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 16.4, Problem 25E
To determine
Find the work done by the field
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A function is defined on the interval (-π/2,π/2) by this multipart rule:
if -π/2 < x < 0
f(x) =
a
if x=0
31-tan x
+31-cot x
if 0 < x < π/2
Here, a and b are constants. Find a and b so that the function f(x) is continuous at x=0.
a=
b= 3
Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.
f(x) = (x + 4x4) 5,
a = -1
lim f(x)
X--1
=
lim
x+4x
X--1
lim
X-1
4
x+4x
5
))"
5
))
by the power law
by the sum law
lim (x) + lim
X--1
4
4x
X-1
-(0,00+(
Find f(-1).
f(-1)=243
lim (x) +
-1 +4
35
4 ([
)
lim (x4)
5
x-1
Thus, by the definition of continuity, f is continuous at a = -1.
by the multiple constant law
by the direct substitution property
1. Compute
Lo
F⚫dr, where
and C is defined by
F(x, y) = (x² + y)i + (y − x)j
r(t) = (12t)i + (1 − 4t + 4t²)j
from the point (1, 1) to the origin.
Chapter 16 Solutions
Thomas' Calculus (14th Edition)
Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Prob. 6ECh. 16.1 - Prob. 7ECh. 16.1 - Match the vector equations in Exercises 1–8 with...Ch. 16.1 - Evaluate ∫C (x + y) ds, where C is the...Ch. 16.1 - Evaluate ∫C (x − y + z − 2) ds, where C is the...
Ch. 16.1 - Evaluate ∫C (xy + y + z) ds along the curve r(t) =...Ch. 16.1 - Evaluate along the curve r(t) = (4 cos t)i + (4...Ch. 16.1 - Prob. 13ECh. 16.1 - Find the line integral of over the curve r(t) =...Ch. 16.1 - Integrate over the path C1 followed by C2 from...Ch. 16.1 - Integrate over the path C1 followed by C2...Ch. 16.1 - Integrate f(x, y, z) = (x + y + z)/(x2 + y2 + z2)...Ch. 16.1 - Integrate over the circle r(t) = (a cos t)j + (a...Ch. 16.1 - Evaluate ∫C x ds, where C is
the straight-line...Ch. 16.1 - Evaluate , where C is
the straight-line segment x...Ch. 16.1 - Find the line integral of along the curve r(t) =...Ch. 16.1 - Find the line integral of f(x, y) = x − y + 3...Ch. 16.1 - Evaluate , where C is the curve x = t2, y = t3,...Ch. 16.1 - Find the line integral of along the curve , 1/2 ≤...Ch. 16.1 - Evaluate ,where C is given in the accompanying...Ch. 16.1 - Evaluate , where C is given in the accompanying...Ch. 16.1 - Prob. 27ECh. 16.1 - Prob. 28ECh. 16.1 - In Exercises 27–30, integrate f over the given...Ch. 16.1 - In Exercises 27–30, integrate f over the given...Ch. 16.1 - Prob. 31ECh. 16.1 - Find the area of one side of the “wall” standing...Ch. 16.1 - Mass of a wire Find the mass of a wire that lies...Ch. 16.1 - Center of mass of a curved wire A wire of density ...Ch. 16.1 - Mass of wire with variable density Find the mass...Ch. 16.1 - Center of mass of wire with variable density Find...Ch. 16.1 - Moment of inertia of wire hoop A circular wire...Ch. 16.1 - Inertia of a slender rod A slender rod of constant...Ch. 16.1 - Two springs of constant density A spring of...Ch. 16.1 - Wire of constant density A wire of constant...Ch. 16.1 - Prob. 41ECh. 16.1 - Center of mass and moments of inertia for wire...Ch. 16.2 - Find the gradient fields of the functions in...Ch. 16.2 - Find the gradient fields of the functions in...Ch. