University Calculus: Early Transcendentals (4th Edition)
University Calculus: Early Transcendentals (4th Edition)
4th Edition
ISBN: 9780134995540
Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.
Publisher: PEARSON
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Optimization
Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the se…
Integration
Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integratio…
Application of Integration
In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.
Volume
In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects th…
Area
Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.
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Chapter 13.3, Problem 55E

In Exercises 55−60, verify that w x y = w y x .

55. w = ln ( 2 x + 3 y )

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Chapter 13 Solutions

University Calculus: Early Transcendentals (4th Edition)

Ch. 13.1 - In Exercises 1–4, find the specific function...Ch. 13.1 - In Exercises 1–4, find the specific function...Ch. 13.1 - In Exercises 1–4, find the specific function...Ch. 13.1 - In Exercises 1–4, find the specific function...Ch. 13.1 - In Exercises 5–12, find and sketch the domain for...Ch. 13.1 - In Exercises 5–12, find and sketch the domain for...Ch. 13.1 - In Exercises 512, find and sketch the domain for...Ch. 13.1 - Prob. 8ECh. 13.1 - In Exercises 5–12, find and sketch the domain for...Ch. 13.1 - Prob. 10E
Ch. 13.1 - In Exercises 512, find and sketch the domain for...Ch. 13.1 - Prob. 12ECh. 13.1 - In Exercises 1316, find and sketch the level...Ch. 13.1 - In Exercises 13–16, find and sketch the level...Ch. 13.1 - In Exercises 13–16, find and sketch the level...Ch. 13.1 - Prob. 16ECh. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - In Exercises 17-30, (a) find the function’s...Ch. 13.1 - Prob. 28ECh. 13.1 - Prob. 29ECh. 13.1 - Prob. 30ECh. 13.1 - Exercises 31–36 show level curves for six...Ch. 13.1 - Exercises 31–36 show level curves for six...Ch. 13.1 - Exercises 31–36 show level curves for six...Ch. 13.1 - Exercises 31–36 show level curves for six...Ch. 13.1 - Exercises 31–36 show level curves for six...Ch. 13.1 - Exercises 31–36 show level curves for six...Ch. 13.1 - Display the values of the functions in Exercises...Ch. 13.1 - Display the values of the functions in Exercises...Ch. 13.1 - Display the values of the functions in Exercises...Ch. 13.1 - Display the values of the functions in Exercises...Ch. 13.1 - Prob. 41ECh. 13.1 - Prob. 42ECh. 13.1 - Display the values of the functions in Exercises...Ch. 13.1 - Prob. 44ECh. 13.1 - Display the values of the functions in Exercises...Ch. 13.1 - Prob. 46ECh. 13.1 - Display the values of the functions in Exercises...Ch. 13.1 - Prob. 48ECh. 13.1 - In Exercises 49–52, find an equation for, and...Ch. 13.1 - In Exercises 49–52, find an equation for, and...Ch. 13.1 - In Exercises 49–52, find an equation for, and...Ch. 13.1 - In Exercises 49–52, find an equation for, and...Ch. 13.1 - In Exercises 53–60, sketch a typical level surface...Ch. 13.1 - Prob. 54ECh. 13.1 - In Exercises 53–60, sketch a typical level surface...Ch. 13.1 - Prob. 56ECh. 13.1 - Prob. 57ECh. 13.1 - Prob. 58ECh. 13.1 - In Exercises 53–60, sketch a typical level surface...Ch. 13.1 - In Exercises 53–60, sketch a typical level surface...Ch. 13.1 - In Exercises 61–64, find an equation for the level...Ch. 13.1 - In Exercises 61–64, find an equation for the level...Ch. 13.1 - Prob. 63ECh. 13.1 - Prob. 64ECh. 13.1 - Prob. 65ECh. 13.1 - Prob. 66ECh. 13.1 - Prob. 67ECh. 13.1 - Prob. 68ECh. 13.2 - Find the limits in Exercises 1–12. 