Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Chapter 9.P, Problem 6P
A subtangent is a portion of the
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(7) (12 points) Let F(x, y, z) = (y, x+z cos yz, y cos yz).
Ꮖ
(a) (4 points) Show that V x F = 0.
(b) (4 points) Find a potential f for the vector field F.
(c) (4 points) Let S be a surface in R3 for which the Stokes' Theorem is valid. Use
Stokes' Theorem to calculate the line integral
Jos
F.ds;
as denotes the boundary of S. Explain your answer.
(3) (16 points) Consider
z = uv,
u = x+y,
v=x-y.
(a) (4 points) Express z in the form z = fog where g: R² R² and f: R² →
R.
(b) (4 points) Use the chain rule to calculate Vz = (2, 2). Show all intermediate
steps otherwise no credit.
(c) (4 points) Let S be the surface parametrized by
T(x, y) = (x, y, ƒ (g(x, y))
(x, y) = R².
Give a parametric description of the tangent plane to S at the point p = T(x, y).
(d) (4 points) Calculate the second Taylor polynomial Q(x, y) (i.e. the quadratic
approximation) of F = (fog) at a point (a, b). Verify that
Q(x,y) F(a+x,b+y).
=
(6) (8 points) Change the order of integration and evaluate
(z +4ry)drdy .
So S√ ²
0
Chapter 9 Solutions
Calculus (MindTap Course List)
Ch. 9.1 - Show that y=23ex+e2x is a solution of the...Ch. 9.1 - Verify that y=tcostt is a solution of the...Ch. 9.1 - a For what values of r does the function y=erx...Ch. 9.1 - Prob. 4ECh. 9.1 - Which of the following functions are solutions of...Ch. 9.1 - a Show that every member of the family of...Ch. 9.1 - a What can you say about a solution of the...Ch. 9.1 - a What can you say about the graph of a solution...Ch. 9.1 - Prob. 9ECh. 9.1 - The Fitzhugh-Nagumo model for the electrical...
Ch. 9.1 - Explain why the functions with the given graphs...Ch. 9.1 - The function with the given graph is a solution of...Ch. 9.1 - Match the differential equations with the solution...Ch. 9.1 - Suppose you have just poured a cup of freshly...Ch. 9.1 - Prob. 15ECh. 9.1 - Von Bertalanffys equation states that the rate of...Ch. 9.1 - Differential equations have been used extensively...Ch. 9.2 - A direction field for the differential equation...Ch. 9.2 - A direction field for the differential equation...Ch. 9.2 - Prob. 3ECh. 9.2 - Prob. 4ECh. 9.2 - 36 Match the differential equation with its...Ch. 9.2 - 36 Match the differential equation with its...Ch. 9.2 - Prob. 7ECh. 9.2 - Prob. 8ECh. 9.2 - Prob. 9ECh. 9.2 - 910 Sketch a direction field for the differential...Ch. 9.2 - Prob. 11ECh. 9.2 - Prob. 12ECh. 9.2 - Prob. 13ECh. 9.2 - Prob. 14ECh. 9.2 - Prob. 15ECh. 9.2 - Prob. 16ECh. 9.2 - Use a computer algebra system to draw a direction...Ch. 9.2 - Make a rough sketch of a direction field for the...Ch. 9.2 - a Use Eulers method with each of the following...Ch. 9.2 - A direction field for a differential equation is...Ch. 9.2 - Prob. 21ECh. 9.2 - Prob. 22ECh. 9.2 - Use Eulers method with step size 0.1 to estimate...Ch. 9.2 - Prob. 24ECh. 9.2 - a Program a calculator or computer to use Eulers...Ch. 9.2 - a Program your computer algebra system, using...Ch. 9.2 - The figure shows a circuit containing an...Ch. 9.2 - In Exercise 9.1.14 we considered a 95C cup of...Ch. 9.3 - 110 Solve the differential equation. dydx=3x2y2Ch. 9.3 - Prob. 2ECh. 9.3 - Prob. 3ECh. 9.3 - 110 Solve the differential equation. y+xey=0Ch. 9.3 - Prob. 5ECh. 9.3 - Prob. 6ECh. 9.3 - Prob. 7ECh. 9.3 - Prob. 8ECh. 9.3 - Prob. 9ECh. 9.3 - Prob. 10ECh. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - Prob. 12ECh. 9.3 - Prob. 13ECh. 9.3 - Prob. 14ECh. 9.3 - 1118 Find the solution of the differential...Ch. 9.3 - Prob. 16ECh. 9.3 - Prob. 17ECh. 9.3 - Prob. 18ECh. 9.3 - Find an equation of the curve that passes through...Ch. 9.3 - Find the function f such that...Ch. 9.3 - Prob. 21ECh. 9.3 - Prob. 22ECh. 9.3 - Prob. 23ECh. 9.3 - Prob. 24ECh. 9.3 - Prob. 25ECh. 9.3 - Prob. 26ECh. 9.3 - a Use a computer algebra system to draw a...Ch. 9.3 - 2728 a Use a computer algebra system to draw a...Ch. 9.3 - 2932 Find the orthogonal trajectories of the...Ch. 9.3 - 2932 Find the orthogonal trajectories of the...Ch. 9.3 - 2932 Find the orthogonal trajectories of the...Ch. 9.3 - 2932 Find the orthogonal trajectories of the...Ch. 9.3 - 3335 An integral equation is an equation that...Ch. 9.3 - 3335 An integral equation is an equation that...Ch. 9.3 - Prob. 35ECh. 9.3 - Find a function f such that f(3)=2 and...Ch. 9.3 - Prob. 37ECh. 9.3 - In Exercise 9.2.28 we discussed a differential...Ch. 9.3 - Prob. 39ECh. 9.3 - In an elementary chemical reaction, single...Ch. 9.3 - In contrast to the situation of Exercise 40,...Ch. 9.3 - Prob. 42ECh. 9.3 - Prob. 43ECh. 9.3 - A certain small country has 10 billion in paper...Ch. 9.3 - Prob. 45ECh. 9.3 - Prob. 46ECh. 9.3 - A vat with 500 gallons of beer contains 4 alcohol...Ch. 9.3 - A tank contains 1000 L of pure water. Brine that...Ch. 9.3 - Prob. 49ECh. 9.3 - An object of mass m is moving horizontally through...Ch. 9.3 - Prob. 51ECh. 9.3 - Prob. 52ECh. 9.3 - Let A(t) be the area of a tissue culture at time t...Ch. 9.3 - Prob. 54ECh. 9.4 - 12 A population grows according to the given...Ch. 9.4 - 1-2 A population grows according to the given...Ch. 9.4 - Suppose that a population develops according to...Ch. 9.4 - Suppose that a population grows according to a...Ch. 9.4 - The Pacific halibut fishery has been modeled by...Ch. 9.4 - Suppose a population P(t) satisfies...Ch. 9.4 - Suppose a population grows according to a logistic...Ch. 9.4 - The table gives the number of yeast cells in a new...Ch. 9.4 - Prob. 9ECh. 9.4 - Prob. 10ECh. 9.4 - One model for the spread of a rumor is that the...Ch. 9.4 - Biologists stocked a lake with 400 fish and...Ch. 9.4 - Prob. 13ECh. 9.4 - Prob. 14ECh. 9.4 - Prob. 15ECh. 9.4 - Prob. 16ECh. 9.4 - Consider a population P=P(t) with constant...Ch. 9.4 - Let c be a positive number. A differential...Ch. 9.4 - Lets modify the logistic differential equation of...Ch. 9.4 - Consider the differential equation...Ch. 9.4 - There is considerable evidence to support the...Ch. 9.4 - Prob. 22ECh. 9.4 - In a seasonal-growth model, a periodic function of...Ch. 9.4 - Prob. 24ECh. 9.4 - Prob. 25ECh. 9.5 - 14 Determine whether the differential equation is...Ch. 9.5 - Prob. 2ECh. 9.5 - Prob. 3ECh. 9.5 - Prob. 4ECh. 9.5 - 514 Solve the differential equation. y+y=1Ch. 9.5 - 514 Solve the differential equation. yy=exCh. 9.5 - 514 Solve the differential equation. y=xyCh. 9.5 - 514 Solve the differential equation....Ch. 9.5 - Prob. 9ECh. 9.5 - Prob. 10ECh. 9.5 - Prob. 11ECh. 9.5 - Prob. 12ECh. 9.5 - Prob. 13ECh. 9.5 - Prob. 