Thomas' Calculus (14th Edition)
14th Edition
ISBN: 9780134438986
Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher: PEARSON
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Chapter 14.3, Problem 58E
To determine
Prove that
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Chapter 14 Solutions
Thomas' Calculus (14th Edition)
Ch. 14.1 - In Exercises 1–4, find the specific function...Ch. 14.1 - In Exercises 1–4, find the specific function...Ch. 14.1 - In Exercises 1–4, find the specific function...Ch. 14.1 - In Exercises 1–4, find the specific function...Ch. 14.1 - In Exercises 5–12, find and sketch the domain for...Ch. 14.1 - In Exercises 5–12, find and sketch the domain for...Ch. 14.1 - In Exercises 5–12, find and sketch the domain for...Ch. 14.1 - In Exercises 5–12, find and sketch the domain for...Ch. 14.1 - In Exercises 5–12, find and sketch the domain for...Ch. 14.1 - In Exercises 5–12, find and sketch the domain for...
Ch. 14.1 - In Exercises 5–12, find and sketch the domain for...Ch. 14.1 - In Exercises 5–12, find and sketch the domain for...Ch. 14.1 - In Exercises 13–16, find and sketch the level...Ch. 14.1 - In Exercises 13–16, find and sketch the level...Ch. 14.1 - In Exercises 13–16, find and sketch the level...Ch. 14.1 - In Exercises 13–16, find and sketch the level...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17-30, (a) find the function’s...Ch. 14.1 - In Exercises 17–30, (a) find the function’s...Ch. 14.1 - In Exercises 17–30, (a) find the function’s...Ch. 14.1 - Exercises 31–36 show level curves for six...Ch. 14.1 - Exercises 31–36 show level curves for six...Ch. 14.1 - Exercises 31–36 show level curves for six...Ch. 14.1 - Exercises 31–36 show level curves for six...Ch. 14.1 - Exercises 31–36 show level curves for six...Ch. 14.1 - Exercises 31–36 show level curves for six...Ch. 14.1 - Display the values of the functions in Exercises...Ch. 14.1 - Display the values of the functions in Exercises...Ch. 14.1 - Display the values of the functions in Exercises...Ch. 14.1 - Display the values of the functions in Exercises...Ch. 14.1 - Display the values of the functions in Exercises...Ch. 14.1 - Prob. 42ECh. 14.1 - Display the values of the functions in Exercises...Ch. 14.1 - Prob. 44ECh. 14.1 - Display the values of the functions in Exercises...Ch. 14.1 - Display the values of the functions in Exercises...Ch. 14.1 - Prob. 47ECh. 14.1 - Prob. 48ECh. 14.1 - In Exercises 49–52, find an equation for, and...Ch. 14.1 - In Exercises 49–52, find an equation for, and...Ch. 14.1 - In Exercises 49–52, find an equation for, and...Ch. 14.1 - In Exercises 49–52, find an equation for, and...Ch. 14.1 - In Exercises 53–60, sketch a typical level surface...Ch. 14.1 - In Exercises 53–60, sketch a typical level surface...Ch. 14.1 - In Exercises 53–60, sketch a typical level surface...Ch. 14.1 - In Exercises 53–60, sketch a typical level surface...Ch. 14.1 - In Exercises 53–60, sketch a typical level surface...Ch. 14.1 - In Exercises 53–60, sketch a typical level surface...Ch. 14.1 - In Exercises 53–60, sketch a typical level surface...Ch. 14.1 - In Exercises 53–60, sketch a typical level surface...Ch. 14.1 - In Exercises 61–64, find an equation for the level...Ch. 14.1 - In Exercises 61–64, find an equation for the level...Ch. 14.1 - In Exercises 61–64, find an equation for the level...Ch. 14.1 - In Exercises 61–64, find an equation for the level...Ch. 14.1 - In Exercises 65–68, find and sketch the domain of...Ch. 14.1 - In Exercises 65–68, find and sketch the domain of...Ch. 14.1 - Prob. 67ECh. 14.1 - In Exercises 65–68, find and sketch the domain of...Ch. 14.2 - Find the limits in Exercises 1–12.
1.
Ch. 14.2 - Find the limits in Exercises 1–12.
2.
Ch. 14.2 - Find the limits in Exercises 1–12.
3.
Ch. 14.2 - Find the limits in Exercises 1–12.
4.
Ch. 14.2 - Find the limits in Exercises 1–12.
5.
Ch. 14.2 - Find the limits in Exercises 1–12.
6.
Ch. 14.2 - Find the limits in Exercises 1–12.
7.