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5ECh. 16.2 - Prob. 6ECh. 16.2 - In Exercises 7−12, find the line integrals of F...Ch. 16.2 - Prob. 8ECh. 16.2 - In Exercises 7−12, find the line integrals of F...Ch. 16.2 - In Exercises 7−12, find the line integrals of F...Ch. 16.2 - Prob. 11ECh. 16.2 - Line Integrals of Vector Fields
In Exercises 7−12,...Ch. 16.2 - In Exercises 13–16, find the line integrals along...Ch. 16.2 - In Exercises 13–16, find the line integrals along...Ch. 16.2 - In Exercises 13–16, find the line integrals along...Ch. 16.2 - In Exercises 13–16, find the line integrals along...Ch. 16.2 - Prob. 17ECh. 16.2 - Along the curve , , evaluate each of the following...Ch. 16.2 - In Exercises 19–22, find the work done by F over...Ch. 16.2 - In Exercises 19–22, find the work done by F over...Ch. 16.2 - In Exercises 19–22, find the work done by F over...Ch. 16.2 - In Exercises 19–22, find the work done by F over...Ch. 16.2 - Evaluate along the curve from (–1, 1) to (2,...Ch. 16.2 - Prob. 24ECh. 16.2 - Evaluate for the vector field along the curve ...Ch. 16.2 - Prob. 26ECh. 16.2 - Prob. 27ECh. 16.2 - Prob. 28ECh. 16.2 - Circulation and flux Find the circulation and flux...Ch. 16.2 - Flux across a circle Find the flux of the...Ch. 16.2 - Prob. 31ECh. 16.2 - In Exercises 31–34, find the circulation and flux...Ch. 16.2 - In Exercises 31–34, find the circulation and flux...Ch. 16.2 - In Exercises 31–34, find the circulation and flux...Ch. 16.2 - Flow integrals Find the flow of the velocity field...Ch. 16.2 - Flux across a triangle Find the flux of the field...Ch. 16.2 - Prob. 37ECh. 16.2 - The flow of a gas with a density of over the...Ch. 16.2 - Find the flow of the velocity field F = y2i + 2xyj...Ch. 16.2 - Prob. 40ECh. 16.2 - Prob. 41ECh. 16.2 - Prob. 42ECh. 16.2 - Prob. 43ECh. 16.2 - Prob. 44ECh. 16.2 - Prob. 45ECh. 16.2 - Prob. 46ECh. 16.2 - Prob. 47ECh. 16.2 - Prob. 48ECh. 16.2 - A field of tangent vectors
Find a field G = P(x,...Ch. 16.2 - A field of tangent vectors
Find a field G = P(x,...Ch. 16.2 - Unit vectors pointing toward the origin Find a...Ch. 16.2 - Prob. 52ECh. 16.2 - Prob. 53ECh. 16.2 - Prob. 54ECh. 16.2 - Prob. 55ECh. 16.2 - Prob. 56ECh. 16.2 - In Exercises 55–58, F is the velocity field of a...Ch. 16.2 - In Exercises 55–58, F is the velocity field of a...Ch. 16.2 - Circulation Find the circulation of F = 2xi + 2zj...Ch. 16.2 - Prob. 60ECh. 16.2 - Flow along a curve The field F = xyi + yj − yzk is...Ch. 16.2 - Flow of a gradient field Find the flow of the...Ch. 16.3 - Which fields in Exercises 1–6 are conservative,...Ch. 16.3 - Which fields in Exercises 1–6 are conservative,...Ch. 16.3 - Which fields in Exercises 1–6 are conservative,...Ch. 16.3 - Which fields in Exercises 1–6 are conservative,...Ch. 16.3 - Which fields in Exercises 1−6 are conservative,...Ch. 16.3 - Which fields in Exercises 1−6 are conservative,...Ch. 16.3 - Finding Potential Functions
