Calculus, Early Transcendentals
9th Edition
ISBN: 9781337613927
Author: Stewart
Publisher: CENGAGE L
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Chapter 14.5, Problem 56E
To determine
To Verify: The function
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Find the partial derivatives of the function
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Second derivatives Let z = f(x, y) =
where x = s + t2² and
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Compute the second partial derivatives -
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f(x, y) = 4 cos(xy?)
Verify the following theorem in this case.
If f(x, y) is of class c² (is twice continuously differentiable), then the mixed partial derivatives are equal; that is,
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Chapter 14 Solutions
Calculus, Early Transcendentals
Ch. 14.1 - If f(x,y)=x2y/2xy2 , find (a) f(1,3) (b) f(2,1)...Ch. 14.1 - If g(x,y)=xsiny+ysinx , find (a) g(,0) (b)...Ch. 14.1 - Let F(x,y,z)=yx2z . (a) Evaluate F(3,4,1) . (b)...Ch. 14.1 - Let f(x,y,z)=lnzx2+y2 . (a) Evaluate f(4,3,6) ....Ch. 14.1 - Find and sketch the domain of the function. 7....Ch. 14.1 - Find and sketch the domain of the function. 8....Ch. 14.1 - Find and sketch the domain of the function. 9....Ch. 14.1 - Find and sketch the domain of the function. 10....Ch. 14.1 - Find and sketch the domain of the function. 11....Ch. 14.1 - Find and sketch the domain of the function. 12....
Ch. 14.1 - Find and sketch the domain of the function. 13....Ch. 14.1 - Find and sketch the domain of the function. 14....Ch. 14.1 - Find and sketch the domain of the function. 16....Ch. 14.1 - A model for the surface area of a human body is...Ch. 14.1 - Prob. 18ECh. 14.1 - The temperature-humidity index I (or humidex, for...Ch. 14.1 - A company makes three sizes of cardboard boxes:...Ch. 14.1 - Sketch the graph of the function. 23. f(x,y)=yCh. 14.1 - Sketch the graph of the function. 24. f(x,y)=x2Ch. 14.1 - Sketch the graph of the function. 25....Ch. 14.1 - Sketch the graph of the function. 26. f(x,y)=cosyCh. 14.1 - Sketch the graph of the function. 27. f(x,y)=sinxCh. 14.1 - Sketch the graph of the function. 28. f(x,y)=2x2y2Ch. 14.1 - Sketch the graph of the function. 29....Ch. 14.1 - Sketch the graph of the function. 30....Ch. 14.1 - Sketch the graph of the function. 31....Ch. 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 - Level curves (isothermals) are shown for the...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...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 - Graph the function using various domains and...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. 68....Ch. 14.1 - Prob. 69ECh. 14.1 - Prob. 70ECh. 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 - Prob. 75ECh. 14.1 - Prob. 76ECh. 14.1 - Investigate the family of functions...Ch. 14.1 - Graph the functions f(x,y)=x2+y2 f(x,y)=ex2+y2...Ch. 14.2 - Prob. 1ECh. 14.2 - Explain why each function is continuous or...Ch. 14.2 - Prob. 3ECh. 14.2 - Prob. 4ECh. 14.2 - Prob. 5ECh. 14.2 - Prob. 6ECh. 14.2 - Prob. 7ECh. 14.2 - Prob. 8ECh. 14.2 - Prob. 9ECh. 14.2 - Prob. 10ECh. 14.2 - Prob. 11ECh. 14.2 - Find the limit. 12. lim(x,y)(,/2)cosysin2ycosxcosyCh. 14.2 - Prob. 13ECh. 14.2 - Prob. 14ECh. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - Show that the limit does not exist. 17....Ch. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Prob. 21ECh. 14.2 - Prob. 22ECh. 14.2 - Prob. 23ECh. 14.2 - Prob. 24ECh. 14.2 - Prob. 25ECh. 14.2 - Prob. 26ECh. 14.2 - Find the limit, if it exists, or show that the...Ch. 14.2 - Prob. 28ECh. 14.2 - Prob. 29ECh. 14.2 - Prob. 30ECh. 14.2 - Prob. 31ECh. 14.2 - Prob. 32ECh. 14.2 - Prob. 33ECh. 14.2 - Prob. 34ECh. 14.2 - Prob. 35ECh. 14.2 - Prob. 36ECh. 14.2 - Prob. 37ECh. 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 - Prob. 48ECh. 14.2 - Determine the set of points at which the function...Ch. 14.2 - Prob. 50ECh. 14.2 - Prob. 51ECh. 14.2 - Prob. 52ECh. 14.2 - Prob. 53ECh. 14.2 - Prove the three special limits in (2).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,...Ch. 14.2 - Show that the function f given by f(x) = |x| is...Ch. 14.2 - If c Vn, show that the function f given by f(x) =...Ch. 14.3 - At the beginning of this section we discussed the...Ch. 14.3 - Prob. 2ECh. 14.3 - Prob. 4ECh. 14.3 - Prob. 5ECh. 14.3 - A contour map is given for a function f. Use it to...Ch. 14.3 - If f(x, y) = 16 4x2 y2, find fx(1, 2) and fy(1,...Ch. 14.3 - If f(x,y)=4x24y2, find fx(1,0) and fy(1, 0) and...Ch. 14.3 - Prob. 9ECh. 14.3 - Prob. 10ECh. 14.3 - Prob. 11ECh. 14.3 - Prob. 12ECh. 14.3 - Prob. 13ECh. 14.3 - Prob. 14ECh. 14.3 - Prob. 15ECh. 14.3 - Prob. 16ECh. 14.3 - Prob. 17ECh. 14.3 - Prob. 18ECh. 14.3 - Prob. 19ECh. 14.3 - Prob. 20ECh. 14.3 - Prob. 21ECh. 14.3 - Prob. 22ECh. 14.3 - Prob. 23ECh. 14.3 - Prob. 24ECh. 14.3 - Prob. 25ECh. 14.3 - Prob. 26ECh. 14.3 - Prob. 27ECh. 14.3 - Prob. 28ECh. 14.3 - Prob. 29ECh. 14.3 - Prob. 30ECh. 14.3 - Prob. 31ECh. 14.3 - Prob. 32ECh. 14.3 - Prob. 33ECh. 14.3 - Prob. 34ECh. 14.3 - Prob. 35ECh. 14.3 - Prob. 36ECh. 14.3 - Prob. 37ECh. 14.3 - Find the indicated partial derivative. 42....Ch. 14.3 - Find the indicated partial derivative. 43....Ch. 14.3 - Find the indicated partial derivative. 44. f(x, 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 - Use implicit differentiation to find z/x and z/y ....Ch. 14.3 - Find z/x and z/y. 51. (a) z = f(x) + g(y) (b) z =...Ch. 14.3 - Find z/x and z/y. 52. (a) z = f(x)g(y) (b) z =...Ch. 14.3 - Prob. 47ECh. 14.3 - Find all the second partial derivatives. 54. f(x,...Ch. 14.3 - Find all the second partial derivatives. 55....Ch. 14.3 - Prob. 50ECh. 14.3 - Prob. 51ECh. 14.3 - Prob. 52ECh. 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 derivative(s). 63. f(x,...Ch. 14.3 - Find the indicated partial derivative(s). 64. f(x,...Ch. 14.3 - Find the indicated partial derivative(s). 65....Ch. 14.3 - Prob. 60ECh. 14.3 - Find the indicated partial derivative(s). 67....Ch. 14.3 - Prob. 62ECh. 14.3 - Find the indicated partial derivative(s). 69....Ch. 14.3 - Find the indicated partial derivative(s). 70. u =...Ch. 14.3 - Use Definition 4 to find fx(x,y) and fy(x,y) . 65....Ch. 14.3 - Use Definition 4 to find fx(x,y) and fy(x,y) . 66....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 gxyz. [Hint: Which...Ch. 14.3 - Use the table of values of f(x, y) to estimate the...Ch. 14.3 - If u=ea1x1+a2x2++anxn, where a12+a22++an2=1, show...Ch. 14.3 - Show that the function u=u(x,t) is a solution of...Ch. 14.3 - Prob. 78ECh. 14.3 - Verify that the function u=1/x2+y2+z2 is a...Ch. 14.3 - Prob. 80ECh. 14.3 - The Diffusion Equation The diffusion equation...Ch. 14.3 - Prob. 82ECh. 14.3 - Prob. 83ECh. 14.3 - Prob. 84ECh. 14.3 - Show that the Cobb-Douglas production function...Ch. 14.3 - Prob. 86ECh. 14.3 - Prob. 87ECh. 14.3 - Prob. 88ECh. 14.3 - In the project following Section 4.7 we expressed...Ch. 14.3 - Prob. 91ECh. 14.3 - Prob. 92ECh. 14.3 - The ellipsoid 4x2+2y2+z2=16 intersects the plane...Ch. 14.3 - Prob. 95ECh. 14.3 - If a, b, c are the sides of a triangle and A, B, C...Ch. 14.3 - Prob. 97ECh. 14.3 - Prob. 98ECh. 14.3 - If f(x,y)=x(x2+y2)3/2esin(x2y) find fx(1,0)....Ch. 14.3 - If f(x,y)=x3+y33 find fx(0, 0).Ch. 14.3 - Prob. 1DPCh. 14.3 - Prob. 2DPCh. 14.4 - The graph of a function f is shown. Find an...Ch. 14.4 - The graph of a function f is shown. Find an...