the points on the hyperboloid where the tangent plane is parallel to the plane

Find du if \(u=\ln \left(1+s e^{2 t}\right)\)

Find the linear approximation of the function \(f(x, \ y, \ z)=x^{3} \sqrt{y^{2}+z^{2}}\) at the point (2, 3, 4) and use it to estimate the number \((1.98)^{3} \sqrt{(3.01)^{2}+(3.97)^{2}}\).

The two legs of a right triangle are measured as and with a possible error in measurement of at most in each. Use differentials to estimate the maximum error in the calculated value of (a) the area of the triangle and (b) the length of the hypotenuse.

If \(u=x^{2} y^{3}+z^{4}\), where \(x=p+3 p^{2}\), \(y=p e^{p}\), and \(z=p \sin p\), use the Chain Rule to find \(d u / d p\). Equation Transcription: Text Transcription: u=x^2 y^3+z^4 x=p+3p^2 y=pe^p z=p sin? p du/dp

If \(v=x^{2} \sin y+y e^{x y}\), where \(x=s+2 t\) and \(y=s t\) , use the Chain Rule to find \(\partial v / \partial s\) and \(\partial v / \partial t\) when \(s=0}\) and \(t=1\). Equation Transcription: Text Transcription: v=x^2 sin? y+ye^xy x=s+2t y=st partial v/partial s partial v/partial t s=0 t=1

Suppose , where , , and Find and when and

Use a tree diagram to write out the Chain Rule for the case where \(w=f(t, \ u, \ v), \ t=t(p, \ q, \ r, \ s), u=u(p, \ q, \ r, \ s)\), and \(v=v(p, \ q, \ r, \ s)\) are all differentiable functions.

If , where is differentiable, show that

The length of a side of a triangle is increasing at a rate of , the length of another side is decreasing at a rate of , and the contained angle is increasing at a rate of radian . How fast is the area of the triangle changing when inches, inches, and

If , where , and has continuous second partial derivatives, show that

If \(\cos (x y z)=1+x^{2} y^{2}+z^{2}\), find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). Equation Transcription: Text Transcription: cos?(xyz)=1+x^2 y^2+z^2 partial z/partial x partial z/partial y

Find the gradient of the function

(a) When is the directional derivative of a maximum? (b) When is it a minimum? (c) When is it ? (d) When is it half of its maximum value?

Find the directional derivative of at the given point in the indicated direction. , in the direction toward the point

Find the directional derivative of f at the given point in the indicated direction. \(f(x, y, z)=x^2 y+x \sqrt{1+z}, \quad(1,2,3)\) in the direction of \(\mathbf{v}=2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}\)

Find the maximum rate of change of at the point In which direction does it occur?

Find the direction in which increases most rapidly at the point . What is the maximum rate of increase?

The contour map shows wind speed in knots during Hurricane Andrew on August Use it to estimate the value of the directional derivative of the wind speed at Homestead, Florida, in the direction of the eye of the hurricane.

Find parametric equations of the tangent line at the point to the curve of intersection of the surface and the plane .

Find the local maximum and minimum values and saddle points of the function. You are encouraged to graph the function with a domain and viewpoint ihai reveals all the important aspects of the function.

Find the local maximum and minimum values and saddle points of the function. You are encouraged to graph the function with a domain and viewpoint ihai reveals all the important aspects of the function.

Find the local maximum and minimum values and saddle points of the function. You are encouraged to graph the function with a domain and viewpoint ihai reveals all the important aspects of the function.

Find the local maximum and minimum values and saddle points of the function. You are encouraged to graph the function with a domain and viewpoint ihai reveals all the important aspects of the function.

Find th? absolute maximum and minimum values of on the set . is the closed triangular region in the -plane with vertices , and

Find th? absolute maximum and minimum values of on the set . is the disk

Use a graph or level curves or both to estimate the local maximum and minimum values and saddle points of . Then use calculus to find these values precisely.

Use a graphing calculator or computer (or Newton's method) to find the critical points of correct to three decimal places. Then classify the critical points and find the highest point on the graph.

Use Lagrange multipliers to find the maximum and minimum values of subject to the given constraint(s).

Use Lagrange multipliers to find the maximum and minimum values of subject to the given constraint(s).

Use Lagrange multipliers to find the maximum and minimum values of subject to the given constraint(s).

Use Lagrange multipliers to find the maximum and minimum values of subject to the given constraint(s). ,

Find the points on the surface that are closest to the origin.

In this problem we identify a point on the line such that the sum of the distances from to and from to is a minimum. (a) Write a function that gives the sum of the distances from to a point and from to . Let Following the method of Lagrange multipliers, we wish to find the minimum value of subject to the constraint Graph the constraint curve along with several level curves of , and then use the graph to estimate the minimum value of . What point on the line minimizes (b) Verify that the gradient vectors and are parallel.

A pentagon is formed by placing an isosceles triangle on a rectangle, as shown in the figure. If the pentagon has fixed perimeter , find the lengths of the sides of the pentagon that maximize the area of the pentagon.

A rectangle with length L and width W is cut into four smaller rectangles by two lines parallel to the sides. Find the maximum and minimum values of the sum of the squares of the areas of the smaller rectangles.

Marine biologists have determined that when a shark detects the presence of blood in the water, it will swim in the direction in which the concentration of the blood increases most rapidly. Based on certain tests, the concentration of blood (in parts per million) at a point on the surface of seawater is approximated by where and are measured in meters in a rectangular coordinate system with the blood source at the origin. (a) Identify the level curves of the concentration function and sketch several members of this family together with a path that a shark will follow to the source. (b) Suppose a shark is at the point when it first detects the presence of blood in the water. Find an equation of the shark's path by setting up and solving a differential equation.

A long piece of galvanized sheet metal with width is to be bent into a symmetric form with three straight sides to make a rain gutter. A cross-section is shown in the figure. (a) Determine the dimensions that allow the maximum possible flow; that is, find the dimensions that give the maximum possible cross-sectional area. (b) Would it be better to bend the metal into a gutter with a semicircular cross-section?

A long piece of galvanized sheet metal with width is to be bent into a symmetric form with three straight sides to make a rain gutter. A cross-section is shown in the figure. (a) Determine the dimensions that allow the maximum possible flow; that is, find the dimensions that give the maximum possible cross-sectional area. (b) Would it be better to bend the metal into a gutter with a semicircular cross-section?

Suppose is a differentiable function of one variable. Show that all tangent planes to the surface intersect in a common point.

(a) Newton's method for approximating a solution of an equation (see Section 4.8) can be adapted to approximating a solution of a system of equations and The surfaces and intersect in a curve that intersects the -plane at the point , which is the solution of the system. If an initial approximation is close to this point, then the tangent planes to the surfaces at intersect in a straight line that intersects the -plane in a point , which should be closer to . (Compare with Figure 4.8.2.) Show that where , and their partial derivatives are evaluated at . If we continue this procedure, we obtain successive approximations . (b) It was Thomas Simpson (1710-1761) who formulated Newton's method as we know it today and who extended it to functions of two variables as in part (a). (See the biography of Simpson in Section 7.7.) The example that he gave to illustrate the method was to solve the system of equations In other words, he found the points of intersection of the curves in the figure. Use the method of part (a) to find the coordinates of the points of intersection correct to six decimal places.

If the ellipse is to enclose the circle , what values of and minimize the area of the ellipse?

Show that the maximum value of the function is Hint: One method for attacking this problem is to use the Cauchy-Schwarz Inequality: (See Exercise 12.3.61.)