Ch. 14.2 - Find the limits in Exercises 1–12.
8.
Ch. 14.2 - Find the limits in Exercises 1–12.
9.
Ch. 14.2 - Find the limits in Exercises 1–12.
10.
Ch. 14.2 - Find the limits in Exercises 1–12.
11.
Ch. 14.2 - Find the limits in Exercises 1–12.
12.
Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 13–24 by rewriting...Ch. 14.2 - Find the limits in Exercises 25–30.
25.
Ch. 14.2 - Find the limits in Exercises 25–30.
26.
Ch. 14.2 - Find the limits in Exercises 25–30.
27.
Ch. 14.2 - Find the limits in Exercises 25–30.
28.
Ch. 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 - Prob. 38ECh. 14.2 - At what points (x, y, z) in space are the...Ch. 14.2 - Prob. 40ECh. 14.2 - By considering different paths of approach, show...Ch. 14.2 - By considering different paths of approach, show...Ch. 14.2 - By considering different paths of approach, show...Ch. 14.2 - By considering different paths of approach, show...Ch. 14.2 - By considering different paths of approach, show...Ch. 14.2 - By considering different paths of approach, show...Ch. 14.2 - By considering different paths of approach, show...Ch. 14.2 - Prob. 48ECh. 14.2 - In Exercises 49–54, show that the limits do not...Ch. 14.2 - In Exercises 49–54, show that the limits do not...Ch. 14.2 - In Exercises 49–54, show that the limits do not...Ch. 14.2 - In Exercises 49–54, show that the limits do not...Ch. 14.2 - In Exercises 49–54, show that the limits do not...Ch. 14.2 - In Exercises 49–54, show that the limits do not...Ch. 14.2 - Let
Find each of the following limits, or explain...Ch. 14.2 - Let .
Find the following limits.
Ch. 14.2 - Show that the function in Example 6 has limit 0...Ch. 14.2 - If f(x0, y0) = 3, what can you say about
if f is...Ch. 14.2 - The Sandwich Theorem for functions of two...Ch. 14.2 - The Sandwich Theorem for functions of two...Ch. 14.2 - The Sandwich Theorem for functions of two...Ch. 14.2 - Prob. 62ECh. 14.2 - Prob. 63ECh. 14.2 - Continuous extension Define f(0, 0) in a way that...Ch. 14.2 - In Exercises 65–70, find the limit of f as (x, y)...Ch. 14.2 - In Exercises 65–70, find the limit of f as (x, y)...Ch. 14.2 - In Exercises 65–70, find the limit of f as (x, y)...Ch. 14.2 - In Exercises 65–70, find the limit of f as (x, y)...Ch. 14.2 - Prob. 69ECh. 14.2 - Prob. 70ECh. 14.2 - In Exercises 71 and 72, define f(0, 0) in a way...Ch. 14.2 - In Exercises 71 and 72, define f(0, 0) in a way...Ch. 14.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 14.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 14.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 14.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 14.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 14.2 - Each of Exercises 73–78 gives a function f(x, y)...Ch. 14.2 - Each of Exercises 79–82 gives a function f(x, y,...Ch. 14.2 - Prob. 80ECh. 14.2 - Each of Exercises 79–82 gives a function f(x, y,...Ch. 14.2 - Each of Exercises 79–82 gives a function f(x, y,...Ch. 14.2 - Show that is continuous at every point .
Ch. 14.2 - Show that is continuous at the origin.
Ch. 14.3 - In Exercises 1–22, find and .
1.
Ch. 14.3 - In Exercises 1–22, find and .
2.
Ch. 14.3 - In Exercises 1–22, find and .
3.
Ch. 14.3 - In Exercises 1–22, find and .
4.
Ch. 14.3 - In Exercises 1–22, find and .
5.
Ch. 14.3 - In Exercises 1–22, find and .
6.
Ch. 14.3 - In Exercises 1–22, find and .
7.
Ch. 14.3 - In Exercises 1–22, find and .
8.
Ch. 14.3 - In Exercises 1–22, find and .
9.
Ch. 14.3 - In Exercises 1–22, find and .
10.
Ch. 14.3 - Prob. 11ECh. 14.3 - In Exercises 1–22, find and .
12.
Ch. 14.3 - In Exercises 1–22, find and .
13.
Ch. 14.3 - In Exercises 1–22, find and .
14.
Ch. 14.3 - In Exercises 1–22, find and .
15.
Ch. 14.3 - In Exercises 1–22, find and .
16.
Ch. 14.3 - In Exercises 1–22, find and .
17.