In Exercises 7–12,...Ch. 16.3 -
In Exercises 7–12, find a potential function f...Ch. 16.3 - In Exercises 7–12, find a potential function f for...Ch. 16.3 - In Exercises 7–12, find a potential function f for...Ch. 16.3 - In Exercises 7–12, find a potential function f for...Ch. 16.3 - In Exercises 7–12, find a potential function f for...Ch. 16.3 - Prob. 13ECh. 16.3 - In Exercises 13–17, show that the differential...Ch. 16.3 - Prob. 15ECh. 16.3 - In Exercises 13–17, show that the differential...Ch. 16.3 - In Exercises 13–17, show that the differential...Ch. 16.3 - Prob. 18ECh. 16.3 -
Although they are not defined on all of space R3,...Ch. 16.3 - Although they are not defined on all of space R3,...Ch. 16.3 - Prob. 21ECh. 16.3 - Prob. 22ECh. 16.3 - Prob. 23ECh. 16.3 - Evaluate
along the line segment C joining (0, 0,...Ch. 16.3 - Independence of path Show that the values of the...Ch. 16.3 - Prob. 26ECh. 16.3 - Prob. 27ECh. 16.3 - In Exercises 27 and 28, find a potential function...Ch. 16.3 - Work along different paths Find the work done by F...Ch. 16.3 - Work along different paths Find the work done by F...Ch. 16.3 - Evaluating a work integral two ways Let F =...Ch. 16.3 - Integral along different paths Evaluate the line...Ch. 16.3 - Exact differential form How are the constants a,...Ch. 16.3 - Prob. 34ECh. 16.3 - Prob. 35ECh. 16.3 - Prob. 36ECh. 16.3 - Prob. 37ECh. 16.3 - Gravitational field
Find a potential function for...Ch. 16.4 - In Exercises 1–6, find the k-component of curl(F)...Ch. 16.4 - Prob. 2ECh. 16.4 - In Exercises 1–6, find the k-component of curl(F)...Ch. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.4 - In Exercises 1–6, find the k-component of curl(F)...Ch. 16.4 - In Exercises 7–10, verify the conclusion of...Ch. 16.4 - In Exercises 7–10, verify the conclusion of...Ch. 16.4 - In Exercises 7–10, verify the conclusion of...Ch. 16.4 - In Exercises 7–10, verify the conclusion of...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 16.4 - Prob. 20ECh. 16.4 - Find the counterclockwise circulation and outward...Ch. 16.4 - Find the counterclockwise circulation and the...Ch. 16.4 - Prob. 23ECh. 16.4 - Find the counterclockwise circulation of around...Ch. 16.4 - In Exercises 25 and 26, find the work done by F in...Ch. 16.4 - Prob. 26ECh. 16.4 - Apply Green’s Theorem to evaluate the integrals in...Ch. 16.4 - Apply Green’s Theorem to evaluate the integrals in...Ch. 16.4 - Apply Green’s Theorem to evaluate the integrals in...Ch. 16.4 - Prob. 30ECh. 16.4 - Prob. 31ECh. 16.4 - Prob. 32ECh. 16.4 - Use the Green’s Theorem area formula given above...Ch. 16.4 - Prob. 34ECh. 16.4 - Prob. 35ECh. 16.4 - Integral dependent only on area Show that the...Ch. 16.4 - Evaluate the integral
for any closed path C.
Ch. 16.4 - Evaluate the integral
for any closed path C.