14ECh. 9.5 - Prob. 15ECh. 9.5 - Prob. 16ECh. 9.5 - Prob. 17ECh. 9.5 - Prob. 18ECh. 9.5 - 1520 Solve the initial-value problem....Ch. 9.5 - Prob. 20ECh. 9.5 - Prob. 21ECh. 9.5 - Prob. 22ECh. 9.5 - A Bernoulli differential equation named after...Ch. 9.5 - 2425 Use the method of Exercise 23 to solve the...Ch. 9.5 - Prob. 25ECh. 9.5 - Prob. 26ECh. 9.5 - Prob. 27ECh. 9.5 - Prob. 28ECh. 9.5 - Prob. 29ECh. 9.5 - Prob. 30ECh. 9.5 - Prob. 31ECh. 9.5 - Prob. 32ECh. 9.5 - In Section 9.3 we looked at mixing problems in...Ch. 9.5 - Prob. 34ECh. 9.5 - An object with mass m is dropped from rest and we...Ch. 9.5 - If we ignore air resistance, we can conclude that...Ch. 9.5 - Prob. 37ECh. 9.5 - To account for seasonal variation in the logistic...Ch. 9.6 - For each predator-prey system, determine which of...Ch. 9.6 - Each system of differential equations is a model...Ch. 9.6 - Prob. 3ECh. 9.6 - Prob. 4ECh. 9.6 - 56 A phase trajectory is shown for populations of...Ch. 9.6 - 56 A phase trajectory is shown for populations of...Ch. 9.6 - 78 Graphs of populations of two species are shown....Ch. 9.6 - Prob. 8ECh. 9.6 - Prob. 9ECh. 9.6 - Populations of aphids and ladybugs are modeled by...Ch. 9.6 - In Example 1 we used Lotka-Volterra equations to...Ch. 9.6 - In Exercise 10 we modeled populations of aphids...Ch. 9.R - Prob. 1CCCh. 9.R - Prob. 2CCCh. 9.R - Prob. 3CCCh. 9.R - Prob. 4CCCh. 9.R - Prob. 5CCCh. 9.R - Prob. 6CCCh. 9.R - Prob. 7CCCh. 9.R - Prob. 8CCCh. 9.R - a Write Lotka-Volterra equations to model...Ch. 9.R - Prob. 1TFQCh. 9.R - Prob. 2TFQCh. 9.R - Prob. 3TFQCh. 9.R - Determine whether the statement is true or false....Ch. 9.R - Prob. 5TFQCh. 9.R - Determine whether the statement is true or false....Ch. 9.R - Prob. 7TFQCh. 9.R - Prob. 1ECh. 9.R - a Sketch a direction field for the differential...Ch. 9.R - a A direction field for the differential equation...Ch. 9.R - Prob. 4ECh. 9.R - Prob. 5ECh. 9.R - Prob. 6ECh. 9.R - 58 Solve the differential equation. 2yey2y=2x+3xCh. 9.R - 58 Solve the differential equation. x2yy=2x3e1/xCh. 9.R - 911 Solve the initial-value problem....Ch. 9.R - 911 Solve the initial-value problem....Ch. 9.R - Prob. 11ECh. 9.R - Prob. 12ECh. 9.R - 1314 Find the orthogonal trajectories of the...Ch. 9.R - Prob. 14ECh. 9.R - Prob. 15ECh. 9.R - a The population of the world was 6.1 billion in...Ch. 9.R - Prob. 17ECh. 9.R - Prob. 18ECh. 9.R - One model for the spread of an epidemic is that...Ch. 9.R - Prob. 20ECh. 9.R - Prob. 21ECh. 9.R - Populations of birds and insects are modeled by...Ch. 9.R - Prob. 23ECh. 9.R - Prob. 24ECh. 9.P - Find all functions f such that f is continuous and...Ch. 9.P - Prob. 2PCh. 9.P - Prob. 3PCh. 9.P - Find all functions f that satisfy the equation...Ch. 9.P - Prob. 5PCh. 9.P - A subtangent is a portion of the x-axis that lies...Ch. 9.P - A peach pie is taken out of the oven at 5:00 PM....Ch. 9.P - Snow began to fall during the morning of February...Ch. 9.P - A dog sees a rabbit running in a straight line...Ch. 9.P - a Suppose that the dog in Problem 9 runs twice as...Ch. 9.P - A planning engineer for a new alum plant must...Ch. 9.P - Prob. 12PCh. 9.P - Prob. 13PCh. 9.P - Prob. 14PCh. 9.P - Prob. 15PCh. 9.P - a An outfielder fields a baseball 280 ft away from...
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