Ch. 14.3 - In Exercises 1–22, find and .
18.
Ch. 14.3 - In Exercises 1–22, find and .
19.
Ch. 14.3 - In Exercises 1–22, find and .
20.
Ch. 14.3 - In Exercises 1–22, find and .
21.
Ch. 14.3 - In Exercises 1–22, find and .
22.
Ch. 14.3 - Prob. 23ECh. 14.3 - Prob. 24ECh. 14.3 - In Exercises 23–34, find fx, fy, and fz.
25.
Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
26. f(x,...Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
27. f(x,...Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
28. f(x,...Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
29. f(x,...Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
30. f(x,...Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
31.
Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
32. f(x,...Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
33. f(x,...Ch. 14.3 - In Exercises 23–34, find fx, fy, and fz.
34. f(x,...Ch. 14.3 - In Exercises 35–40, find the partial derivative of...Ch. 14.3 - Prob. 36ECh. 14.3 - Prob. 37ECh. 14.3 - Prob. 38ECh. 14.3 - In Exercises 35–40, find the partial derivative of...Ch. 14.3 - In Exercises 35–40, find the partial derivative of...Ch. 14.3 - Prob. 41ECh. 14.3 - Prob. 42ECh. 14.3 - Prob. 43ECh. 14.3 - Prob. 44ECh. 14.3 - Find all the second-order partial derivatives of...Ch. 14.3 - Prob. 46ECh. 14.3 - Find all the second-order partial derivatives of...Ch. 14.3 - Find all the second-order partial derivatives of...Ch. 14.3 - Find all the second-order partial derivatives of...Ch. 14.3 - Prob. 50ECh. 14.3 - Find all the second-order partial derivatives of...Ch. 14.3 - Prob. 52ECh. 14.3 - Find all the second-order partial derivatives of...Ch. 14.3 - Prob. 54ECh. 14.3 - In Exercises 55–60, verify that .
55.
Ch. 14.3 - Prob. 56ECh. 14.3 - In Exercises 55–60, verify that .
57.
Ch. 14.3 - In Exercises 55–60, verify that .
58.
Ch. 14.3 - In Exercises 55–60, verify that .
59.
Ch. 14.3 - In Exercises 55–60, verify that .
60.
Ch. 14.3 - Which order of differentiation enables one to...Ch. 14.3 - Prob. 62ECh. 14.3 - Prob. 63ECh. 14.3 - Prob. 64ECh. 14.3 - Prob. 65ECh. 14.3 - Prob. 66ECh. 14.3 - Let f(x, y) = 2x + 3y = 4. Find the slope of the...Ch. 14.3 - Let f(x, y) = x2 + y3. Find the slope of the line...Ch. 14.3 - Prob. 69ECh. 14.3 - Prob. 70ECh. 14.3 - Prob. 71ECh. 14.3 - In Exercises 77-80, find a function z = f(x, y)...Ch. 14.3 - In Exercises 77-80, find a function z = f(x, y)...Ch. 14.3 - Prob. 74ECh. 14.3 - Prob. 75ECh. 14.3 - Prob. 76ECh. 14.3 - Exercises 71 and 72 are about the triangle shown...Ch. 14.3 - Prob. 78ECh. 14.3 - Two dependent variables Express vx in terms of u...Ch. 14.3 - Prob. 80ECh. 14.3 - Let
Find fx, fy, fxy, and fyx, state the domain...Ch. 14.3 - Let
The graph of ƒ is shown on page 799.
