Thomas' Calculus
4th Edition
ISBN: 9780134439099
Author: Hass, Joel., Heil, Christopher , WEIR, Maurice D.
Publisher: Pearson,
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Chapter 14.3, Problem 58E
To determine
Prove that
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Class, the class silues, and the class notes, whether the series does alternate and the absolute
values of the terms decrease), and if the test does apply, determine whether the series converges
or diverges. For the ith series, if the test does not apply the let Mi = 2, while if the test
determines divergence then M¿ = 4, and if it determines convergence then M¿ = 8.
1:
2:
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4:
5 :
∞
n=1
∞
(−1)n+1.
Σ(-1) +1
n=1
∞
п
3m² +2
Σ(-1)+1 sin(2n).
n=1
∞
2n² + 2n +3
4n2 +6
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e-n + n²
3n23n+1
9n² +3
In(n + 1)
2n+1
Σ(-1) +1
n=1
∞
Σ(-1)".
n=1
Then the value of cos(M₁) + cos(2M2) + cos(3M3) + sin(2M4) + sin(M5) is
1.715
0.902
0.930
-1.647
-0.057
● 2.013
1.141
4.274
3. FCX14) = x²+3xx-y3
+.
A cylindrical chemical storage tank with a capacity of 950m3 is going to be constructed in a warehouse that is 11m by 14m with a height of 10m. The specifications call for the case to be made of sheet metal that costs $90/m2, the top to be made from sheet metal that costs $45/m2 and the wall to be made of sheet metal that costs $80/m2. If you want to minimize the cost to make the storage house, how much would you end up spending to build the tank?
Chapter 14 Solutions
Thomas' Calculus
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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