1. Ch. 13.2 - Find the limits in Exercises 1–12. 2. Ch. 13.2 - Find the limits in Exercises 1–12. 3. Ch. 13.2 - Find the limits in Exercises 1–12. 4. Ch. 13.2 - Find the limits in Exercises 1–12. 5. Ch. 13.2 - Find the limits in Exercises 1–12. 6. Ch. 13.2 - Find the limits in Exercises 1–12. 7. Ch. 13.2 - Prob. 8ECh. 13.2 - Find the limits in Exercises 1–12. 9. Ch. 13.2 - Prob. 10ECh. 13.2 - Find the limits in Exercises 1–12. 11. Ch. 13.2 - Prob. 12ECh. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Prob. 18ECh. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Prob. 20ECh. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 13.2 - Find the limits in Exercises 25–30. 25. Ch. 13.2 - Prob. 26ECh. 13.2 - Prob. 27ECh. 13.2 - Prob. 28ECh. 13.2 - Prob. 29ECh. 13.2 - Prob. 30ECh. 13.2 - Prob. 31ECh. 13.2 - Prob. 32ECh. 13.2 - Prob. 33ECh. 13.2 - Prob. 34ECh. 13.2 - Prob. 35ECh. 13.2 - Prob. 36ECh. 13.2 - Prob. 37ECh. 13.2 - Prob. 38ECh. 13.2 - At what points (x, y, z) in space are the...Ch. 13.2 - Prob. 40ECh. 13.2 - By considering different paths of approach, show...Ch. 13.2 - By considering different paths of approach, show...Ch. 13.2 - By considering different paths of approach, show...Ch. 13.2 - Prob. 44ECh. 13.2 - By considering different paths of approach, show...Ch. 13.2 - By considering different paths of approach, show...Ch. 13.2 - By considering different paths of approach, show...Ch. 13.2 - Prob. 48ECh. 13.2 - In Exercises 49–54, show that the limits do not...Ch. 13.2 - In Exercises 49–54, show that the limits do not...Ch. 13.2 - Prob. 51ECh. 13.2 - Prob. 52ECh. 13.2 - In Exercises 49–54, show that the limits do not...Ch. 13.2 - Prob. 54ECh. 13.2 - Let Find each of the following limits, or explain...Ch. 13.2 - Let . Find the following limits. Ch. 13.2 - Show that the function in Example 6 has limit 0...Ch. 13.2 - Prob. 58ECh. 13.2 - The Sandwich Theorem for functions of two...Ch. 13.2 - The Sandwich Theorem for functions of two...Ch. 13.2 - The Sandwich Theorem for functions of two...Ch. 13.2 - The Sandwich Theorem for functions of two...Ch. 13.2 - Prob. 63ECh. 13.2 - Prob. 64ECh. 13.2 - In Exercises 65–70, find the limit of f as (x, y)...Ch. 13.2 - In Exercises 65–70, find the limit of f as (x, y)...Ch. 13.2 - In Exercises 65–70, find the limit of f as (x, y)...Ch. 13.2 - Prob. 68ECh. 13.2 - Prob. 69ECh. 13.2 - In Exercises 65–70, find the limit of f as (x, y)...Ch. 13.2 - In Exercises 71 and 72, define f(0, 0) in a way...Ch. 13.2 - In Exercises 71 and 72, define f(0, 0) in a way...Ch. 13.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 13.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 13.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 13.