Show...Ch. 14.3 - Show that each function in Exercises 83-90...Ch. 14.3 - Show that each function in Exercises 83-90...Ch. 14.3 - Show that each function in Exercises 83-90...Ch. 14.3 - Show that each function in Exercises 83-90...Ch. 14.3 - Prob. 87ECh. 14.3 - Prob. 88ECh. 14.3 - Prob. 89ECh. 14.3 - Prob. 90ECh. 14.3 - Prob. 91ECh. 14.3 - Show that the functions in Exercises 91-97 are all...Ch. 14.3 - Prob. 93ECh. 14.3 - Prob. 94ECh. 14.3 - Prob. 95ECh. 14.3 - Prob. 96ECh. 14.3 - Prob. 97ECh. 14.3 - Prob. 98ECh. 14.3 - Prob. 99ECh. 14.3 - Prob. 100ECh. 14.3 - Prob. 101ECh. 14.3 -
Show that fx(0, 0) and fy(0, 0) exist, but f is...Ch. 14.3 - Prob. 103ECh. 14.3 - Show that satisfies the equation Txx + Tyy = T3
Ch. 14.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 14.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 14.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 14.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 14.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 14.4 - In Exercises 1–6, (a) express dw/dt as a function...Ch. 14.4 - In Exercises 7 and 8, (a) express and as...Ch. 14.4 - In Exercises 7 and 8, (a) express and as...Ch. 14.4 - In Exercises 9 and 10, (a) express and as...Ch. 14.4 - In Exercises 9 and 10, (a) express and as...Ch. 14.4 - In Exercises 11 and 12, (a) express and as...Ch. 14.4 - In Exercises 11 and 12, (a) express ∂u/∂x, ∂u/∂y,...Ch. 14.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 14.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 14.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 14.4 - Prob. 16ECh. 14.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 14.4 - Prob. 18ECh. 14.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 14.4 - Prob. 20ECh. 14.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 14.4 - Prob. 22ECh. 14.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 14.4 - In Exercises 13–24, draw a dependency diagram and...Ch. 14.4 - Assuming that the equations in Exercises 25–30...Ch. 14.4 - Assuming that the equations in Exercises 25–30...Ch. 14.4 - Assuming that the equations in Exercises 25–30...Ch. 14.4 - Assuming that the equations in Exercises 25–30...Ch. 14.4 - Assuming that the equations in Exercises 25–30...Ch. 14.4 - Assuming that the equations in Exercises 25–30...Ch. 14.4 - Find the values of ∂z/∂x and ∂z/∂y at the points...Ch. 14.4 - Find the values of ∂z/∂x and ∂z/∂y at the points...Ch. 14.4 - Find the values of ∂z/∂x and ∂z/∂y at the points...Ch. 14.4 - Prob. 34ECh. 14.4 - Prob. 35ECh. 14.4 - Prob. 36ECh. 14.4 - Prob. 37ECh. 14.4 - Prob. 38ECh. 14.4 - Find ∂z/∂u and ∂z/∂v when u = ln 2, v = 1 if z = 5...Ch. 14.4 - Prob. 40ECh. 14.4 - Assume that w = f(s3 + t2) and f′(x) = ex. Find ...Ch. 14.4 - Assume that , , and . Find and .
Ch. 14.4 - Assume that z = f(x, y), x = g(t), y = h(t), fx(2,...Ch. 14.4 - Assume that z = f(x, y)2, x = g(t), y = h(t),...Ch. 14.4 - Assume that z = f(w), w = g(x, y), x = 2r3 − s2,...Ch. 14.4 - Assume that z = ln (f(w)), w = g(x, y), , and y =...Ch. 14.4 - Changing voltage in a circuit The voltage V in a...Ch. 14.4 - Changing dimensions in a box The lengths a, b, and...Ch. 14.4 - Prob. 49ECh. 14.4 - Polar coordinates Suppose that we substitute polar...Ch. 14.4 - Laplace equations Show that if satisfies the...Ch. 14.4 - Laplace equations Let , where , , and . Show that...Ch. 14.4 - Extreme values on a helix Suppose that the partial...Ch. 14.4 - A space curve Let w = x2e2y cos 3z. Find the value...Ch. 14.4 - Temperature on a circle Let T = f(x, y) be the...Ch. 14.4 - Temperature on an ellipse Let T = g(x, y) be the...Ch. 14.4 - The temperature T = T(x, y) in °C at point (x, y)...Ch. 14.4 - Prob. 58ECh. 14.4 - Prob. 59ECh. 14.4 - Find the derivatives of the functions in Exercises...Ch. 14.5 - In Exercises 1–6, find the gradient of the...Ch. 14.5 - In Exercises 1–6, find the gradient of the...Ch. 14.5 - In Exercises 1–6, find the gradient of the...Ch. 14.5 - In Exercises 1–6, find the gradient of the...Ch. 14.5 - In Exercises 1–6, find the gradient of the...Ch. 14.5 - In Exercises 1–6, find the gradient of the...Ch. 14.5 - In Exercises 7–10, find f at the given point.
7.
Ch. 14.5 - In Exercises 7–10, find f at the given point.
8.
Ch. 14.5 - In Exercises 7–10, find f at the given point.
9.
Ch. 14.5 - In Exercises 7–10, find f at the given point.
