Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 9.3, Problem 24E
Solve the equation e−yy′ + cos x = 0 and graph several members of the family of solutions. How does the solution curve change as the constant C varies?
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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.
temperature 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.)
Find the area of the shaded region.
(a)
5-
y
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(1,4)
(5,0)
1
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4
5
6
(b)
3 y
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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
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as…
Chapter 9 Solutions
Calculus: Early Transcendentals
Ch. 9.1 - Show that y=23ex+e2x is a solution of the...Ch. 9.1 - Verify that y = t cos t t is a solution of the...Ch. 9.1 - (a) For what values of r does the function y = erx...Ch. 9.1 - (a) For what values of k does the function y = cos...Ch. 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 - A population is modeled by the differential...Ch. 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 - Psychologists interested in learning theory study...Ch. 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 y'...Ch. 9.2 - A direction field for the differential equation...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Match the differential equation with its direction...Ch. 9.2 - Prob. 7ECh. 9.2 - Use the direction field labeled III (above) to...Ch. 9.2 - Sketch a direction field for the differential...Ch. 9.2 - 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 - (a) Use Eulers method with each of the following...Ch. 9.2 - A direction field for a differential equation is...Ch. 9.2 - Use Eulers method with step size 0.5 to compute...Ch. 9.2 - Prob. 22ECh. 9.2 - Prob. 23ECh. 9.2 - (a) Use Eulers method with step size 0.2 to...Ch. 9.2 - The figure shows a circuit containing an...Ch. 9.3 - Solve the differential equation. 1. dydx=3x2y2Ch. 9.3 - Solve the differential equation. 2. dydx=xyCh. 9.3 - Solve the differential equation. 3. xyy=x2+1Ch. 9.3 - Solve the differential equation. 4. y+xey=0Ch. 9.3 - Solve the differential equation. 5. (ey1)y=2+cosxCh. 9.3 - Solve the differential equation. 6....Ch. 9.3 - Solve the differential equation. 7. ddt=tsecet2Ch. 9.3 - Solve the differential equation. 8....Ch. 9.3 - Solve the differential equation. 9. dpdt=t2pp+t21Ch. 9.3 - Solve the differential equation. 10. dzdt+et+z=0Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find the solution of the differential equation...Ch. 9.3 - Find an equation of the curve that passes through...Ch. 9.3 - Find the function f such that f(x) = xf(x) x and...Ch. 9.3 - Solve the differential equation y = x + y by...Ch. 9.3 - Solve the differential equation xy = y + xey/x by...Ch. 9.3 - (a) Solve the differential equation y=2x1y2. (b)...Ch. 9.3 - Solve the equation eyy + cos x = 0 and graph...Ch. 9.3 - Find the orthogonal trajectories of the family of...Ch. 9.3 - Find the orthogonal trajectories of the family of...Ch. 9.3 - Find the orthogonal trajectories of the family of...Ch. 9.3 - Find the orthogonal trajectories of the family of...Ch. 9.3 - An integral equation is an equation that contains...Ch. 9.3 - An integral equation is an equation that contains...Ch. 9.3 - An integral equation is an equation that contains...Ch. 9.3 - Find a function f such that f(3) = 2 and (t2 +...Ch. 9.3 - Solve the initial-value problem in Exercise 9.2.27...Ch. 9.3 - In Exercise 9.2.28 we discussed a differential...Ch. 9.3 - In Exercise 9.1.15 we formulated a model for...Ch. 9.3 - In an elementary chemical reaction, single...Ch. 9.3 - In contrast to the situation of Exercise 40,...Ch. 9.3 - A sphere with radius 1 m has temperature 15C. It...Ch. 9.3 - A glucose solution is administered intravenously...Ch. 9.3 - A certain small country has 10 billion in paper...Ch. 9.3 - A tank contains 1000 L of brine with 15 kg of...Ch. 9.3 - The air in a room with volume 180 m3 contains...Ch. 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 - When a raindrop falls, it increases in size and so...Ch. 9.3 - An object of mass m is moving horizontally through...Ch. 9.3 - Allometric growth in biology refers to...Ch. 9.3 - A model for tumor growth is given by the Gompertz...Ch. 9.3 - According to Newtons Law of Universal Gravitation,...Ch. 9.4 - A population grows according to the given logistic...Ch. 9.4 - A population grows according to the given logistic...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 - Prob. 8ECh. 9.4 - The population of the world was about 6.1 billion...Ch. 9.4 - Prob. 10ECh. 9.4 - Prob. 11ECh. 9.4 - Biologists stocked a lake with 400 fish and...Ch. 9.4 - (a) Show that if P satisfies the logistic equation...Ch. 9.4 - For a fixed value of M (say M = 10), the family of...Ch. 