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 13.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 13.2 - Prob. 78ECh. 13.2 - Each of Exercises 79–82 gives a function f(x, y,...Ch. 13.2 - Prob. 80ECh. 13.2 - Each of Exercises 79–82 gives a function f(x, y,...Ch. 13.2 - Prob. 82ECh. 13.2 - Prob. 83ECh. 13.2 - Prob. 84ECh. 13.3 - In Exercises 1–22, find and . 1. Ch. 13.3 - In Exercises 1–22, find and . 2. Ch. 13.3 - In Exercises 1–22, find and . 3. Ch. 13.3 - In Exercises 1–22, find and . 4. Ch. 13.3 - In Exercises 1–22, find and . 5. Ch. 13.3 - In Exercises 1–22, find and . 6. Ch. 13.3 - In Exercises 1–22, find and . 7. Ch. 13.3 - In Exercises 1–22, find and . 8. Ch. 13.3 - In Exercises 1–22, find and . 9. Ch. 13.3 - In Exercises 1–22, find and . 10. Ch. 13.3 - In Exercises 1–22, find and . 11. Ch. 13.3 - In Exercises 1–22, find and . 12. Ch. 13.3 - In Exercises 1–22, find and . 13. Ch. 13.3 - In Exercises 1–22, find and . 14. Ch. 13.3 - In Exercises 122, find f/x and f/y . 15....Ch. 13.3 - In Exercises 1–22, find and . 16. Ch. 13.3 - In Exercises 1–22, find and . 17. Ch. 13.3 - Prob. 18ECh. 13.3 - In Exercises 1–22, find and . 19. Ch. 13.3 - Prob. 20ECh. 13.3 - In Exercises 1–22, find and . 21. Ch. 13.3 - In Exercises 1–22, find and . 22. Ch. 13.3 - In Exercises 23–34, find fx, fy, and fz. 23. f(x,...Ch. 13.3 - Prob. 24ECh. 13.3 - In Exercises 23–34, find fx, fy, and fz. 25. Ch. 13.3 - In Exercises 23–34, find fx, fy, and fz. 26. f(x,...Ch. 13.3 - In Exercises 23–34, find fx, fy, and fz. 27. f(x,...Ch. 13.3 - Prob. 28ECh. 13.3 - In Exercises 23–34, find fx, fy, and fz. 29. f(x,...Ch. 13.3 - In Exercises 23–34, find fx, fy, and fz. 30. f(x,...Ch. 13.3 - In Exercises 23–34, find fx, fy, and fz. 31. Ch. 13.3 - In Exercises 23–34, find fx, fy, and fz. 32. f(x,...Ch. 13.3 - In Exercises 23–34, find fx, fy, and fz. 33. f(x,...Ch. 13.3 - Prob. 34ECh. 13.3 - In Exercises 35–40, find the partial derivative of...Ch. 13.3 - In Exercises 35–40, find the partial derivative of...Ch. 13.3 - Prob. 37ECh. 13.3 - Prob. 38ECh. 13.3 - Prob. 39ECh. 13.3 - Prob. 40ECh. 13.3 - Prob. 41ECh. 13.3 - Prob. 42ECh. 13.3 - Find all the second-order partial derivatives of...Ch. 13.3 - Prob. 44ECh. 13.3 - Find all the second-order partial derivatives of...Ch. 13.3 - Prob. 46ECh. 13.3 - Prob. 47ECh. 13.3 - Find all the second-order partial derivatives of...Ch. 13.3 - Find all the second-order partial derivatives of...Ch. 13.3 - Prob. 50ECh. 13.3 - Find all the second-order partial derivatives of...Ch. 13.3 - Prob. 52ECh. 13.3 - Prob. 53ECh. 13.3 - Find all the second-order partial derivatives of...Ch. 13.3 - In Exercises 5560, verify that wxy=wyx . 55....Ch. 13.3 - Prob. 56ECh. 13.3 - Prob. 57ECh. 13.3 - In Exercises 55–60, verify that . 58. Ch. 13.3 - In Exercises 55–60, verify that . 