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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

In Exercises sketch the region bounded by the graphs of the equations and find the area of the region. y = -x² + 3x + 1, y = -x + 1
In Exercises find the area of the region by integrating (a) With respect to x and (b) With respect to y. (c) Compare your results. Which method is simpler? In general, will this method always be simpler than the other one? Why or why not? x² y = y = 6 x 10 8 6 4 + -6-4-2 -2 2 46 X
In Exercises sketch the region bounded by the graphs of the equations and find the area of the region. f(x) = x² + 2x, g(x) = x + 2
In Exercises sketch the region bounded by the graphs of the equations and find the area of the region. y = -x³ + 2, y = x - 3, x= -1, x = 1
In Exercises find the area of the region by integrating (a) With respect to x and (b) Withrespect to y. (c) Compare your results. Which method issimpler? In general, will this method always be simpler thanthe other one? Why or why not? x = 4 - y² x=y=2 + -6-4-2 6 4 -4 -6+ 4 6 X
In Exercises the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. [₂[(2 - y) - y²] dy -2
In Exercises the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. [* (2√y-y) dy 0
In Exercises the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. [*[ (x + 1) = 1/1] 4 - dx 2 0
In Exercises the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. *π/4 J-1/4 (sec² x cos x) dx
In Exercises the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. [[(3-x)-x] X 2 X dx
In Exercises determine which value best approximates the area of the region bounded by the graphs of ƒ and g. Make your selection on the basis of a sketch of the region and not by performing any calculations.)ƒ(x) = 2 - 1/2x, g(x) = 2 - √x(a) 1(b) 6 (c) -3(d) 3(e) 4
In Exercises the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. J-1 [(2 - x²) - x²] dx
In Exercises determine which value best approximates the area of the region bounded by the graphs of ƒ and g. Make your selection on the basis of a sketchof the region and not by performing any calculations.)ƒ(x) = x + 1, g(x) = (x - 1)²(a) -2(b) 2 (c) 10(d) 4(e) 8
In Exercises set up the definite integral that gives the area of the region. y, = (x - 1)3 Y2 y2 = x - 1 y 1 # + 2 V2 X
In Exercises set up the definite integral that gives the area of the region. y₁ = x² - 6x Y₂ = 0 -2 -4 -8 2 4 Y1₁ Y/₂ 8 X
In Exercises set up the definite integral that gives the area of the region. Y₁ = 3(x³ - x) y₂ = 0 AV -1 Y2 X
In Exercises set up the definite integral that gives the area of the region. y₁ = x² Y₂ = x³ 1 Y₁ 1/2 1 X
In Exercises set up the definite integral that gives the area of the region. y₁ = x² - 4x +3 1/₂ = x² + 2x + 3 - y 4 3 AV 4 5
In Exercises set up the definite integral that gives the area of the region. y₁ = x² + 2x + 1 y₂ = 2x + 5 -4/-2 00 8 6 y ₂/ Y₁ 2 4 tex
In Exercises solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the formthat can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation isy’ — y = y3 y' + P(x)y = Q(x)y"
In Exercises solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the formthat can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation isyy' - 2y² = ex y' + P(x)y = Q(x)y"
In Exercises solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the formthat can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is y' + P(x)y = Q(x)y"
In Exercises solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the formthat can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is y' + P(x)y = Q(x)y"
In Exercises solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the formthat can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation isxy' + y = xy³ y' + P(x)y = Q(x)y"
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.y' + xy = ex y is a first-order linear differential equation.
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.y' + x √y = x² is a first-order linear differential equation.
In Exercises solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the formthat can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation is y' + P(x)y = Q(x)y"
In Exercises solve the Bernoulli differential equation. The Bernoulli equation is a well-known nonlinear equation of the formthat can be reduced to a linear form by a substitution. The general solution of a Bernoulli equation isy' + xy = xy-¹ y' + P(x)y = Q(x)y"
In Exercises solve the first-order differential equation by any appropriate method. y' = 2x√1 - y²
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential 3 (²³) y' Equation y = 2x³ 213 Initial Condition y(1) = 1
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation y' + 5y = e5x Initial Condition y(0) = 3
In Exercises solve the first-order differential equation by any appropriate method.x dx + (y + ey)(x² + 1) dy = 0
In Exercises solve the first-order differential equation by any appropriate method.3(y - 4x²) dx + x dy = 0
In Exercises solve the first-order differential equation by any appropriate method.(x + y) dx - x dy = 0
In Exercises solve the first-order differential equation by any appropriate method. dy y cos x cos x + = 0 dx
In Exercises solve the first-order differential equation by any appropriate method.(2y - ex) dx + x dy = 0
In Exercises solve the first-order differential equation by any appropriate method. dy dx || x-3 y(y + 4)
In Exercises solve the first-order linear differential equation. dy dx Sy -18
In Exercises solve the first-order linear differential equation.(x + 3)y' + 2y = 2(x + 3)²
In Exercises solve the first-order differential equation by any appropriate method. dy dx e2x+y ex-y
