Math

CalculusCalculus (MindTap Course List)Equation 6 is a formula for the derivative d y / d x of a function defined implicitly by an equation F ( x , y ) = 0 , provided that F is differentiable and F y ≠ 0 . Prove that if F has continuous second derivatives, then a formula for the second derivative of y is d 2 y d x 2 = − F x x F y 2 − 2 F x y F x F y + F y y F x 2 F y 3BuyFind*launch*

8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 14.5, Problem 59E

Textbook Problem

Equation 6 is a formula for the derivative
*F* is differentiable and
*F* has continuous second derivatives, then a formula for the second derivative of *y* is

Calculus (MindTap Course List)

Ch. 14.1 - In Example 2 we considered the function W=f(T,v),...Ch. 14.1 - The temperature-humidity index I or humidex, for...Ch. 14.1 - A manufacturer has modeled its yearly production...Ch. 14.1 - Verify for the Cobb-Douglas production function...Ch. 14.1 - A model for the surface area of a human body is...Ch. 14.1 - The wind-chill index W discussed in Example 2 has...Ch. 14.1 - The wave heights h in the open sea depend on the...Ch. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Let g(x,y)=cos(x+2y). aEvaluate g(2,1). bFind the...Ch. 14.1 - Let F(x,y)=1+4y2. aEvaluate F(3,1). bFind and...

Ch. 14.1 - Let f(x,y,z)=x+y+z+ln(4x2y2z2). aEvaluate...Ch. 14.1 - Let g(x,y,z)=x3y2z10xyz. aEvaluate g(1,2,3). bFind...Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Find and sketch the domain of the function....Ch. 14.1 - Sketch the graph of the function. f(x,y)=yCh. 14.1 - Sketch the graph of the function. f(x,y)=x2Ch. 14.1 - Sketch the graph of the function. f(x,y)=104x5yCh. 14.1 - Sketch the graph of the function. f(x,y)=cosyCh. 14.1 - Sketch the graph of the function. f(x,y)=sinxCh. 14.1 - Sketch the graph of the function. f(x,y)=2x2y2Ch. 14.1 - Sketch the graph of the function. f(x,y)=x2+4y2+1Ch. 14.1 - Sketch the graph of the function. f(x,y)=4x2+y2Ch. 14.1 - Sketch the graph of the function. f(x,y)=44x2y2Ch. 14.1 - Match the function with its graph labeled IVI....Ch. 14.1 - A contour map for a function f is shown. Use it to...Ch. 14.1 - Shown is a contour map of atmospheric pressure in...Ch. 14.1 - Level curves isothermals are shown for the typical...Ch. 14.1 - Two contour maps are shown. One is for a function...Ch. 14.1 - Locate the points A and B on the map of Lonesome...Ch. 14.1 - Make a rough sketch of a contour map for the...Ch. 14.1 - The body mass index BMI of a person is defined by...Ch. 14.1 - The body mass index is defined in Exercise 39....Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - A contour map of a function is shown. Use it to...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Draw a contour map of the function showing several...Ch. 14.1 - Sketch both a contour map and a graph of the...Ch. 14.1 - Sketch both a contour map and a graph of the...Ch. 14.1 - A thin metal plate, located in the xy-plane, has...Ch. 14.1 - If V(x,y) is the electric potential at a point...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Match the function a with its graph labeled AF...Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Describe the level surfaces of the function....Ch. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Describe how the graph of g is obtained from the...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Use a computer to graph the function using various...Ch. 14.1 - Graph the function using various domains and...Ch. 14.1 - Graph the function using various domains and...Ch. 14.1 - Investigate the family of functions...Ch. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Use a computer to investigate the family of...Ch. 14.1 - Graph the functions f(x,y)=x2+y2 f(x,y)=ex2+y2...Ch. 14.1 - aShow that, by taking logarithms, the general...Ch. 14.2 - Suppose that lim(x,y)(3,1)f(x,y)=6. What can you...Ch. 14.2 - Explain why each function is continuous or...Ch. 14.2 - Use a table of numerical values of f(x,y) for...Ch. 14.2 - Use a table of numerical values of f(x,y) for...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Use a computer graph of the function to explain...Ch. 14.2 - Use a computer graph of the function to explain...Ch. 14.2 - Find h(x,y)=g(f(x,y)) and the set of points at...Ch. 14.2 - Find h(x,y)=g(f(x,y)) and the set of points at...Ch. 14.2 - Graph the function and observe where it is...Ch. 14.2 - Graph the function and observe where it is...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Use polar coordinates to find the limit. If (r,)...Ch. 14.2 - Use polar coordinates to find the limit. If (r,)...Ch. 14.2 - Use polar coordinates to find the limit. If (r,)...Ch. 14.2 - At the beginning of this section we considered the...Ch. 14.2 - Graph and discuss the continuity of the function...Ch. 14.2 - Let f(x,y)={0ify0oryx41if0yx4 a Show that f(x,y)0...Ch. 14.2 - Show that the function f given by f(x)=|x| is...Ch. 14.2 - If cVn, show that the function f given by f(x)=cx...Ch. 14.3 - The temperature T in C at a location in the...Ch. 14.3 - At the beginning of this section we discussed the...Ch. 14.3 - The wind-chill index W is the perceived...Ch. 14.3 - The wave heights h in the open sea depend on the...Ch. 14.3 - Determine the signs of the partial derivatives for...Ch. 14.3 - Determine the signs of the partial derivatives for...Ch. 14.3 - Determine the signs of the partial derivatives for...Ch. 14.3 - Determine the signs of the partial derivatives for...Ch. 14.3 - The following surfaces, labeled a, b, and c, are...Ch. 14.3 - A contour map is given for a function f. Use it to...Ch. 14.3 - If f(x,y)=164x2y2, find fx(1,2) and fy(1,2) and...Ch. 14.3 - If f(x,y)=4x24y2, find fx(1,0) and fy(1,0) and...Ch. 14.3 - Find fx and fy and graph f, fx, and fy with...Ch. 14.3 - Find fx and fy and graph f, fx, and fy with...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the first partial derivatives of the...Ch. 14.3 - Find the indicated partial derivative....Ch. 14.3 - Find the indicated partial derivative....Ch. 14.3 - Find the indicated partial derivative....Ch. 14.3 - Find the indicated partial derivative....Ch. 14.3 - Use the definition of partial derivatives as...Ch. 14.3 - Use the definition of partial derivatives as...Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Use implicit differentiation to find z/x and z/y....Ch. 14.3 - Find z/y and z/y. a z=f(x)+g(y) b z=f(x+y)Ch. 14.3 - Find z/y and z/y. a z=f(x)g(y) b z=f(xy) c...Ch. 14.3 - Find all the second partial derivatives....Ch. 14.3 - Find all the second partial derivatives....Ch. 14.3 - Find all the second partial derivatives. z=y2x+3yCh. 14.3 - Find all the second partial derivatives. T=e2rcosCh. 14.3 - Find all the second partial derivatives....Ch. 14.3 - Find all the second partial derivatives. w=1+uv2Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Verify that the conclusion of Clairauts Theorem...Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives. W=u+v2;...Ch. 14.3 - Find the indicated partial derivatives....Ch. 14.3 - Find the indicated partial derivatives. w=xy+2z;...Ch. 14.3 - Find the indicated partial derivatives. u=xaybzc;...Ch. 14.3 - If f(x,y,z)=xy2z3+arcsin(xz), find fxzy. Hint:...Ch. 14.3 - If g(x,y,z)=1+xz+1xy, find gxzy. Hint: Use a...Ch. 14.3 - Use the table of values of f(x,y) to estimate the...Ch. 14.3 - Level curves are shown for a function f. Determine...Ch. 14.3 - Verify that the function u=e2k2tsinkx is a...Ch. 14.3 - Determine whether each of the following functions...Ch. 14.3 - Verify that the function u=1/x2+y2+z2 is a...Ch. 14.3 - Show that each of the following functions is a...Ch. 14.3 - If f and g are twice differentiable functions of a...Ch. 14.3 - If u=ea1x1+a2x2+...