59. Ch. 13.3 - Prob. 60ECh. 13.3 - Which order of differentiation enables one to...Ch. 13.3 - Prob. 62ECh. 13.3 - Prob. 63ECh. 13.3 - Prob. 64ECh. 13.3 - Prob. 65ECh. 13.3 - Prob. 66ECh. 13.3 - Prob. 67ECh. 13.3 - Prob. 68ECh. 13.3 - Prob. 69ECh. 13.3 - Prob. 70ECh. 13.3 - Exercises 71 and 72 are about the triangle shown...Ch. 13.3 - Prob. 72ECh. 13.3 - Two dependent variables Express vx in terms of u...Ch. 13.3 - Prob. 74ECh. 13.3 - Let f(x, y) = 2x + 3y = 4. Find the slope of the...Ch. 13.3 - Prob. 76ECh. 13.3 - In Exercises 77-80, find a function z = f(x, y)...Ch. 13.3 - In Exercises 77-80, find a function z = f(x, y)...Ch. 13.3 - In Exercises 77-80, find a function z = f(x, y)...Ch. 13.3 - In Exercises 77-80, find a function z = f(x, y)...Ch. 13.3 - Let Find fx, fy, fxy, and fyx, state the domain...Ch. 13.3 - Let Show that for all x, and for all y. Show...Ch. 13.3 - Show that each function in Exercises 83-90...Ch. 13.3 - Show that each function in Exercises 83-90...Ch. 13.3 - Show that each function in Exercises 83-90...Ch. 13.3 - Prob. 86ECh. 13.3 - Prob. 87ECh. 13.3 - Prob. 88ECh. 13.3 - Prob. 89ECh. 13.3 - Prob. 90ECh. 13.3 - Show that the functions in Exercises 91-97 are all...Ch. 13.3 - Prob. 92ECh. 13.3 - Show that the functions in Exercises 91-97 are all...Ch. 13.3 - Prob. 94ECh. 13.3 - Prob. 95ECh. 13.3 - Prob. 96ECh. 13.3 - Prob. 97ECh. 13.3 - Prob. 98ECh. 13.3 - Prob. 99ECh. 13.3 - Prob. 100ECh. 13.3 - Prob. 101ECh. 13.3 - Show that fx(0, 0) and fy(0, 0) exist, but f is...Ch. 13.3 - Prob. 103ECh. 13.3 - Prob. 104ECh. 13.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 13.4 - In Exercises 16, (a) express dw/dt as a function...Ch. 13.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 13.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 13.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 13.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 13.4 - In Exercises 7 and 8, (a) express and as...Ch. 13.4 - In Exercises 7 and 8, (a) express and as...Ch. 13.4 - In Exercises 9 and 10, (a) express and as...Ch. 13.4 - In Exercises 9 and 10, (a) express and as...Ch. 13.4 - In Exercises 11 and 12, (a) express and as...Ch. 13.4 - In Exercises 11 and 12, (a) express ∂u/∂x, ∂u/∂y,...Ch. 13.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 13.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 13.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 13.4 - Prob. 16ECh. 13.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 13.4 - Prob. 18ECh. 13.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 13.4 - Prob. 20ECh. 13.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 13.4 - Prob. 22ECh. 13.4 - Prob. 23ECh. 13.4 - Prob. 24ECh. 13.4 - Assuming that the equations in Exercises 25–30...Ch. 13.4 - Prob. 26ECh. 13.4 - Assuming that the equations in Exercises 25–30...Ch. 13.4 - Assuming that the equations in Exercises 25–30...Ch. 13.4 - Prob. 29ECh. 13.4 - Assuming that the equations in Exercises 25–30...Ch. 13.4 - Find the values of ∂z/∂x and ∂z/∂y at the points...Ch. 13.4 - Prob. 32ECh. 13.4 - Prob. 33ECh. 13.4 - Prob. 34ECh. 13.4 - Prob. 35ECh. 13.4 - Prob. 36ECh. 13.4 - Prob. 37ECh. 13.4 - Prob. 38ECh. 13.4 - Prob. 39ECh. 13.4 - Prob. 40ECh. 13.4 - Assume that w = f(s3 + t2) and f′(x) = ex. Find ...Ch. 13.4 - Assume that , , and . Find and . Ch. 13.4 - Assume that z = f(x, y), x = g(t), y = h(t), fx(2,...Ch. 13.4 - Prob. 44ECh. 13.4 - Assume that z = f(w), w = g(x, y), x = 2r3 − s2,...Ch. 13.4 - Prob. 46ECh. 13.4 - Changing voltage in a circuit