Ch. 16.4 - Prob. 39ECh. 16.4 - Definite integral as a line integral Suppose that...Ch. 16.4 - Prob. 41ECh. 16.4 - Prob. 42ECh. 16.4 - Green’s Theorem and Laplace’s equation Assuming...Ch. 16.4 - Maximizing work Among all smooth, simple closed...Ch. 16.4 - Regions with many holes Green’s Theorem holds for...Ch. 16.4 - Prob. 46ECh. 16.4 - Prob. 47ECh. 16.4 - Prob. 48ECh. 16.5 - In Exercises 1–16, find a parametrization of the...Ch. 16.5 - In Exercises 1–16, find a parametrization of the...Ch. 16.5 - In Exercises 1–16, find a parametrization of the...Ch. 16.5 - In Exercises 1–16, find a parametrization of the...Ch. 16.5 - Prob. 5ECh. 16.5 - Prob. 6ECh. 16.5 - Prob. 7ECh. 16.5 - In Exercises 1–16, find a parametrization of the...Ch. 16.5 - Prob. 9ECh. 16.5 - Prob. 10ECh. 16.5 - Prob. 11ECh. 16.5 - Prob. 12ECh. 16.5 - Prob. 13ECh. 16.5 - In Exercises 1–16, find a parametrization of the...Ch. 16.5 - Prob. 15ECh. 16.5 - Prob. 16ECh. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - Prob. 19ECh. 16.5 - Prob. 20ECh. 16.5 - Prob. 21ECh. 16.5 - Prob. 22ECh. 16.5 - Prob. 23ECh. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - In Exercises 17–26, use a parametrization to...Ch. 16.5 - Prob. 27ECh. 16.5 - Prob. 28ECh. 16.5 - Prob. 29ECh. 16.5 - Prob. 30ECh. 16.5 - A torus of revolution (doughnut) is obtained by...Ch. 16.5 - Prob. 32ECh. 16.5 - Prob. 33ECh. 16.5 - Prob. 34ECh. 16.5 - Prob. 35ECh. 16.5 - Prob. 36ECh. 16.5 - Find the area of the surface cut from the...Ch. 16.5 - Find the area of the band cut from the paraboloid...Ch. 16.5 - Find the area of the region cut from the plane x +...Ch. 16.5 - Find the area of the portion of the surface x2 –...Ch. 16.5 - Prob. 41ECh. 16.5 - Prob. 42ECh. 16.5 - Find the area of the ellipse cut from the plane z...Ch. 16.5 - Find the area of the upper portion of the cylinder...Ch. 16.5 - Prob. 45ECh. 16.5 - Prob. 46ECh. 16.5 - Prob. 47ECh. 16.5 - Find the area of the surface 2x3/2 + 2y3/2 – 3z =...Ch. 16.5 - Prob. 49ECh. 16.5 - Prob. 50ECh. 16.5 - Prob. 51ECh. 16.5 - Prob. 52ECh. 16.5 - Prob. 53ECh. 16.5 - Find the area of the surfaces in Exercises...Ch. 16.5 - Use the parametrization
and Equation (5) to...Ch. 16.5 - Prob. 56ECh. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - Prob. 5ECh. 16.6 - Prob. 6ECh. 16.6 - Prob. 7ECh. 16.6 - In Exercises 1–8, integrate the given function...Ch. 16.6 - Integrate G(x, y, z) = x + y + z over the surface...Ch. 16.6 - Integrate G(x, y, z) = y + z over the surface of...Ch. 16.6 - Prob. 11ECh. 16.6 - Prob. 12ECh. 16.6 - Integrate G(x, y, z) = x + y + z over the portion...Ch. 16.6 - Prob. 14ECh. 16.6 - Prob. 15ECh. 16.6 - Prob. 16ECh. 16.6 - Prob. 17ECh. 16.6 - Prob. 18ECh. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - Prob. 21ECh. 16.6 - Prob. 22ECh. 16.6 - Prob. 23ECh. 16.6 - Prob. 24ECh. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - In Exercises 19–28, use a parametrization to find...Ch. 16.6 - Prob. 28ECh. 16.6 - Prob. 29ECh. 16.6 - Prob. 30ECh. 16.6 - In Exercises 31–36, use Equation (7) to find the...Ch. 16.6 - Prob. 32ECh. 16.6 - Prob. 33ECh. 16.6 - Prob. 34ECh. 16.6 - Prob. 35ECh. 16.6 - In Exercises 31–36, use Equation (7) to find the...Ch. 16.6 - Find the flux of the field through the surface...Ch. 16.6 - Find the flux of the field F(x, y, z) = 4xi + 4yj...Ch. 16.6 - Let S be the portion of the cylinder y = ex in the...Ch. 16.6 - Let S be the portion of the cylinder y = ln x in...Ch. 16.6 - Find the outward flux of the field F = 2xyi+ 2yzj...Ch. 16.6 - Find the outward flux of the field F = xzi + yzj +...Ch. 16.6 - Prob. 43ECh. 16.6 - Prob. 44ECh. 16.6 - Prob. 45ECh. 16.6 - Conical surface of constant density Find the...Ch. 16.6 - Prob. 47ECh. 16.6 - Prob. 48ECh. 16.6 - Prob. 49ECh. 16.6 - A surface S lies on the paraboloid directly above...Ch. 16.7 - In Exercises 1–6, find the curl of each vector...Ch. 16.7 - Prob. 2ECh. 16.7 - In Exercises 1–6, find the curl of each vector...Ch. 16.7 - Prob. 4ECh. 16.7 - Prob. 5ECh. 16.7 - Prob. 6ECh. 16.7 - In Exercises 7–12, use the surface integral in...Ch. 16.7 - Prob. 8ECh. 16.7 - In Exercises 7–12, use the surface integral in...Ch. 16.7 - Prob. 10ECh. 16.7 - In Exercises 7–12, use the surface integral in...Ch. 16.7 - In Exercises 7–12, use the surface integral in...Ch. 16.7 - Let n be the unit normal in the direction away...Ch. 16.7 - Prob. 14ECh. 16.7 - Prob. 15ECh. 16.7 - Prob. 16ECh. 16.7 - Prob. 17ECh. 16.7 - Prob. 18ECh. 16.7 - In Exercises 19–24, use the surface integral in...Ch. 16.7 - In Exercises 19–24, use the surface integral in...Ch. 16.7 - In Exercises 19–24, use the surface integral in...Ch. 16.7 - Prob. 22ECh. 16.7 - Prob. 23ECh. 16.7 - In Exercises 19–24, use the surface integral in...Ch. 16.7 - Prob. 25ECh. 16.7 - Verify Stokes’ Theorem for the vector field F =...Ch. 16.7 - Prob. 27ECh. 16.7 - Prob. 28ECh. 16.7 - Prob. 29ECh. 16.7 - Prob. 30ECh. 16.7 - Prob. 31ECh. 16.7 - Prob. 32ECh. 16.7 - Prob. 33ECh. 16.7 - Prob. 34ECh. 16.8 - In Exercises 1–8, find the divergence of the...Ch. 16.8 - Prob. 2ECh. 16.8 - Prob. 3ECh. 16.8 - Prob. 4ECh. 16.8 - Prob. 5ECh. 16.8 - Prob. 6ECh. 16.8 - Prob. 7ECh. 16.8 - Prob. 8ECh. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - Prob. 13ECh. 16.8 - Prob. 14ECh. 16.8 - Prob. 15ECh. 16.8 - Prob. 16ECh. 16.8 - Prob. 17ECh. 16.8 - Prob. 18ECh. 16.8 - Prob. 19ECh. 16.8 - In Exercises 9–20, use the Divergence Theorem to...Ch. 16.8 - Prob. 21ECh. 16.8 - Prob. 22ECh. 16.8 - Prob. 23ECh. 16.8 - Prob. 24ECh. 16.8 - Prob. 25ECh. 16.8 - Prob. 26ECh. 16.8 - Prob. 27ECh. 16.8 - Prob. 28ECh. 16.8 - Prob. 29ECh. 16.8 - Prob. 30ECh. 16.8 - Prob. 31ECh. 16.8 - Prob. 32ECh. 16.8 - Prob. 33ECh. 16.8 - Prob. 34ECh. 16.8 - Prob. 35ECh. 16.8 - Prob. 36ECh. 16 - Prob. 1GYRCh. 16 - How can you use line integrals to find the centers...Ch. 16 - Prob. 3GYRCh. 16 - Prob. 4GYRCh. 16 - Prob. 5GYRCh. 16 - Prob. 6GYRCh. 16 - Prob. 7GYRCh. 16 - Prob. 8GYRCh. 16 - Prob. 9GYRCh. 16 - Prob. 10GYRCh. 16 - How do you calculate the area of a parametrized...Ch. 16 - Prob. 12GYRCh. 16 - What is an oriented surface? What is the surface...Ch. 16 - Prob. 14GYRCh. 16 - Prob. 15GYRCh. 16 - Prob. 16GYRCh. 16 - Prob. 17GYRCh. 16 - Prob. 18GYRCh. 16 - The accompanying figure shows two polygonal paths...Ch. 16 - The accompanying figure shows three polygonal...Ch. 16 - Integrate over the circle r(t) = (a cos t)j + (a...Ch. 16 - Prob. 4PECh. 16 - Evaluate the integrals in Exercises 5 and 6.