+anxn, where a12+a22+...+an2=1,...Ch. 14.3 - The diffusion equation ct=D2cx2 where D is a...Ch. 14.3 - The temperature at a point x, y on a flat metal...Ch. 14.3 - The total resistance R produced by three...Ch. 14.3 - Show that the Cobb-Douglas production function...Ch. 14.3 - Show that the Cobb-Douglas production function...Ch. 14.3 - Cobb and Douglas used the equation...Ch. 14.3 - The van der Waals equation for n moles of a gas is...Ch. 14.3 - The gas law for a fixed mass m of an ideal gas at...Ch. 14.3 - For the ideal gas of Exercise 88, show that...Ch. 14.3 - The wind-chill index is modeled by the function...Ch. 14.3 - A model for the surface area of a human body is...Ch. 14.3 - One of Poiseuilles laws states that the resistance...Ch. 14.3 - In the project on page 271 we expressed the power...Ch. 14.3 - The average energy E in kcal needed for a lizard...Ch. 14.3 - The kinetic energy of a body with mass m and...Ch. 14.3 - If a, b, c are the sides of a triangle and A, B, C...Ch. 14.3 - You are told that there is a function f whose...Ch. 14.3 - The paraboloid z=6xx22y2 intersects the plane x=1...Ch. 14.3 - The ellipsoid 4x2+2y2+z2=16 intersects the plane...Ch. 14.3 - In a study of frost penetration it was found that...Ch. 14.3 - Use Clairauts Theorem to show that if the...Ch. 14.3 - a How many nth-order partial derivatives does a...Ch. 14.3 - If f(x,y)=x(x2+y2)3/2esin(x2y) find fx(1,0). Hint:...Ch. 14.3 - If f(x,y)=x3+y33, find fx(0,0).Ch. 14.3 - Let f(x,y)={x3yxy3x2+y2if(x,y)(0,0)0if(x,y)=(0,0)...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Find an equation of the tangent plane to the given...Ch. 14.4 - Graph the surface and the tangent plane at the...Ch. 14.4 - Graph the surface and the tangent plane at the...Ch. 14.4 - Draw the graph of f and its tangent plane at the...Ch. 14.4 - Draw the graph of f and its tangent plane at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Explain why the function is differentiable at the...Ch. 14.4 - Verify the linear approximation at (0,0)....Ch. 14.4 - Verify the linear approximation at (0,0)....Ch. 14.4 - Given that f is a differentiable function with...Ch. 14.4 - Find the linear approximation of the function...Ch. 14.4 - Find the linear approximation of the function...Ch. 14.4 - The wave heights h in the open sea depend on the...Ch. 14.4 - Use the table in Example 3 to find a linear...Ch. 14.4 - The wind-chill index W is the perceived...Ch. 14.4 - Find the differential of the function. z=e2xcos2tCh. 14.4 - Find the differential of the function. u=x2+3y2Ch. 14.4 - Find the differential of the function. m=p5q3Ch. 14.4 - Find the differential of the function. T=v1+uvwCh. 14.4 - Find the differential of the function. R=2cosCh. 14.4 - Find the differential of the function. L=xzey2z2Ch. 14.4 - If z=5x2+y2 and (x,y) changes from (1,2) to...Ch. 14.4 - If z=x2xy+3y2 and (x,y) changes from (3,1) to...Ch. 14.4 - The length and width of a rectangle are measured...Ch. 14.4 - Use differentials to estimate the amount of metal...Ch. 14.4 - Use differentials to estimate the amount of tin in...Ch. 14.4 - The wind-chill index is modeled by the function...Ch. 14.4 - The tension T in the string of the yo-yo in the...Ch. 14.4 - The pressure, volume, and temperature of a mole of...Ch. 14.4 - If R is the total resistance of three resistors,...Ch. 14.4 - A model for the surface area of a human body is...Ch. 14.4 - In Exercise 14.1.39 and Example 14.3.3, the body...Ch. 14.4 - Suppose you need to know an equation of the...Ch. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Show that the function is differentiable by...Ch. 14.4 - Prove that if f is a function of two variables...Ch. 14.4 - a The function...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Use the Chain Rule to find z/s and z/t....Ch. 14.5 - Let p(t)=f(x,y), where f is differentiable,...Ch. 14.5 - Let R(s,t)=G(u(s,t),v(s,t)), where G, u and v are...Ch. 14.5 - Suppose f is a differentiable function of x and y,...Ch. 14.5 - Suppose f is a differentiable function of x and y,...Ch. 14.5 - Use a tree diagram to write out the Chain Rule for...Ch. 14.5 - Use a tree diagram to write out the Chain Rule for...Ch. 14.5 - Use a tree diagram to write out the Chain Rule for...Ch. 14.5 - Use a tree diagram to write out the Chain Rule for...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use the Chain Rule to find the indicated partial...Ch. 14.5 - Use Equation 6 to find dy/dx. ycosx=x2+y2Ch. 14.5 - Use Equation 6 to find dy/dx. cos(xy)=1+sinyCh. 14.5 - Use Equation 6 to find dy/dx. tan1(x2y)=x+xy2Ch. 14.5 - Use Equation 6 to find dy/dx. eysinx=x+xyCh. 14.5 - Use Equations 7 to find z/x and z/y. x2+2y2+3z2=1Ch. 14.5 - Use Equations 7 to find z/x and z/y. x2y2+z22z=4Ch. 14.5 - Use Equations 7 to find z/x and z/y. ez=xyzCh. 14.5 - Use Equations 7 to find z/x and z/y. yz+xlny=z2Ch. 14.5 - The temperature at a point x, y is Tx, y, measured...Ch. 14.5 - Wheat production W in a given year depends on the...Ch. 14.5 - The speed of sound traveling through ocean water...Ch. 14.5 - The radius of a right circular cone is increasing...Ch. 14.5 - The length l, width w, and height h of a box...Ch. 14.5 - The voltage V in a simple electrical circuit is...Ch. 14.5 - The pressure of 1 mole of an ideal gas is...Ch. 14.5 - A manufacturer has modeled its yearly production...Ch. 14.5 - One side of a triangle is increasing at a rate of...Ch. 14.5 - A sound with frequency fs is produced by a source...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - Assume that all the given functions have...Ch. 14.5 - A function f is called homogeneous of degree n if...Ch. 14.5 - If f is homogeneous of degree n, show that...Ch. 14.5 - If f is homogeneous of degree n, show that...Ch. 14.5 - Suppose that the equation F(x,y,z)=0 implicitly...Ch. 14.5 - Equation 6 is a formula for the derivative dy/dx...Ch. 14.6 - Level curves for barometric pressure in millibars...Ch. 14.6 - The contour map shows the average maximum...Ch. 14.6 - A table of values for the wind-chill index...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - Find the directional derivative of f at the given...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - a Find the gradient of f. b Evaluate the gradient...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Find the directional derivative of the function at...Ch. 14.6 - Use the figure to estimate Duf(2,2).Ch. 14.6 - Find the directional derivative of f(x,y)=xy at...Ch. 14.6 - Find the directional derivative of f(x,y,z)=xy2z3...