10....Ch. 14.5 - In Exercises 11–18, find the derivative of the...Ch. 14.5 - In Exercises 11-18, find the derivative of the...Ch. 14.5 - In Exercises 11-18, find the derivative of the...Ch. 14.5 - In Exercises 11-18, find the derivative of the...Ch. 14.5 - In Exercises 11-18, find the derivative of the...Ch. 14.5 - In Exercises 11-18, find the derivative of the...Ch. 14.5 - In Exercises 11-18, find the derivative of the...Ch. 14.5 - In Exercises 11-18, find the derivative of the...Ch. 14.5 - In Exercises 19–24, find the directions in which...Ch. 14.5 - In Exercises 19–24, find the directions in which...Ch. 14.5 - In Exercises 19–24, find the directions in which...Ch. 14.5 - In Exercises 19–24, find the directions in which...Ch. 14.5 - In Exercises 19–24, find the directions in which...Ch. 14.5 - In Exercises 19–24, find the directions in which...Ch. 14.5 - In Exercises 25–28, sketch the curve f(x, y) = c,...Ch. 14.5 - In Exercises 25–28, sketch the curve f(x, y) = c,...Ch. 14.5 - In Exercises 25–28, sketch the curve f(x, y) = c,...Ch. 14.5 - In Exercises 25–28, sketch the curve f(x, y) = c,...Ch. 14.5 - Let f(x, y) = x2 − xy + y2 − y. Find the...Ch. 14.5 - Let Find the directions u and the values of for...Ch. 14.5 - Zero directional derivative In what direction is...Ch. 14.5 - Zero directional derivative In what directions is...Ch. 14.5 - Is there a direction u in which the rate of change...Ch. 14.5 - Changing temperature along a circle Is there a...Ch. 14.5 - The derivative of f(x, y) at P0(1, 2) in the...Ch. 14.5 - The derivative of f(x, y, z) at a point P is...Ch. 14.5 - Prob. 37ECh. 14.5 - Prob. 38ECh. 14.5 - Lines in the xy-plane Show that A(x – x0) + B(y –...Ch. 14.5 - The algebra rules for gradients Given a constant k...Ch. 14.5 - In Exercises 41–44, find a parametric equation for...Ch. 14.5 - Prob. 42ECh. 14.5 - In Exercises 41–44, find a parametric equation for...Ch. 14.5 - In Exercises 41–44, find a parametric equation for...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 1–10, find equations for the
tangent...Ch. 14.6 - In Exercises 11–14, find an equation for the plane...Ch. 14.6 - In Exercises 11–14, find an equation for the plane...Ch. 14.6 - In Exercises 11–14, find an equation for the plane...Ch. 14.6 - In Exercises 11–14, find an equation for the plane...Ch. 14.6 - In Exercises 15–20, find parametric equations for...Ch. 14.6 - In Exercises 15–20, find parametric equations for...Ch. 14.6 - In Exercises 15–20, find parametric equations for...Ch. 14.6 - In Exercises 15–20, find parametric equations for...Ch. 14.6 - In Exercises 15–20, find parametric equations for...Ch. 14.6 - In Exercises 15–20, find parametric equations for...Ch. 14.6 - Prob. 21ECh. 14.6 - Prob. 22ECh. 14.6 - By about how much will
change if the point P(x,...Ch. 14.6 - Prob. 24ECh. 14.6 - Temperature change along a circle Suppose that the...Ch. 14.6 - Changing temperature along a space curve The...Ch. 14.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 14.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 14.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 14.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 14.