5.
Ch. 16 - Prob. 6PECh. 16 - Prob. 7PECh. 16 - Integrate F = 3x2yi + (x3 + l)j + 9z2k around the...Ch. 16 - Prob. 9PECh. 16 - Evaluate the integrals in Exercises 9 and...Ch. 16 - Prob. 11PECh. 16 - Prob. 12PECh. 16 - Prob. 13PECh. 16 - Hemisphere cut by cylinder Find the area of the...Ch. 16 - Prob. 15PECh. 16 - Prob. 16PECh. 16 - Circular cylinder cut by planes Integrate g(x, y,...Ch. 16 - Prob. 18PECh. 16 - Prob. 19PECh. 16 - Prob. 20PECh. 16 - Prob. 21PECh. 16 - Prob. 22PECh. 16 - Prob. 23PECh. 16 - Prob. 24PECh. 16 - Prob. 25PECh. 16 - Prob. 26PECh. 16 - Prob. 27PECh. 16 - Prob. 28PECh. 16 - Which of the fields in Exercises 29–32 are...Ch. 16 - Prob. 30PECh. 16 - Which of the fields in Exercises 29–32 are...Ch. 16 - Prob. 32PECh. 16 - Prob. 33PECh. 16 - Prob. 34PECh. 16 - In Exercises 35 and 36, find the work done by each...Ch. 16 - In Exercises 35 and 36, find the work done by each...Ch. 16 - Finding work in two ways Find the work done...Ch. 16 - Flow along different paths Find the flow of the...Ch. 16 - Prob. 39PECh. 16 - Prob. 40PECh. 16 - Prob. 41PECh. 16 - Prob. 42PECh. 16 - Prob. 43PECh. 16 - Prob. 44PECh. 16 - Prob. 45PECh. 16 - Prob. 46PECh. 16 - Prob. 47PECh. 16 - Moment of inertia of a cube Find the moment of...Ch. 16 - Use Green’s Theorem to find the counterclockwise...Ch. 16 - Prob. 50PECh. 16 - Prob. 51PECh. 16 - Prob. 52PECh. 16 - In Exercises 53–56, find the outward flux of F...Ch. 16 - In Exercises 53–56, find the outward flux of F...Ch. 16 - Prob. 55PECh. 16 - Prob. 56PECh. 16 - Prob. 57PECh. 16 - Prob. 58PECh. 16 - Prob. 59PECh. 16 - Prob. 60PECh. 16 - Prob. 1AAECh. 16 - Use the Green’s Theorem area formula in Exercises...Ch. 16 - Prob. 3AAECh. 16 - Use the Green's Theorem area formula in Exercises...Ch. 16 - Prob. 5AAECh. 16 - Prob. 6AAECh. 16 - Prob. 7AAECh. 16 - Prob. 8AAECh. 16 - Prob. 9AAECh. 16 - Prob. 10AAECh. 16 - Prob. 11AAECh. 16 - Prob. 12AAECh. 16 - Archimedes’ principle If an object such as a ball...Ch. 16 - Prob. 14AAECh. 16 - Faraday’s law If E(t, x, y, z) and B(t, x, y, z)...Ch. 16 - Prob. 16AAECh. 16 - Prob. 17AAECh. 16 - Prob. 18AAECh. 16 - Prob. 19AAECh. 16 - Prob. 20AAECh. 16 - Prob. 21AAE
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- 2. Consider the vector force: F(x, y, z) = 2xye²i + (x²e² + y)j + (x²ye² — z)k. (A) [80%] Show that F satisfies the conditions for a conservative vector field, and find a potential function (x, y, z) for F. Remark: To find o, you must use the method explained in the lecture. (B) [20%] Use the Fundamental Theorem for Line Integrals to compute the work done by F on an object moves along any path from (0,1,2) to (2, 1, -8).arrow_forwardhelp pleasearrow_forwardIn each of Problems 1 through 4, draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t → ∞. If this behavior depends on the initial value of y at t = 0, describe the dependency.1. y′ = 3 − 2yarrow_forward