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 - 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 - 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 - 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). 17. ex...Ch. 14.4 - Verify the linear approximation at (0, 0). 18....Ch. 14.4 - Given that f is a differentiable function with...Ch. 14.4 - Find the linear approximation of the function f(x,...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. 27. m =...Ch. 14.4 - Find the differential of the function. 35....Ch. 14.4 - Find the differential of the function. 25. z = e2x...Ch. 14.4 - Find the differential of the function. 26....Ch. 14.4 - Find the differential of the function. 35....Ch. 14.4 - Find the differential of the function. 36....Ch. 14.4 - Find the differential of the function. 29. R = 2...Ch. 14.4 - Find the differential of the function. 28....Ch. 14.4 - If z = 5x2 + y2 and (x, y) changes from (1, 2) to...Ch. 14.4 - If z = x2 xy + 3y2 and (x, y) changes from (3, 1)...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 - Prob. 44ECh. 14.4 - The radius of a right circular cylinder is...Ch. 14.4 - The wind-chill index is modeled by the function...Ch. 14.4 - Prob. 47ECh. 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 - Prob. 51ECh. 14.4 - Suppose you need to know an equation of the...Ch. 14.4 - Prove that if f is a function of two variables...Ch. 14.4 - (a) The function...Ch. 14.5 - Find dz/dt in two ways: by using the Chain Rule,...Ch. 14.5 - Find dz/dt in two ways: by using the Chain Rule,...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 1. z =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 2....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 3. z =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 4....Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 5. w =...Ch. 14.5 - Use the Chain Rule to find dz/dt or dw/dt. 6. w =...Ch. 14.5 - Find z/s and z/t in two ways: by using the Chain...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 7. z = (x ...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 8. z =...Ch. 14.5 - Use the Chain Rule to find z/s and z/t. 9. z =...Ch. 14.5 - Prob. 14ECh. 14.5 - Prob. 16ECh. 14.5 - Suppose f is a differentiable function of x and y...Ch. 14.5 - Let R(s, t) = G(u(s, t), v(s, t)), where G, u, and...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 - Prob. 21ECh. 14.5 - Prob. 22ECh. 14.5 - Prob. 23ECh. 14.5 - Prob. 24ECh. 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 5 to find dy/dx . 31. ycosx=x2+y2Ch. 14.5 - Use Equation 5 to find dy/dx . 32. cos(xy)=1+sinyCh. 14.5 - Use Equation 5 to find dy/dx . 33. tan1x2y=x+xy2Ch. 14.5 - Use Equation 5 to find dy/dx . 34. eysinx=x+xyCh. 14.5 - Use Equations 6 to find z/x and z/y . 35....Ch. 14.5 - Prob. 36ECh. 14.5 - Prob. 37ECh. 14.5 - Prob. 38ECh. 14.5 - The temperature at a point (x, y) is T(x, y),...Ch. 14.5 - Prob. 40ECh. 14.5 - The speed of sound traveling through ocean water...Ch. 14.5 - Prob. 42ECh. 14.5 - Prob. 43ECh. 