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - Find the maximum rate of change of f at the given...Ch. 14.6 - a Show that a differentiable function f decreases...Ch. 14.6 - Find the directions in which the directional...Ch. 14.6 - Find all points at which the direction of fastest...Ch. 14.6 - Near a buoy, the depth of a lake at the point with...Ch. 14.6 - The temperature T in a metal ball is inversely...Ch. 14.6 - The temperature at a point (x,y,z) is given by...Ch. 14.6 - Suppose that over a certain region of space the...Ch. 14.6 - Suppose you are climbing a hill whose shape is...Ch. 14.6 - Let f be a function of two variables that has...Ch. 14.6 - Shown is a topographic map of Blue River Pine...Ch. 14.6 - Show that the operation of taking the gradient of...Ch. 14.6 - Sketch the gradient vector f(4,6) for the function...Ch. 14.6 - The second directional derivative of f(x,y) is...Ch. 14.6 - a If u=a,b is a unit vector and f has continuous...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Find equations of a the tangent plane and b the...Ch. 14.6 - Use a computer to graph the surface, the tangent...Ch. 14.6 - Use a computer to graph the surface, the tangent...Ch. 14.6 - If f(x,y)=xy, find the gradient vector f(3,2) and...Ch. 14.6 - If g(x,y)=x2+y24x, find the gradient vector g(1,2)...Ch. 14.6 - Show that the equation of the tangent plane to the...Ch. 14.6 - Find the equation of the tangent plane to the...Ch. 14.6 - Show that the equation of the tangent plane to the...Ch. 14.6 - At what point on the ellipsoid x2+y2+2z2=1 is the...Ch. 14.6 - Are there any points on the hyperboloid x2y2z2=1...Ch. 14.6 - Show that the ellipsoid 3x2+2y2+z2=9 and the...Ch. 14.6 - Show that every plane that is tangent to the cone...Ch. 14.6 - Show that every normal line to the sphere...Ch. 14.6 - Where does the normal line to the paraboloid...Ch. 14.6 - At what points does the normal line through the...Ch. 14.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 14.6 - Show that the pyramids cut off from the first...Ch. 14.6 - Find parametric equations for the tangent line to...Ch. 14.6 - a The plane y+z=3 intersects the cylinder x2+y2=5...Ch. 14.6 - Where does the helix r(t)=cost,sint,t intersect...Ch. 14.6 - The helix r(t)=cos(t/2),sin(t/2),t intersects the...Ch. 14.6 - a Two surfaces are called orthogonal at a point of...Ch. 14.6 - a Show that the function f(x,y)=xy3 is continuous...Ch. 14.6 - Suppose that the directional derivatives of f(x,y)...Ch. 14.6 - Show that if z=f(x,y) is differentiable at...Ch. 14.7 - Suppose 1, 1 is a critical point of a function f...Ch. 14.7 - Suppose 0, 2 is a critical point of a function g...Ch. 14.7 - Use the level curves in the figure to predict the...Ch. 14.7 - Use the level curves in the figure to predict the...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Find the local maximum and minimum values and...Ch. 14.7 - Show that f(x,y)=x2+4y24xy+2 has an infinite...Ch. 14.7 - Show that f(x,y)=x2yex2y2 has minimum values at...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graph or level curves or both to estimate...Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Use a graphing device as in Example 4 or Newtons...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - Find the absolute maximum and minimum values of f...Ch. 14.7 - For functions of one variable it is impossible for...Ch. 14.7 - If a function of one variable is continuous on an...Ch. 14.7 - Find the shortest distance from the point (2,0,3)...Ch. 14.7 - Find the point on the plane x2y+3z=6 that is...Ch. 14.7 - Find the points on the cone z2=x2+y2 that are...Ch. 14.7 - Find the points on the surface y2=9+xz that are...Ch. 14.7 - Find three positive numbers whose sum is 100 and...Ch. 14.7 - Find three positive numbers whose sum is 12 and...Ch. 14.7 - Find the maximum volume of a rectangular box that...Ch. 14.7 - Find the dimensions of the box with volume 1000...Ch. 14.7 - Find the volume of the largest rectangular box in...Ch. 14.7 - Find the dimensions of the rectangular box with...Ch. 14.7 - Find the dimensions of a rectangular