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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation (7)x y' + 0 Initial Condition y(2) = 2
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation y' + y tan x = sec x + cos x Initial Condition y(0) = 1
In Exercises solve the differential equation. xy' (x + 1)y = 0
In Exercises solve the differential equation. dy dx 105
In Exercises solve the differential equation. (2 + x)y' xy = 0
In Exercises write and solve the differential equation that models the verbal statement.The rate of change of y with respect to t is proportional to 50 - t.
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation y' cos²x + y − 1 = 0 Initial Condition y(0) = 5
In Exercises write and solve the differential equation that models the verbal statement.The rate of change of y with respect to t is inversely proportional to the cube of t.
In Exercises solve the differential equation. dy dx = y + 8
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation x³y² + 2y = el/x² Initial Condition y(1) = e
In Exercises solve the differential equation. dy dx (3 + y)²
In Exercises use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use n steps of size h. y' = 5x - 2y, y(0) = 2,
In Exercises (a) Sketch an approximate solution of the differential equation satisfying the given initial condition by hand on the slope field(b) Find the particular solution that satisfies the
In Exercises solve the differential equation. dy dx || 2x - 5x²
In Exercises use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use n steps of size h. y' = x - y, y(0) = 4, n
In Exercises (a) Sketch an approximate solution of the differential equation satisfying the given initial condition by hand on the slope field(b) Find the particular solution that satisfies the
In Exercises(a) Sketch the slope field for the differential equation(b) Use the slope field to sketch the solution that passes through the given point. Use a graphing utility to verify your results
In Exercises a medical researcher wants to determine the concentration C (in moles per liter) of a tracer drug injected into a moving fluid. Solve this problem by considering a single-compartment
In Exercises(a) Sketch the slope field for the differential equation(b) Use the slope field to sketch the solution that passes through the given point. Use a graphing utility to verify your results
In Exercises solve the first-order linear differential equation.y' + y tan x = sec x
In Exercises solve the first-order linear differential equation.y' - 3x²y = ex³
In Exercises a differential equation and its slope field are given. Complete the table by determining the slopes (if possible) in the slope field at the given points. X y dy/dx -4 -2 2 0 02 4 4 6 8 48
In Exercises a medical researcher wants to determine the concentration C (in moles per liter) of a tracer drug injected into a moving fluid. Solve this problem by considering a single-compartment
In Exercises solve the first-order linear differential equation.y' + 3y = e3x
In Exercises a differential equation and its slope field are given. Complete the table by determining the slopes (if possible) in the slope field at the given points. X y dy/dx -4 -2 2 0 02 4 4 6 8 48
In Exercises use integration to find a general solution of the differential equation. dy dx = e2-x
Biomass is a measure of the amount of living matter in an ecosystem. Suppose the biomass s(t) in a given ecosystem increases at a rate of about 3.5 tons per year, and decreases by about 1.9% per
In Exercises solve the first-order linear differential equation.(x - 1)y' + y = x² - 1
In Exercises use integration to find a general solution of the differential equation. dy dx 2e³x
Show that the logistic equationcan be written asWhat can you conclude about the graph of the logistic equation? y L 1 + be-kt
In Exercises solve the first-order linear differential equation.(y - 1) sin x dx - dy = 0
The cylindrical water tank shown in the figure has a height of 18 feet. When the tank is full, a circular valve is opened at the bottom of the tank. After 30 minutes, the depth of the water is 12
In Exercises solve the first-order linear differential equation.(y + 1) cos x dx - dy = 0
In Exercises use integration to find a general solution of the differential equation. dy dx 2 sin x
In Exercises solve the first-order linear differential equation.y' + 2xy = 10x
In Exercises solve the first-order linear differential equation. dy dx + (1) y = = 6x + 2
In Exercises solve the first-order linear differential equation. dy + dx X 3x - 5 = 3x
Suppose the tank in Exercise 6 has a height of 20 feet and a radius of 8 feet, and the valve is circular with a radius of 2 inches. The tank is full when the valve is opened. How long will it take
In Exercises use integration to find a general solution of the differential equation. dy dx = cos 2x
In Exercises determine whether the differential equation is linear. Explain your reasoning. 2-у y y 5x
Torricelli’s Law states that water will flow from an opening at the bottom of a tank with the same speed that it would attain falling from the surface of the water to the opening. One of the forms
In Exercises solve the first-order linear differential equation.y' - y = 16
In Exercises use integration to find a general solution of the differential equation. dy dx = 3x² 8x
Another model that can be used to represent population growth is the Gompertz equation, which is the solution of the differential equationwhere k is a constant and L is the carrying capacity.(a)
In Exercises determine whether the differential equation is linear. Explain your reasoning.y' - y sin x = xy²
Determine whether the function y = 2 sin 2x is a solution of the differential equation y'" - 8y = 0.