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 14.6 - In Exercises 27–32, find the linearization L(x, y)...Ch. 14.6 - Wind chill factor Wind chill, a measure of the...Ch. 14.6 - Find the linearization L(v, T) of the function...Ch. 14.6 - In Exercises 35-40, find the linearization L(x, y)...Ch. 14.6 - In Exercises 35-40, find the linearization L(x, y)...Ch. 14.6 - In Exercises 35-40, find the linearization L(x, y)...Ch. 14.6 - In Exercises 35-40, find the linearization L(x, y)...Ch. 14.6 - Prob. 39ECh. 14.6 - In Exercises 35-40, find the linearization L(x, y)...Ch. 14.6 - Find the linearizations L(x, y, z) of the...Ch. 14.6 - Find the linearizations L(x, y, z) of the...Ch. 14.6 - Prob. 43ECh. 14.6 - Prob. 44ECh. 14.6 - Prob. 45ECh. 14.6 - Prob. 46ECh. 14.6 - Prob. 47ECh. 14.6 - In Exercises 47-50, find the linearization L(x, y,...Ch. 14.6 - Prob. 49ECh. 14.6 - Prob. 50ECh. 14.6 - Estimating maximum error Suppose that T is to be...Ch. 14.6 - Prob. 52ECh. 14.6 - Prob. 53ECh. 14.6 - Prob. 54ECh. 14.6 - Prob. 55ECh. 14.6 - Prob. 56ECh. 14.6 - Prob. 57ECh. 14.6 - Change along the involute of a circle Find the...Ch. 14.6 - Tangent curves A smooth curve is tangent to the...Ch. 14.6 - Normal curves A smooth curve is normal to a...Ch. 14.6 - Consider a closed rectangular box with a square...Ch. 14.7 - Prob. 1ECh. 14.7 - Prob. 2ECh. 14.7 - Prob. 3ECh. 14.7 - Prob. 4ECh. 14.7 - Prob. 5ECh. 14.7 - Prob. 6ECh. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Prob. 10ECh. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Prob. 18ECh. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Prob. 20ECh. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Find all the local maxima, local minima, and...Ch. 14.7 - Prob. 31ECh. 14.7 - Prob. 32ECh. 14.7 - Prob. 33ECh. 14.7 - In Exercises 31–38, find the absolute maxima and...Ch. 14.7 - Prob. 35ECh. 14.7 - Prob. 36ECh. 14.7 - In Exercises 31–38, find the absolute maxima and...Ch. 14.7 - Prob. 38ECh. 14.7 - Find two numbers a and b with such that
has its...Ch. 14.7 - Find two numbers a and b with such that
has its...Ch. 14.7 - Temperatures A flat circular plate has the shape...Ch. 14.7 - Prob. 42ECh. 14.7 - Find the maxima, minima, and saddle points of f(x,...Ch. 14.7 - The discriminant fxxfyy − fxv2 is zero at the...Ch. 14.7 - Show that (0, 0) is a critical point of f(x, y) =...Ch. 14.7 - For what values of the constant k does the Second...Ch. 14.7 - If fx(a, b) = fy(a, b) = 0, must f have a local...Ch. 14.7 - Prob. 48ECh. 14.7 - Among all the points on the graph of that lie...Ch. 14.7 - Find the point on the graph of nearest the plane...Ch. 14.7 - Find the point on the plane 3x + 2y + z = 6 that...Ch. 14.7 - Prob. 52ECh. 14.7 - Find three numbers whose sum is 9 and whose sum of...Ch. 14.7 - Prob. 54ECh. 14.7 - Find the maximum value of where .
Ch. 14.7 - Find the minimum distance from the cone to the...Ch. 14.7 - Find the dimensions of the rectangular box of...Ch. 14.7 - Prob. 58ECh. 14.7 - You are to construct an open rectangular box from...Ch. 14.7 - Consider the function f(x, y) = x2 + y2 + 2xy – x...Ch. 14.7 - Find the point on the graph of nearest the...Ch. 14.7 - Prob. 62ECh. 14.7 - Extreme Values on Parametrized Curves To find the...Ch. 14.7 - Extreme Values on Parametrized Curves To find the...Ch. 14.7 - Extreme Values on Parametrized Curves To find the...Ch. 14.7 - Extreme Values on Parametrized Curves To find the...Ch. 14.7 - Least squares and regression lines When we try to...Ch. 14.7 - Prob. 68ECh. 14.7 - In Exercises 68–70, use Equations (2) and (3) to...Ch. 14.7 - In Exercises 68–70, use Equations (2) and (3) to...Ch. 14.8 - Extrema on an ellipse Find the points on the...Ch. 14.8 - Extrema on a circle Find the extreme values of...Ch. 14.8 - Maximum on a line Find the maximum value of f(x,...Ch. 14.8 - Extrema on a line Find the local extreme values of...Ch. 14.8 - Constrained minimum Find the points on the curve...Ch. 14.8 - Constrained minimum Find the points on the curve...Ch. 14.8 - Use the method of Lagrange multipliers to...Ch. 14.8 - Extrema on a curve Find the points on