box of...Ch. 14.7 - The base of an aquarium with given volume V is...Ch. 14.7 - A cardboard box without a lid is to have a volume...Ch. 14.7 - A rectangular building is being designed to...Ch. 14.7 - If the length of the diagonal of a rectangular box...Ch. 14.7 - A model for the yield Y of an agricultural crop as...Ch. 14.7 - The Shannon index sometimes called the...Ch. 14.7 - Three alleles alternative versions of a gene A, B,...Ch. 14.7 - Suppose that a scientist has reason to believe...Ch. 14.7 - Find an equation of the plane that passes through...Ch. 14.8 - Pictured are a contour map of f and a curve with...Ch. 14.8 - a Use a graphing calculator or computer to graph...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - Each of these extreme value problems has a...Ch. 14.8 - The method of Lagrange multipliers assumes that...Ch. 14.8 - Find the minimum value of f(x,y,z)=x2+2y2+3z2...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f subject to both...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Find the extreme values of f on the region...Ch. 14.8 - Consider the problem of maximizing the function...Ch. 14.8 - Consider the problem of minimizing the function...Ch. 14.8 - a If your computer algebra system plots implicitly...Ch. 14.8 - The total production P of a certain product...Ch. 14.8 - Referring to Exercise 27, we now suppose that the...Ch. 14.8 - Use Lagrange multipliers to prove that the...Ch. 14.8 - Use Lagrange multipliers to prove that the...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Find the maximum and minimum volumes of a...Ch. 14.8 - The plane x+y+2z=2 intersects the paraboloid...Ch. 14.8 - The plane 4x3y+8z=5 intersects the cone z2=x2+y2...Ch. 14.8 - Find the maximum and minimum values of f subject...Ch. 14.8 - Find the maximum and minimum values of f subject...Ch. 14.8 - a Find the maximum value of...Ch. 14.8 - a Maximize i=1nxiyi subject to the constraints...Ch. 14.R - aWhat is a function of two variables? bDescribe...Ch. 14.R - What is a function of three variables? How can you...Ch. 14.R - What does lim(x,y)(a,b)f(x,y)=L mean? How can you...Ch. 14.R - a What does it mean to say that f is continuous at...Ch. 14.R - a Write expressions for the partial derivatives...Ch. 14.R - What does Clairauts Theorem say?Ch. 14.R - How do you find a tangent plane to each of the...Ch. 14.R - Define the linearization of f at (a,b). What is...Ch. 14.R - aWhat does it mean to say that f is differentiable...Ch. 14.R - If z=f(x,y), what are the differentials dx, dy,...Ch. 14.R - State the Chain Rule for the case where z=f(x,y)...Ch. 14.R - If z is defined implicitly as a function of x and...Ch. 14.R - aWrite an expression as a limit for the...Ch. 14.R - aDefine the gradient vector f for a function f of...Ch. 14.R - What do the following statements mean? af has a...Ch. 14.R - aIf f has a local maximum at a, b, what can you...Ch. 14.R - State the Second Derivatives Test.Ch. 14.R - aWhat is a closed set in 2? What is a bounded set?...Ch. 14.R - Explain how the method of Lagrange multipliers...Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Determine whether the statement is true or false....Ch. 14.R - Find and sketch the domain of the function....Ch. 14.R - Find and sketch the domain of the function....Ch. 14.R - Sketch the graph of the function. f(x,y)=1y2Ch. 14.R - Sketch the graph of the function. f(x,y)=x2+(y2)2Ch. 14.R - Sketch several level curves of the function....Ch. 14.R - Sketch several level curves of the function....Ch. 14.R - Make a rough sketch of a contour map for the...Ch. 14.R - The contour map of a function f is shown. a...Ch. 14.R - Evaluate the limit or show