The voltage V in a...Ch. 13.4 - Changing dimensions in a box The lengths a, b, and...Ch. 13.4 - Prob. 49ECh. 13.4 - Polar coordinates Suppose that we substitute polar...Ch. 13.4 - Laplace equations Show that if satisfies the...Ch. 13.4 - Prob. 52ECh. 13.4 - Extreme values on a helix Suppose that the partial...Ch. 13.4 - A space curve Let w = x2e2y cos 3z. Find the value...Ch. 13.4 - Prob. 55ECh. 13.4 - Temperature on an ellipse Let T = g(x, y) be the...Ch. 13.4 - Prob. 57ECh. 13.4 - Prob. 58ECh. 13.4 - Find the derivatives of the functions in Exercises...Ch. 13.4 - Find the derivatives of the functions in Exercises...Ch. 13.5 - In Exercises 1–6, find the gradient of the...Ch. 13.5 - In Exercises 1–6, find the gradient of the...Ch. 13.5 - In Exercises 1–6, find the gradient of the...Ch. 13.5 - In Exercises 1–6, find the gradient of the...Ch. 13.5 - In Exercises 1–6, find the gradient of the...Ch. 13.5 - Prob. 6ECh. 13.5 - In Exercises 7–10, find f at the given point. 7. Ch. 13.5 - Prob. 8ECh. 13.5 - In Exercises 7–10, find f at the given point. 9. Ch. 13.5 - In Exercises 7–10, find f at the given point. 10....Ch. 13.5 - In Exercises 11–18, find the derivative of the...Ch. 13.5 - Prob. 12ECh. 13.5 - In Exercises 11–18, find the derivative of the...Ch. 13.5 - In Exercises 11–18, find the derivative of the...Ch. 13.5 - In Exercises 11–18, find the derivative of the...Ch. 13.5 - In Exercises 11–18, find the derivative of the...Ch. 13.5 - In Exercises 11–18, find the derivative of the...Ch. 13.5 - Prob. 18ECh. 13.5 - In Exercises 19–24, find the directions in which...Ch. 13.5 - Prob. 20ECh. 13.5 - In Exercises 19–24, find the directions in which...Ch. 13.5 - In Exercises 19–24, find the directions in which...Ch. 13.5 - In Exercises 19–24, find the directions in which...Ch. 13.5 - Prob. 24ECh. 13.5 - In Exercises 25–28, sketch the curve f(x, y) = c,...Ch. 13.5 - Prob. 26ECh. 13.5 - In Exercises 25–28, sketch the curve f(x, y) = c,...Ch. 13.5 - Prob. 28ECh. 13.5 - Let f(x, y) = x2 − xy + y2 − y. Find the...Ch. 13.5 - Prob. 30ECh. 13.5 - Zero directional derivative In what direction is...Ch. 13.5 - Zero directional derivative In what directions is...Ch. 13.5 - Is there a direction u in which the rate of change...Ch. 13.5 - Changing temperature along a circle Is there a...Ch. 13.5 - Prob. 35ECh. 13.5 - Prob. 36ECh. 13.5 - Directional derivatives and scalar components How...Ch. 13.5 - Directional derivatives and partial derivatives...Ch. 13.5 - Lines in the xy-plane Show that A(x – x0) + B(y –...Ch. 13.5 - The algebra rules for gradients Given a constant k...Ch. 13.5 - In Exercises 41–44, find a parametric equation for...Ch. 13.5 - Prob. 42ECh. 13.5 - Prob. 43ECh. 13.5 - Prob. 44ECh. 13.6 - In Exercises 1–10, find equations for the tangent...Ch. 13.6 - Prob. 2ECh. 13.6 - In Exercises 1–10, find equations for the tangent...Ch. 13.6 - In Exercises 1–10, find equations for the tangent...Ch. 13.6 - In Exercises 1–10, find equations for the tangent...Ch. 13.6 - In Exercises 1–10, find equations for the tangent...Ch. 13.6 - In Exercises 1–10, find equations for the tangent...Ch. 13.6 - Prob. 8ECh. 13.6 - In Exercises 1–10, find