the curve x2...Ch. 14.8 - Minimum surface area with fixed volume Find the...Ch. 14.8 - Cylinder in a sphere Find the radius and height of...Ch. 14.8 - Rectangle of greatest area in an ellipse Use the...Ch. 14.8 - Rectangle of longest perimeter in an ellipse Find...Ch. 14.8 - Extrema on a circle Find the maximum and minimum...Ch. 14.8 - Extrema on a circle Find the maximum and minimum...Ch. 14.8 - Ant on a metal plate The temperature at a point...Ch. 14.8 - Prob. 16ECh. 14.8 - Minimum distance to a point Find the point on the...Ch. 14.8 - Maximum distance to a point Find the point on the...Ch. 14.8 - Minimum distance to the origin Find the minimum...Ch. 14.8 - Minimum distance to the origin Find the point on...Ch. 14.8 - Minimum distance to the origin Find the points on...Ch. 14.8 - Minimum distance to the origin Find the point(s)...Ch. 14.8 - Extrema on a sphere Find the maximum and minimum...Ch. 14.8 - Extrema on a sphere Find the points on the sphere...Ch. 14.8 - Minimizing a sum of squares Find three real...Ch. 14.8 - Maximizing a product Find the largest product the...Ch. 14.8 - Prob. 27ECh. 14.8 - Prob. 28ECh. 14.8 - Hottest point on a space probe A space probe in...Ch. 14.8 - Extreme temperatures on a sphere Suppose that the...Ch. 14.8 - Cobb-Douglas production function During the 1920s,...Ch. 14.8 - Prob. 32ECh. 14.8 - Prob. 33ECh. 14.8 - Prob. 34ECh. 14.8 - Prob. 35ECh. 14.8 - Prob. 36ECh. 14.8 - Maximize the function subject to the constraints...Ch. 14.8 - Prob. 38ECh. 14.8 - Minimum distance to the origin Find the point...Ch. 14.8 - Find the extreme values of on the intersection of...Ch. 14.8 - Extrema on a curve of intersection Find the...Ch. 14.8 - Prob. 42ECh. 14.8 - Prob. 43ECh. 14.8 - Minimum distance to the origin Find the point...Ch. 14.8 - Prob. 45ECh. 14.8 - Prob. 46ECh. 14.8 - Prob. 47ECh. 14.8 - Sum of products Let a1, a2,..., an be n positive...Ch. 14.9 - In Exercises 1–10, use Taylor’s formula for f(x,...Ch. 14.9 - In Exercises 1–10, use Taylor’s formula for f(x,...Ch. 14.9 - Prob. 3ECh. 14.9 - Prob. 4ECh. 14.9 - Prob. 5ECh. 14.9 - Prob. 6ECh. 14.9 - Prob. 7ECh. 14.9 - Prob. 8ECh. 14.9 - Prob. 9ECh. 14.9 - Prob. 10ECh. 14.9 - Use Taylor’s formula to find a quadratic...Ch. 14.9 - Prob. 12ECh. 14.10 - In Exercises 1–3, begin by drawing a diagram that...Ch. 14.10 - In Exercises 1–3, begin by drawing a diagram that...Ch. 14.10 - Prob. 3ECh. 14.10 - Find
at the point (x, y, z) = (0, 1, π) if
w =...Ch. 14.10 - Prob. 5ECh. 14.10 - Prob. 6ECh. 14.10 - Prob. 7ECh. 14.10 - Suppose that
w = x2 − y2 + 4z + t and x + 2z + t =...Ch. 14.10 - Prob. 9ECh. 14.10 - Prob. 10ECh. 14.10 - Suppose that the equation g(x, y, z) = 0...Ch. 14.10 - Prob. 12ECh. 14 - Prob. 1GYRCh. 14 - What does it mean for sets in the plane or in...Ch. 14 - Prob. 3GYRCh. 14 - Prob. 4GYRCh. 14 - Prob. 5GYRCh. 14 - Prob. 6GYRCh. 14 - Prob. 7GYRCh. 14 - Prob. 8GYRCh. 14 - Prob. 9GYRCh. 14 - Prob. 10GYRCh. 14 - Prob. 11GYRCh. 14 - Prob. 12GYRCh. 14 - Prob. 13GYRCh. 14 - Prob. 14GYRCh. 14 - Prob. 15GYRCh. 14 - Prob. 16GYRCh. 14 - Prob. 17GYRCh. 14 - How do you linearize a function f(x, y) of two...Ch. 14 - Prob. 19GYRCh. 14 - Prob. 20GYRCh. 14 - Prob. 21GYRCh. 14 - Prob. 22GYRCh. 14 - Prob. 23GYRCh. 14 - Prob. 24GYRCh. 14 - Prob. 25GYRCh. 14 - If , where the variables x, y, and z are...Ch. 14 - Prob. 1PECh. 14 - Prob. 2PECh. 14 - Prob. 3PECh. 14 - Prob. 4PECh. 14 - In Exercises 5–8, find the domain and range of the...Ch. 14 - Prob. 6PECh. 14 - Prob. 7PECh. 14 - Prob. 8PECh. 14 - Find the limits in Exercises 9–14.
9.
Ch. 14 - Prob. 10PECh. 14 - Prob. 11PECh. 14 - Find the limits in Exercises 9–14.
12.