In Exercises determine whether the differential equation is linear. Explain your reasoning.2xy - y' In x = y
In Exercises determine whether the differential equation is linear. Explain your reasoning.x³y' + xy = ex + 1
Determine whether the function y = x³ is a solution of the differential equation 2xy' + 4y= 10x³.
Let ƒ be a twice-differentiable real-valued function satisfyingwhere g(x) ≥ 0 for all real x. Prove that |ƒ(x)| is bounded. (x),ƒ (x)8x— = (x),£ + (x)ƒ
It is known that y = ekt is a solution of the differential equation y" - 16y = 0. Find the values of k.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.A slope field shows that the slope at the point (1, 1) is 6. This
Prove that if the family of integral curves of the differential equationis cut by the line x = k, the tangents at the points of intersection are concurrent. dy dx + p(x)y= g(x), p(x) · g(x) = 0 .
Repeat Exercise 93 for which the exact solution of the differential equationwhere y(0) = 1, is y = x - 1 + 2e-x.Data from in Exercises 93The exact solution of the differential equationwhere y(0) = 4,
The exact solution of the differential equationwhere y(0) = 4, is y = 4e-2x.(a) Use a graphing utility to complete the table, where y is the exact value of the solution, y1 is the approximate
It is known that y = A sin ωt is a solutionof the differential equation y" + 16y = 0. Find the values of ω.
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Slope fields represent the general solutions of differential
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The general solution of a differential equation is y = -4.9x² +
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If y = ƒ(x) is a solution of a first-order differential equation,
The graph shows a solution of one of the following differential equations. Determine the correct equation. Explain your reasoning.(a) y' = xy(b) y' = 4x/y(c) y' = - 4xy (d) y' = 4 - xy y X
In Exercises complete the table using the exact solution of the differential equation and two approximations obtained using Euler's Method to approximate the particular solution of the differential
Describe how to use Euler’s Method to approximate a particular solution of a differential equation.
It is known that y = Cekx is a solution of the differential equation y' = 0.07y. Is it possible to determine C or k from the information given? If so, find its value.
Explain how to interpret a slope field.
In Exercises complete the table using the exact solution of the differential equation and two approximations obtained using Euler's Method to approximate the particular solution of the differential
In your own words, describe the difference between a general solution of a differential equation and a particular solution.