- B 2- The figure gives four points and some corresponding rays in the xy-plane. Which of the following is true? A B Angle COB is in standard position with initial ray OB and terminal ray OC. Angle COB is in standard position with initial ray OC and terminal ray OB. C Angle DOB is in standard position with initial ray OB and terminal ray OD. D Angle DOB is in standard position with initial ray OD and terminal ray OB.arrow_forwardtemperature in degrees Fahrenheit, n hours since midnight. 5. The temperature was recorded at several times during the day. Function T gives the Here is a graph for this function. To 29uis a. Describe the overall trend of temperature throughout the day. temperature (Fahrenheit) 40 50 50 60 60 70 5 10 15 20 25 time of day b. Based on the graph, did the temperature change more quickly between 10:00 a.m. and noon, or between 8:00 p.m. and 10:00 p.m.? Explain how you know. (From Unit 4, Lesson 7.) 6. Explain why this graph does not represent a function. (From Unit 4, Lesson 8.)arrow_forwardFind the area of the shaded region. (a) 5- y 3 2- (1,4) (5,0) 1 3 4 5 6 (b) 3 y 2 Decide whether the problem can be solved using precalculus, or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, use a graphical or numerical approach to estimate the solution. STEP 1: Consider the figure in part (a). Since this region is simply a triangle, you may use precalculus methods to solve this part of the problem. First determine the height of the triangle and the length of the triangle's base. height 4 units units base 5 STEP 2: Compute the area of the triangle by employing a formula from precalculus, thus finding the area of the shaded region in part (a). 10 square units STEP 3: Consider the figure in part (b). Since this region is defined by a complicated curve, the problem seems to require calculus. Find an approximation of the shaded region by using a graphical approach. (Hint: Treat the shaded regi as…arrow_forward
- Solve this differential equation: dy 0.05y(900 - y) dt y(0) = 2 y(t) =arrow_forwardSuppose that you are holding your toy submarine under the water. You release it and it begins to ascend. The graph models the depth of the submarine as a function of time. What is the domain and range of the function in the graph? 1- t (time) 1 2 4/5 6 7 8 -2 -3 456700 -4 -5 -6 -7 d (depth) -8 D: 00 t≤ R:arrow_forward0 5 -1 2 1 N = 1 to x = 3 Based on the graph above, estimate to one decimal place the average rate of change from x =arrow_forwardComplete the description of the piecewise function graphed below. Use interval notation to indicate the intervals. -7 -6 -5 -4 30 6 5 4 3 0 2 1 -1 5 6 + -2 -3 -5 456 -6 - { 1 if x Є f(x) = { 1 if x Є { 3 if x Єarrow_forwardComplete the description of the piecewise function graphed below. 6 5 -7-6-5-4-3-2-1 2 3 5 6 -1 -2 -3 -4 -5 { f(x) = { { -6 if -6x-2 if -2< x <1 if 1 < x <6arrow_forwardLet F = V where (x, y, z) x2 1 + sin² 2 +z2 and let A be the line integral of F along the curve x = tcost, y = t sint, z=t, starting on the plane z = 6.14 and ending on the plane z = 4.30. Then sin(3A) is -0.598 -0.649 0.767 0.278 0.502 0.010 -0.548 0.960arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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