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 - Prob. 48ECh. 14.5 - Assume that all the given functions are...Ch. 14.5 - Assume that all the given functions are...Ch. 14.5 - Prob. 51ECh. 14.5 - Prob. 52ECh. 14.5 - Prob. 53ECh. 14.5 - Prob. 54ECh. 14.5 - Prob. 55ECh. 14.5 - Prob. 56ECh. 14.5 - Homogeneous Functions A function f is called...Ch. 14.5 - Prob. 58ECh. 14.5 - Prob. 59ECh. 14.5 - Equation 5 is a formula for the derivative dy/dx...Ch. 14.6 - Level curves for barometric pressure (in...Ch. 14.6 - The contour map shows the average maximum...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 - Find the directional derivative of f at the given...Ch. 14.6 - (a) Find the gradient of f . (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...Ch. 14.6 - (a) Find the gradient of f. (b) Evaluate the...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 - Prob. 16ECh. 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 Du, f(2, 2).Ch. 14.6 - Prob. 21ECh. 14.6 - Find the directional derivative of the function at...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 - 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 - Prob. 38ECh. 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 - Prob. 41ECh. 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...Ch. 14.6 - Prob. 45ECh. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Find equations of (a) the tangent plane and (b)...Ch. 14.6 - Prob. 53ECh. 14.6 - Prob. 54ECh. 14.6 - If f(x, y) = xy, find the gradient vector f(3, 2)...Ch. 14.6 - If g(x, y) = x2 + y2 4x, find the gradient vector...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 - Prob. 59ECh. 14.6 - At what point on the ellipsoid x2 + y2 + 2z2 = 1...Ch. 14.6 - Are there any points on the hyperboloid x2 y2 z2...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 x2 + y2...Ch. 14.6 - Where does the normal line to the paraboloid z =...Ch. 14.6 - Prob. 66ECh. 14.6 - Show that the sum of the x-, y-, and z-intercepts...Ch. 14.6 - Prob. 68ECh. 14.6 - Find parametric equations tor the tangent line to...Ch. 14.6 - (a) The plane y + z = 3 intersects the cylinder x2...Ch. 14.6 - Where does the helix r(t) = cos t, sin t, t...Ch. 14.6 - Prob. 72ECh. 14.6 - Orthogonal Surfaces Two surfaces are called...Ch. 14.6 - Orthogonal Surfaces Two surfaces are called...Ch. 14.6 - Suppose that the directional derivatives of f(x,...Ch. 14.6 - Prob. 76ECh. 14.6 - Prob. 77ECh. 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 y...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 - Prob. 14ECh. 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 + 4y2 4xy + 2 has an...Ch. 14.7 - Show that f(x,y)=x2yex2y2 has maximum 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 - 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,...Ch. 14.7 - Find the point on the plane x 2y + 3z = 6 that is...Ch. 14.7 - Prob. 45ECh. 14.7 - Prob. 46ECh. 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,...Ch. 14.7 - Method of Least Squares Suppose that a scientist...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 - Prob. 4ECh. 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 - Prob. 16ECh. 14.8 - Prob. 17ECh. 14.8 - Prob. 18ECh. 14.8 - Prob. 19ECh. 14.8 - Use Lagrange multipliers to give an alternate...Ch. 14.8 - Find the extreme values of f on the region...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 - Consider the problem of maximizing the function...Ch. 14.8 - Consider the problem of minimizing the function...