A not uncommon calculus mistake is to believe that the prod- uct rule for derivatives says that (ƒg)' = ƒ'g'. If ƒ(x) = ex²,determine, with proof, whether there exists an open interval(a, b) and
In Exercises use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size h. y' = = cos x + sin y, y(0)
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The families x² + y² = 2Cy and x² + y² = 2Kx are mutually
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The differential equation y' = xy - 2y + x - 2 can be written in
In Exercises complete the table using the exact solution of the differential equation and two approximations obtained using Euler's Method to approximate the particular solution of the differential
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The function y = 0 is always a solution of a differential equation
In Exercises use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size h. y' = 0.5x(3-y),
In Exercises solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M(x, y) dx + N(x, y) dy = 0, where M and N are homogeneous
In Exercises use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size h. y' = exy, y(0) = 1, n =
In Exercises solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M(x, y) dx + N(x, y) dy = 0, where M and N are homogeneous
In Exercises solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M(x, y) dx + N(x, y) dy = 0, where M and N are homogeneous
In Exercises use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size h. h 20, h 0.05 = y' = x + y,
In Exercises use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size h. y' = 3x - 2y, y(0) = 3, n
In Exercises solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M(x, y) dx + N(x, y) dy = 0, where M and N are homogeneous
In Exercises determine whether the function is homogeneous, and if it is, determine its degree. A function ƒ(x, y) is homogeneous of degree n if ƒ(tx, ty) = tnƒ(x, y). f(x, y) = tan y X
In Exercises solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M(x, y) dx + N(x, y) dy = 0, where M and N are homogeneous
In Exercises use a computer algebra system to (a) Graph the slope field for the differential equation(b) Graph the solution satisfying the specified initial condition = = dy dx 1 IC 2 ؛
In Exercises determine whether the function is homogeneous, and if it is, determine its degree. A function ƒ(x, y) is homogeneous of degree n if ƒ(tx, ty) = tnƒ(x, y). X f(x, y) = 2 In- y
In Exercises determine whether the function is homogeneous, and if it is, determine its degree. A function ƒ(x, y) is homogeneous of degree n if ƒ(tx, ty) = tnƒ(x, y). f(x, y) = 2 In xy
In Exercises solve the homogeneous differential equation in terms of x and y. A homogeneous differential equation is an equation of the form M(x, y) dx + N(x, y) dy = 0, where M and N are homogeneous
In Exercises use Euler’s Method to make a table of values for the approximate solution of the differential equation with the specified initial value. Use steps of size h. y' = x + y, y(0) = 2, n =
In Exercises use a computer algebra system to (a) Graph the slope field for the differential equation(b) Graph the solution satisfying the specified initial condition dy dx 0.4y(3x), y(0) = 1
In Exercises determine whether the function is homogeneous, and if it is, determine its degree. A function ƒ(x, y) is homogeneous of degree n if ƒ(tx, ty) = tnƒ(x, y). f(x, y) = tan(x + y)
In Exercises determine whether the function is homogeneous, and if it is, determine its degree. A function ƒ(x, y) is homogeneous of degree n if ƒ(tx, ty) = tnƒ(x, y). f(x, y) = ху √x² + y²
In Exercises use a computer algebra system to (a) Graph the slope field for the differential equation(b) Graph the solution satisfying the specified initial condition dy dx = 0.2x(2y), y(0) = 9
In Exercises determine whether the function is homogeneous, and if it is, determine its degree. A function ƒ(x, y) is homogeneous of degree n if ƒ(tx, ty) = tnƒ(x, y). f(x, y) x²y² √x² + y²
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The diffeTruerential equation modeling exponential growth is dy/dx
In Exercises use a computer algebra system to (a) Graph the slope field for the differential equation(b) Graph the solution satisfying the specified initial condition dy dx 0.02y(10 - y), y(0) =
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If prices are rising at a rate of 0.5% per month, then they
Use the slope field for the differential equation y' = 1/x, where x > 0, to sketch the graph of the solution that satisfies each given initial condition. Then make a conjecture about the behavior
In Exercises determine whether the function is homogeneous, and if it is, determine its degree. A function ƒ(x, y) is homogeneous of degree n if ƒ(tx, ty) = tnƒ(x, y). f(x, y) = x³ + 3x²y² -
In Exercises use a computer algebra system to (a) Graph the slope field for the differential equation(b) Graph the solution satisfying the specified initial condition dy dx = 4 y, y(0) = 6
In Exercises determine whether the function is homogeneous, and if it is, determine its degree. A function ƒ(x, y) is homogeneous of degree n if ƒ(tx, ty) = tnƒ(x, y). f(x, y) = x³ 4xy² + y²
In Exercises use a computer algebra system to (a) Graph the slope field for the differential equation(b) Graph the solution satisfying the specified initial condition dy dx = 0.25y, y(0) 4

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