that it does not exist....Ch. 14.R - Evaluate the limit or show that it does not exist....Ch. 14.R - A metal plate is situated in the xy-plane and...Ch. 14.R - Find a linear approximation to the temperature...Ch. 14.R - Find the first partial derivatives....Ch. 14.R - Find the first partial derivatives....Ch. 14.R - Find the first partial derivatives. F(,)=2ln(2+2)Ch. 14.R - Find the first partial derivatives....Ch. 14.R - Find the first partial derivatives....Ch. 14.R - The speed of sound traveling through ocean water...Ch. 14.R - Find all second partial derivatives of f....Ch. 14.R - Find all second partial derivatives of f. z=xe2yCh. 14.R - Find all second partial derivatives of f....Ch. 14.R - Find all second partial derivatives of f....Ch. 14.R - If z=xy+xey/x, show that xzx+yzy=xy+z.Ch. 14.R - If z=sin(x+sint), show that zx2zxt=zt2zx2Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Find equations of a the tangent plane and b the...Ch. 14.R - Use a computer to graph the surface z=x2+y4 and...Ch. 14.R - Find the points on the hyperboloid x2+4y2z2=4...Ch. 14.R - Find du if u=ln(1+se2t).Ch. 14.R - Find the linear approximation of the function...Ch. 14.R - The two legs of a right triangle are measured as 5...Ch. 14.R - If u=x2y3+z4, where x=p+3p2, y=pep, and z=psinp,...Ch. 14.R - If v=x2siny+yexy, where x=s+2t and y=st, use the...Ch. 14.R - Suppose z=f(x,y), where x=g(s,t), y=h(s,t),...Ch. 14.R - Use a tree diagram to write out the Chain Rule for...Ch. 14.R - If z=y+f(x2y2), where f is differentiable, show...Ch. 14.R - The length x of a side of a triangle is increasing...Ch. 14.R - If z=f(u,v), where u=xy,v=y/x, and f has...Ch. 14.R - If cos(xyz)=1+x2y2+z2, find zx and zy.Ch. 14.R - Find the gradient of the function f(x,y,z)=x2eyz2.Ch. 14.R - a When is the directional derivative of f a...Ch. 14.R - 4546 Find the directional derivative of f at the...Ch. 14.R - 4546 Find the directional derivative of f at the...Ch. 14.R - Find the maximum rate of change of f(x,y)=x2y+y at...Ch. 14.R - Find the direction in which f(x,y,z)=zexy...Ch. 14.R - The contour map shows wind speed in knots during...Ch. 14.R - Find parametric equations of the tangent line at...Ch. 14.R - 5154 Find the local maximum and minimum values and...Ch. 14.R - 5154 Find the local maximum and minimum values and...Ch. 14.R - 5154 Find the local maximum and minimum values and...Ch. 14.R - 5154 Find the local maximum and minimum values and...Ch. 14.R - 5556 Find the absolute maximum and minimum values...Ch. 14.R - 5556 Find the absolute maximum and minimum values...Ch. 14.R - Use a graph or level curves or both to estimate...Ch. 14.R - Use a graphing calculator or computer or Newtons...Ch. 14.R - 5962 Use Lagrange multipliers to find the maximum...Ch. 14.R - 5962 Use Lagrange multipliers to find the maximum...Ch. 14.R - 5962 Use Lagrange multipliers to find the maximum...Ch. 14.R - 5962 Use Lagrange multipliers to find the maximum...Ch. 14.R - Find the points on the surface xy2z3=2 that are...Ch. 14.R - A package in the shape of a rectangular box can be...Ch. 14.R - A pentagon is formed by placing an isosceles...Ch. 14.P - A rectangle with length L and width W is cut into...Ch. 14.P - Marine biologists have determined that when a...Ch. 14.P - A long piece of galvanized sheet metal with width...Ch. 14.P - For what values of the number r is the function...Ch. 14.P - Suppose f is a differentiable function of one...Ch. 14.P - a Newtons method for approximating a root of an...Ch. 14.P - If the ellipse x2/a2+y2/b2=1 is to enclose the...Ch. 14.P - Show that the maximum value of the function...

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