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 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 4x 3y + 8z = 5 intersects the cone z2 =...Ch. 14.8 - (a) Maximize i=1nxiyi subject to the constraints...Ch. 14 - (a) What is a function of two variables? (b)...Ch. 14 - Prob. 2CCCh. 14 - Prob. 3CCCh. 14 - (a) What does it mean to say that f is continuous...Ch. 14 - Prob. 5CCCh. 14 - Prob. 6CCCh. 14 - Prob. 7CCCh. 14 - Prob. 8CCCh. 14 - Prob. 9CCCh. 14 - If z = f(x, y), what arc the differentials dx, dy,...Ch. 14 - Prob. 11CCCh. 14 - If z is defined implicitly as a function of x and...Ch. 14 - Prob. 13CCCh. 14 - Prob. 14CCCh. 14 - Prob. 15CCCh. 14 - Prob. 16CCCh. 14 - Prob. 17CCCh. 14 - Prob. 18CCCh. 14 - Prob. 19CCCh. 14 - Prob. 1TFQCh. 14 - Prob. 2TFQCh. 14 - Prob. 3TFQCh. 14 - Prob. 4TFQCh. 14 - Prob. 5TFQCh. 14 - Determine whether the statement is true or false....Ch. 14 - Prob. 7TFQCh. 14 - Prob. 8TFQCh. 14 - Determine whether the statement is true or false....Ch. 14 - Prob. 10TFQCh. 14 - Prob. 11TFQCh. 14 - Prob. 12TFQCh. 14 - Prob. 1ECh. 14 - Find and sketch the domain of the function. 2....Ch. 14 - Prob. 3ECh. 14 - Prob. 4ECh. 14 - Sketch several level curves of the function. 5....Ch. 14 - Sketch several level curves of the function. 6....Ch. 14 - Make a rough sketch of a contour map for the...Ch. 14 - The contour map of a function f is shown, (a)...Ch. 14 - Prob. 9ECh. 14 - Prob. 10ECh. 14 - Prob. 11ECh. 14 - Prob. 12ECh. 14 - Prob. 13ECh. 14 - Prob. 14ECh. 14 - Prob. 15ECh. 14 - Prob. 16ECh. 14 - Prob. 17ECh. 14 - The speed of sound traveling through ocean water...Ch. 14 - Prob. 19ECh. 14 - Prob. 20ECh. 14 - Prob. 21ECh. 14 - Prob. 22ECh. 14 - Prob. 23ECh. 14 - Prob. 24ECh. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Prob. 26ECh. 14 - Prob. 27ECh. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Find equations of (a) the tangent plane and (b)...Ch. 14 - Prob. 30ECh. 14 - Find the points on the hyperboloid x2 + 4y2 z2 =...Ch. 14 - Prob. 32ECh. 14 - Find the linear approximation of the function...Ch. 14 - The two legs of a right triangle are measured as 5...Ch. 14 - Prob. 35ECh. 14 - If v = x2sin y + yexy, where x = s + 2t and y =...Ch. 14 - Suppose z = f(x, y), where x = g(s, t), y = h(s,...Ch. 14 - Prob. 38ECh. 14 - Prob. 39ECh. 14 - Prob. 40ECh. 14 - Prob. 41ECh. 14 - If cos(xyz) = 1 + .x2y2 + z2, find zx and zy.Ch. 14 - Find the gradient of the function f(x,y,z)=x2eyz2.Ch. 14 - Prob. 44ECh. 14 - Prob. 45ECh. 14 - Find the directional derivative of f at the given...Ch. 14 - Prob. 47ECh. 14 - Prob. 48ECh. 14 - The contour map shows wind speed in knots during...Ch. 14 - Find parametric equations of the tangent line at...Ch. 14 - Prob. 51ECh. 14 - Prob. 52ECh. 14 - Prob. 53ECh. 14 - Prob. 54ECh. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Find the absolute maximum and minimum values of f...Ch. 14 - Prob. 57ECh. 14 - Use a graphing calculator or computer (or Newtons...Ch. 14 - Prob. 59ECh. 14 - Prob. 60ECh. 14 - Prob. 61ECh. 14 - Use Lagrange multipliers to find the maximum and...Ch. 14 - Prob. 63ECh. 14 - A pentagon is formed by placing an isosceles...Ch. 14 - A rectangle with length L and width W is cut into...Ch. 14 - Marine biologists have determined that when a...Ch. 14 - A long piece of galvanized sheet metal with width...Ch. 14 - For what values of the number r is the function...Ch. 14 - Suppose f is a differentiable function of one...Ch. 14 - If the ellipse x2/a2 + y2/b2 = 1 is to enclose the...Ch. 14 - Prob. 8PP
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