Ch. 14 - Prob. 13PECh. 14 - Prob. 14PECh. 14 - Prob. 15PECh. 14 - Prob. 16PECh. 14 - Prob. 17PECh. 14 - Continuous extension Let
Is f continuous at the...Ch. 14 - Prob. 19PECh. 14 - Prob. 20PECh. 14 - Prob. 21PECh. 14 - In Exercises 19–24, find the partial derivative of...Ch. 14 - In Exercises 19–24, find the partial derivative of...Ch. 14 - Prob. 24PECh. 14 - Prob. 25PECh. 14 - Prob. 26PECh. 14 - Find the second-order partial derivatives of the...Ch. 14 - Prob. 28PECh. 14 - Prob. 29PECh. 14 - Prob. 30PECh. 14 - Prob. 31PECh. 14 - Find and when if and .
Ch. 14 - Prob. 33PECh. 14 - Prob. 34PECh. 14 - Assuming that the equations in Exercises 35 and 36...Ch. 14 - Prob. 36PECh. 14 - Prob. 37PECh. 14 - Prob. 38PECh. 14 - In Exercises 37–40, find the directions in which f...Ch. 14 - Prob. 40PECh. 14 - Derivative in velocity direction Find the...Ch. 14 - Prob. 42PECh. 14 - Prob. 43PECh. 14 - Prob. 44PECh. 14 - Prob. 45PECh. 14 - Prob. 46PECh. 14 - In Exercises 47 and 48, find an equation for the...Ch. 14 - Prob. 48PECh. 14 - Prob. 49PECh. 14 - Prob. 50PECh. 14 - Prob. 51PECh. 14 - Prob. 52PECh. 14 - Prob. 53PECh. 14 - In Exercises 53 and 54, find parametric equations...Ch. 14 - Prob. 55PECh. 14 - Prob. 56PECh. 14 - Prob. 57PECh. 14 - Find the linearizations of the functions in...Ch. 14 - Prob. 59PECh. 14 - Prob. 60PECh. 14 - Prob. 61PECh. 14 - Prob. 62PECh. 14 - Prob. 63PECh. 14 - Cardiac index To make different people comparable...Ch. 14 - Prob. 65PECh. 14 - Prob. 66PECh. 14 - Test the functions in Exercises 65–70 for local...Ch. 14 - Test the functions in Exercises 65–70 for local...Ch. 14 - Prob. 69PECh. 14 - Prob. 70PECh. 14 - Prob. 71PECh. 14 - In Exercises 71–78, find the absolute maximum and...Ch. 14 - Prob. 73PECh. 14 - Prob. 74PECh. 14 - Prob. 75PECh. 14 - Prob. 76PECh. 14 - In Exercises 71–78, find the absolute maximum and...Ch. 14 - Prob. 78PECh. 14 - Prob. 79PECh. 14 - Prob. 80PECh. 14 - Extrema in a disk Find the extreme values of on...Ch. 14 - Prob. 82PECh. 14 - Prob. 83PECh. 14 - Minimum distance to origin Find the points on the...Ch. 14 - Minimizing cost of a box A closed rectangular box...Ch. 14 - Prob. 86PECh. 14 - Prob. 87PECh. 14 - Prob. 88PECh. 14 - Prob. 89PECh. 14 - Prob. 90PECh. 14 - Prob. 91PECh. 14 - Prob. 92PECh. 14 - Prob. 93PECh. 14 - Prob. 94PECh. 14 - Normal line parallel to a plane Find the points on...Ch. 14 - Prob. 96PECh. 14 - Prob. 97PECh. 14 - Prob. 98PECh. 14 - Prob. 99PECh. 14 - Prob. 100PECh. 14 - In Exercises 101 and 102, begin by drawing a...Ch. 14 - Prob. 102PECh. 14 - Prob. 1AAECh. 14 - Prob. 2AAECh. 14 - Prob. 3AAECh. 14 - Prob. 4AAECh. 14 - Prob. 5AAECh. 14 - Surface in polar coordinates Let
where r and θ...Ch. 14 - Prob. 7AAECh. 14 - Prob. 8AAECh. 14 - Prob. 9AAECh. 14 - Prob. 10AAECh. 14 - Prob. 11AAECh. 14 - Prob. 12AAECh. 14 - Prob. 13AAECh. 14 - Prob. 14AAECh. 14 - Prob. 15AAECh. 14 - Prob. 16AAECh. 14 - Prob. 17AAECh. 14 - Prob. 18AAECh. 14 - Prob. 19AAECh. 14 - Prob. 20AAECh. 14 - Directional derivatives tangent to a surface Let S...Ch. 14 - Prob. 22AAECh. 14 - Prob. 23AAECh. 14 - Prob. 24AAE
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