equations for the tangent...Ch. 13.6 - Prob. 10ECh. 13.6 - In Exercises 11–14, find an equation for the plane...Ch. 13.6 - Prob. 12ECh. 13.6 - In Exercises 11–14, find an equation for the plane...Ch. 13.6 - In Exercises 11–14, find an equation for the plane...Ch. 13.6 - In Exercises 15–20, find parametric equations for...Ch. 13.6 - In Exercises 15–20, find parametric equations for...Ch. 13.6 - Prob. 17ECh. 13.6 - Prob. 18ECh. 13.6 - Prob. 19ECh. 13.6 - Prob. 20ECh. 13.6 - Prob. 21ECh. 13.6 - Prob. 22ECh. 13.6 - By about how much will change if the point P(x,...Ch. 13.6 - Prob. 24ECh. 13.6 - Prob. 25ECh. 13.6 - Changing temperature along a space curve The...Ch. 13.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 13.6 - Prob. 28ECh. 13.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 13.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 13.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 13.6 - Prob. 32ECh. 13.6 - Wind chill factor Wind chill, a measure of the...Ch. 13.6 - Prob. 34ECh. 13.6 - Prob. 35ECh. 13.6 - Prob. 36ECh. 13.6 - Prob. 37ECh. 13.6 - Prob. 38ECh. 13.6 - Prob. 39ECh. 13.6 - Prob. 40ECh. 13.6 - Prob. 41ECh. 13.6 - Prob. 42ECh. 13.6 - Prob. 43ECh. 13.6 - Find the linearizations L(x, y, z) of the...Ch. 13.6 - Prob. 45ECh. 13.6 - Prob. 46ECh. 13.6 - Prob. 47ECh. 13.6 - Prob. 48ECh. 13.6 - Prob. 49ECh. 13.6 - Prob. 50ECh. 13.6 - Estimating maximum error Suppose that T is to be...Ch. 13.6 - Variation in electrical resistance The resistance...Ch. 13.6 - Prob. 53ECh. 13.6 - Prob. 54ECh. 13.6 - Value of a 2 × 2 determinant If |a| is much...Ch. 13.6 - The Wilson lot size formula The Wilson lot size...Ch. 13.6 - The linearization of f(x, y) is a tangent-plane...Ch. 13.6 - Prob. 58ECh. 13.6 - Tangent curves A smooth curve is tangent to the...Ch. 13.6 - Normal curves A smooth curve is normal to a...Ch. 13.6 - Consider a closed rectangular box with a square...Ch. 13.7 - Prob. 1ECh. 13.7 - Prob. 2ECh. 13.7 - Prob. 3ECh. 13.7 - Prob. 4ECh. 13.7 - Prob. 5ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 8ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 10ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 12ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 14ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 16ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 18ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 20ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 22ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 24ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 26ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 28ECh. 13.7 - Find all the local maxima, local minima, and...Ch. 13.7 - Prob. 30ECh. 13.7 - Prob. 31ECh. 13.7 - Prob. 32ECh. 13.7 - Prob. 33ECh. 13.7 - Prob. 34ECh. 13.7 - Prob. 35ECh. 13.7 - Prob. 36ECh. 13.7 - In Exercises 31–38, find the absolute maxima and...Ch. 13.7 - Prob. 38ECh. 13.7 - Find two numbers a and b with such that has its...Ch. 13.7 - Find two numbers a and b with such that has its...Ch. 13.7 - Temperatures A flat circular plate has the shape...Ch. 13.7 - Find the critical point of in the open first...Ch. 13.7 - Find the