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 - Prob. 21ECh. 9.4 - Another model for a growth function for a limited...Ch. 9.4 - In a seasonal-growth model, a periodic function of...Ch. 9.4 - Prob. 24ECh. 9.4 - Prob. 25ECh. 9.5 - Determine whether the differential equation is...Ch. 9.5 - Determine whether the differential equation is...Ch. 9.5 - Determine whether the differential equation is...Ch. 9.5 - Determine whether the differential equation is...Ch. 9.5 - Solve the differential equation. 5. y' + y = 1Ch. 9.5 - Solve the differential equation. 6. y' y = exCh. 9.5 - Solve the differential equation. 7. y' = x yCh. 9.5 - Solve the differential equation. 8. 4x3y + x4y' =...Ch. 9.5 - Solve the differential equation. 9. xy+y=xCh. 9.5 - Solve the differential equation. 10. 2xy+y=2xCh. 9.5 - Solve the differential equation. 11. xy2y=x2,x0Ch. 9.5 - Solve the differential equation. 12. y+2xy=1Ch. 9.5 - Solve the differential equation. 13....Ch. 9.5 - Solve the differential equation. 14....Ch. 9.5 - Solve the initial-value problem. 15....Ch. 9.5 - Solve the initial-value problem. 16....Ch. 9.5 - Solve the initial-value problem. 17....Ch. 9.5 - Solve the initial-value problem. 18....Ch. 9.5 - Solve the initial-value problem. 19....Ch. 9.5 - Solve the initial-value problem. 20....Ch. 9.5 - Solve the differential equation and use a...Ch. 9.5 - Prob. 22ECh. 9.5 - A Bernoulli differential equation (named after...Ch. 9.5 - Use the method of Exercise 23 to solve the...Ch. 9.5 - Use the method of Exercise 23 to solve the...Ch. 9.5 - Solve the second-order equation xy" + 2y' = 12x2...Ch. 9.5 - Prob. 27ECh. 9.5 - Prob. 28ECh. 9.5 - The figure shows a circuit containing an...Ch. 9.5 - Prob. 30ECh. 9.5 - Let P(t) be the performance level of someone...Ch. 9.5 - Prob. 32ECh. 9.5 - In Section 9.3 we looked at mixing problems in...Ch. 9.5 - A tank with a capacity of 400 L is full of a...Ch. 9.5 - An object with mass m is dropped from rest and we...Ch. 9.5 - Prob. 36ECh. 9.5 - Show that the substitution z = 1/P transforms the...Ch. 9.5 - Prob. 38ECh. 9.6 - For each predator-prey system, determine which of...Ch. 9.6 - Each system of differential equations is a model...Ch. 9.6 - The system of differential equations...Ch. 9.6 - Prob. 4ECh. 9.6 - Prob. 5ECh. 9.6 - Prob. 6ECh. 9.6 - Prob. 7ECh. 9.6 - Graphs of populations of two species are shown....Ch. 9.6 - Populations of aphids and ladybugs are modeled by...Ch. 9.6 - Prob. 11ECh. 9 - (a) What is a differential equation? (b) What is...Ch. 9 - What can you say about the solutions of the...Ch. 9 - What is a direction field for the differential...Ch. 9 - Explain how Euler's method works.Ch. 9 - What is a separable differential equation? How do...Ch. 9 - What is a first-order linear differential...Ch. 9 - (a) Write a differential equation that expresses...Ch. 9 - (a) Write the logistic differential equation. (b)...Ch. 9 - (a) Write Lotka-Volterra equations to model...Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Determine whether the statement is true or false....Ch. 9 - Prob. 6RQCh. 9 - Determine whether the statement is true or false....Ch. 9 - (a) A direction field for the differential...Ch. 9 - (a) Sketch a direction field for the differential...Ch. 9 - (a) A direction field for the differential...Ch. 9 - (a) Use Euler's method with step size 0.2 to...Ch. 9 - Solve the differential equation. 5. y=xesinxycosxCh. 9 - Solve the differential equation. 6. dxdy=1t+xtxCh. 9 - Solve the differential equation. 7. 2yey2y=2x+3xCh. 9 - Solve the differential equation. 8. x2yy=2x3e1/xCh. 9 - Solve the initial-value problem. 9....Ch. 9 - Solve the initial-value problem. 10. (1 + cos x)...Ch. 9 - Solve the initial-value problem. 11. xy' y = x ln...Ch. 9 - Solve the initial-value problem y' = 3x2ey, y(0) =...Ch. 9 - Find the orthogonal trajectories of the family of...Ch. 9 - Find the orthogonal trajectories of the family of...Ch. 9 - (a) Write the solution of the initial-value...Ch. 9 - (a) The population of the world was 6.1 billion in...Ch. 9 - The von Bertalanffy growth model is used to...Ch. 9 - A tank contains 100 L of pure water. Brine that...Ch. 9 - One model for the spread of an epidemic is that...Ch. 9 - The Brentano-Stevens Law in psychology models the...Ch. 9 - The transport of a substance across a capillary...Ch. 9 - Populations of birds and insects are modeled by...Ch. 9 - Suppose the model of Exercise 22 is replaced by...Ch. 9 - Barbara weighs 60 kg and is on a diet of 1600...Ch. 9 - Find all functions f such that f' is continuous...Ch. 9 - A student forgot the Product Rule for...Ch. 9 - Let f be a function with the property that f(0) =...Ch. 9 - Find all functions f that satisfy the equation...Ch. 9 - Find the curve y = f(x) such that f(x) 0, f(0) =...Ch. 9 - A subtangent is a portion of the x-axis that lies...Ch. 9 - A peach pie is taken out of the oven at 5:00 pm....Ch. 9 - Snow began to fall during the morning of February...Ch. 9 - A dog sees a rabbit running in a straight line...Ch. 9 - (a) Suppose that the dog in Problem 9 runs twice...Ch. 9 - A planning engineer for a new alum plant must...Ch. 9 - Find the curve that passes through the point (3,...Ch. 9 - Recall that the normal line to a curve at a point...Ch. 9 - Find all curves with the properly that if the...Ch. 9 - Find all curves with the property that if a line...
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