maxima, minima, and saddle points of f(x,...Ch. 13.7 - The discriminant fxxfyy − fxv2 is zero at the...Ch. 13.7 - Show that (0, 0) is a critical point of f(x, y) =...Ch. 13.7 - For what values of the constant k does the Second...Ch. 13.7 - If fx(a, b) = fy(a, b) = 0, must f have a local...Ch. 13.7 - Can you conclude anything about f(a, b) if f and...Ch. 13.7 - Among all the points on the graph of that lie...Ch. 13.7 - Prob. 50ECh. 13.7 - Find the point on the plane 3x + 2y + z = 6 that...Ch. 13.7 - Prob. 52ECh. 13.7 - Find three numbers whose sum is 9 and whose sum of...Ch. 13.7 - Prob. 54ECh. 13.7 - Find the maximum value of where . Ch. 13.7 - Prob. 56ECh. 13.7 - Find the dimensions of the rectangular box of...Ch. 13.7 - Prob. 58ECh. 13.7 - You are to construct an open rectangular box from...Ch. 13.7 - Prob. 60ECh. 13.7 - Find the point on the graph of nearest the...Ch. 13.7 - Prob. 62ECh. 13.7 - Extreme Values on Parametrized Curves To find the...Ch. 13.7 - Prob. 64ECh. 13.7 - Extreme Values on Parametrized Curves To find the...Ch. 13.7 - Prob. 66ECh. 13.7 - Least squares and regression lines When we try to...Ch. 13.7 - Prob. 68ECh. 13.7 - In Exercises 68–70, use Equations (2) and (3) to...Ch. 13.7 - Prob. 70ECh. 13.8 - Extrema on an ellipse Find the points on the...Ch. 13.8 - Extrema on a circle Find the extreme values of...Ch. 13.8 - Maximum on a line Find the maximum value of f(x,...Ch. 13.8 - Extrema on a line Find the local extreme values of...Ch. 13.8 - Constrained minimum Find the points on the curve...Ch. 13.8 - Prob. 6ECh. 13.8 - Use the method of Lagrange multipliers to...Ch. 13.8 - Prob. 8ECh. 13.8 - Minimum surface area with fixed volume Find the...Ch. 13.8 - Prob. 10ECh. 13.8 - Rectangle of greatest area in an ellipse Use the...Ch. 13.8 - Prob. 12ECh. 13.8 - Extrema on a circle Find the maximum and minimum...Ch. 13.8 - Prob. 14ECh. 13.8 - Ant on a metal plate The temperature at a point...Ch. 13.8 - Prob. 16ECh. 13.8 - Minimum distance to a point Find the point on the...Ch. 13.8 - Prob. 18ECh. 13.8 - Minimum distance to the origin Find the minimum...Ch. 13.8 - Prob. 20ECh. 13.8 - Minimum distance to the origin Find the points on...Ch. 13.8 - Prob. 22ECh. 13.8 - Extrema on a sphere Find the maximum and minimum...Ch. 13.8 - Prob. 24ECh. 13.8 - Minimizing a sum of squares Find three real...Ch. 13.8 - Prob. 26ECh. 13.8 - Rectangular box of largest volume in a sphere Find...Ch. 13.8 - Prob. 28ECh. 13.8 - Hottest point on a space probe A space probe in...Ch. 13.8 - Extreme temperatures on a sphere Suppose that the...Ch. 13.8 - Cobb-Douglas production function During the 1920s,...Ch. 13.8 - (Continuation of Exercise 31.) If the cost of a...Ch. 13.8 - Maximizing a utility function: an example from...Ch. 13.8 - Prob. 34ECh. 13.8 - Length of a beam In Section 4.6, Exercise 45, we...Ch. 13.8 - Prob. 36ECh. 13.8 - Maximize the function subject to the constraints...Ch. 13.8 - Prob. 38ECh. 13.8 - Minimum distance to the origin Find the point...Ch. 13.8 - Prob. 40ECh. 13.8 - Extrema on a curve of intersection Find the...Ch. 13.8 - Maximum on line of intersection Find the maximum...Ch. 13.8 - Extrema on a circle of intersection Find the...Ch. 13.8 - Prob. 44ECh. 13.8 - Prob. 45ECh. 13.8 - Prob. 46ECh. 13.8 - Prob. 47ECh. 13.8 - Sum of products Let a1, a2,..., an be n positive...Ch. 13 - Prob. 1GYRCh. 13 - Prob. 2GYRCh. 13 - Prob. 3GYRCh. 13 - Prob. 4GYRCh. 13 - Prob. 5GYRCh. 13 - Prob. 6GYRCh. 13 - Prob. 7GYRCh. 13 - Prob. 8GYRCh. 13 - Prob. 9GYRCh. 13 - Prob. 10GYRCh. 13 - Prob. 11GYRCh. 13 - Prob. 12GYRCh. 13 - What is the general Chain Rule? What form does it...Ch. 13 - Prob. 14GYRCh. 13 - Prob. 15GYRCh. 13 - Prob. 16GYRCh. 13 - Prob. 17GYRCh. 13 - Prob. 18GYRCh. 13 - Prob. 19GYRCh. 13 - Prob. 20GYRCh. 13 - Prob. 21GYRCh. 13 - Prob. 22GYRCh. 13 - Prob. 23GYRCh. 13 - Prob. 24GYRCh. 13 - Prob. 1PECh. 13 - Prob. 2PECh. 13 - Prob. 3PECh. 13 - Prob. 4PECh. 13 - Prob. 5PECh. 13 - Prob. 6PECh. 13 - Prob. 7PECh. 13 - Prob. 8PECh. 13 - Prob. 9PECh. 13 - Prob. 10PECh. 13 - Prob. 11PECh. 13 - Prob. 12PECh. 13 - Prob. 13PECh. 13 - Prob. 14PECh. 13 - Prob. 15PECh. 13 - Prob. 16PECh. 13 - Prob. 17PECh. 13 - Prob. 18PECh. 13 - Prob. 19PECh. 13 - Prob. 20PECh. 13 - Prob. 21PECh. 13 - Prob. 22PECh. 13 - Prob. 23PECh. 13 - Prob. 24PECh. 13 - Prob. 25PECh. 13 - Prob. 26PECh. 13 - Prob. 27PECh. 13 - Prob. 28PECh. 13 - Prob. 29PECh. 13 - Prob. 30PECh. 13 - Prob. 31PECh. 13 - Prob. 32PECh. 13 - Prob. 33PECh. 13 - Prob. 34PECh. 13 - Assuming that the equations in Exercises 35 and 36...Ch. 13 - Prob. 36PECh. 13 - Prob. 37PECh. 13 - Prob. 38PECh. 13 - Prob. 39PECh. 13 - Prob. 40PECh. 13 - Prob. 41PECh. 13 - Prob. 42PECh. 13 - Prob. 43PECh. 13 - Prob. 44PECh. 13 - Prob. 45PECh. 13 - Prob. 46PECh. 13 - Prob. 47PECh. 13 - Prob. 48PECh. 13 - Prob. 49PECh. 13 - Prob. 50PECh. 13 - Prob. 51PECh. 13 - Prob. 52PECh. 13 - Prob. 53PECh. 13 - Prob. 54PECh. 13 - Prob. 55PECh. 13 - Prob. 56PECh. 13 - Prob. 57PECh. 13 - Prob. 58PECh. 13 - Prob. 59PECh. 13 - Prob. 60PECh. 13 - Prob. 61PECh. 13 - Prob. 62PECh. 13 - Prob. 63PECh. 13 - Prob. 64PECh. 13 - Prob. 65PECh. 13 - Prob. 66PECh. 13 - Prob. 67PECh. 13 - Prob. 68PECh. 13 - Prob. 69PECh. 13 - Prob. 70PECh. 13 - Prob. 71PECh. 13 - Prob. 72PECh. 13 - Prob. 73PECh. 13 - Prob. 74PECh. 13 - Prob. 75PECh. 13 - Prob. 76PECh. 13 - Prob. 77PECh. 13 - Prob. 78PECh. 13 - Prob. 79PECh. 13 - Prob. 80PECh. 13 - Prob. 81PECh. 13 - Prob. 82PECh. 13 - Prob. 83PECh. 13 - Prob. 84PECh. 13 - Prob. 85PECh. 13 - Prob. 86PECh. 13 - Prob. 87PECh. 13 - Prob. 88PECh. 13 - Prob. 89PECh. 13 - Prob. 90PECh. 13 - Prob. 91PECh. 13 - Prob. 92PECh. 13 - Prob. 93PECh. 13 - Prob. 94PECh. 13 - Prob. 95PECh. 13 - Prob. 96PECh. 13 - Prob. 97PECh. 13 - Prob. 98PECh. 13 - Prob. 99PECh. 13 - Prob. 100PECh. 13 - Prob. 1AAECh. 13 - Prob. 2AAECh. 13 - Prob. 3AAECh. 13 - Prob. 4AAECh. 13 - Prob. 5AAECh. 13 - Prob. 6AAECh. 13 - Prob. 7AAECh. 13 - Prob. 8AAECh. 13 - Curve tangent to a surface Show that the curve is...Ch. 13 - Prob. 10AAECh. 13 - Prob. 11AAECh. 13 - Prob. 12AAECh. 13 - Prob. 13AAECh. 13 - Prob. 14AAECh. 13 - Prob. 15AAECh. 13 - Prob. 16AAECh. 13 - Prob. 17AAECh. 13 - Prob. 18AAECh. 13 - Prob. 19AAECh. 13 - Prob. 20AAECh. 13 - Prob. 21AAECh. 13 - Prob. 22AAECh. 13 - Prob. 23AAECh. 13 - Prob. 24AAE

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