In Exercises(a) Use a graphing utility to graph the slope field for the differential equation(b) Find the particular solutions of the differential equation passing through the given points(c) Use a graphing utility to graph the particular solutions on the slope field Differential Equation + (cot
In Exercises(a) Use a graphing utility to graph the slope field for the differential equation(b) Find the particular solutions of the differential equation passing through the given points(c) Use a graphing utility to graph the particular solutions on the slope field Differential Equation + 4x³y =
In Exercises(a) Use a graphing utility to graph the slope field for the differential equation(b) Find the particular solutions of the differential equation passing through the given points(c) Use a graphing utility to graph the particular solutions on the slope field Differential Equation + 2xy =
In Exercises solve the first-order linear differential equation.(x - 2)y' + y = 1
In Exercises solve the first-order linear differential equation.4y' = ex/4 + y
In Exercises find the logistic equation that passes through the given point. dy dt = 1.76 1 -²). 8 - (0, 3)
In Exercises(a) Use a graphing utility to graph the slope field for the differential equation(b) Find the particular solutions of the differential equation passing through the given points (c) Use a graphing utility to graph the particular solutions on the slope field Differential Equation 1 -
In Exercises solve the first-order linear differential equation.exy' +4ex y = 1
In Exercises solve the first-order linear differential equation.y' - y = 10
Write a logistic differential equation that models the growth rate of the brook trout population in Exercise 47. Then repeat part (b) using Euler's Method with a step size of h = 1. Compare the approximation with the exact answer.Data from in Exercise 47A conservation department releases 1200 brook
In Exercises find the logistic equation that passes through the given point. dy dt 1 80 (0,8)
In Exercises the logistic equation models the growth of a population. Use the equation to (a) Find the value of k(b) Find the carrying capacity(c) Find the initial population(d) Determine when the population will reach 50% of its carrying capacity(e) Write a logistic differential equation that
A conservation department releases 1200 brook trout into a lake. It is estimated that the carrying capacity of the lake for the species is 20,400. After the first year, there are 2000 brook trout in the lake.(a) Write a logistic equation that models the number of brook trout in the lake.(b) Find
In Exercises sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. -X 11 77 177 -4+1 141 \\ \ \X\\\\ \ \ / / | | 1 1 1 1 1 1 1 1+1|||| 1 1 1 1 1 1 1 | | | | | | | | | | | | | | ! ! ! ! - | | | | | | || y 14 1. = 3 - 2y dx dy
In Exercises the logistic equation models the growth of a population. Use the equation to (a) Find the value of k(b) Find the carrying capacity(c) Find the initial population(d) Determine when the population will reach 50% of its carrying capacity(e) Write a logistic differential equation that
In Exercises match the differential equation with its solution.y' - 2xy = x(a) y = Cex²(b) y = - 1/2 + Cex²(c) y = x² + C(d) y = Ce2x
In Exercises match the differential equation with its solution.y' - 2xy = 0(a) y = Cex²(b) y = - 1/2 + Cex²(c) y = x² + C(d) y = Ce2x
In Exercises match the differential equation with its solution.y' - 2y = 0(a) y = Cex²(b) y = - 1/2 + Cex²(c) y = x² + C(d) y = Ce2x
In Exercises find the particular solution that satisfies the initial condition. Differential Equation yy' - x cos x² = 0 Initial Condition y(0) = -2
In Exercises sketch a few solutions of the differential equation on the slope field and then find the general solution analytically. dy dx /// 1117777 1111///. | |/ / 11 4x y || | \ | | | | / | | \ \\ \ \ \ / / | 4 !! 1117 77777 -X
In Exercises match the differential equation with its solution.y' - 2x = 0(a) y = Cex²(b) y = - 1/2 + Cex²(c) y = x² + C(d) y = Ce2x
The graph shows the amount of concentrate Q (in pounds) in a solution in a tank at time t (in minutes) as a solution with concentrate enters the tank, is well stirred, and is withdrawn from the tank.(a) How much concentrate is in the tank at time t = 0?(b) Which is greater, the rate of solution
What does the term “first-order” refer to in a first-order linear differential equation?
In Exercises find the particular solution that satisfies the initial condition. Differential Equation y³(x + 1)y' - x²(x + 1) = 0 Initial Condition y(0) = 1
In Exercises find the particular solution that satisfies the initial condition. Differential Equation y³y' 3x = 0 - Initial Condition y(2) = 2
Give the standard form of a first-order linear differential equation. What is its integrating factor?
In Exercises find the particular solution that satisfies the initial condition. Differential Equation yy' - 5e2x = 0 Initial Condition y(0) = -3
In Exercises consider a tank that at time t = 0 contains v0 gallons of a solution of which, by weight, q0 pounds is soluble concentrate. Another solution containing q1 pounds of the concentrate per gallon is running into the tank at the rate of r1, gallons per minute. The solution in the tank
In Exercises consider a tank that at time t = 0 contains v0 gallons of a solution of which, by weight, q0 pounds is soluble concentrate. Another solution containing q1 pounds of the concentrate per gallon is running into the tank at the rate of r1, gallons per minute. The solution in the tank is
In Exercises consider a tank that at time t = 0 contains v0 gallons of a solution of which, by weight, q0 pounds is soluble concentrate. Another solution containing q1 pounds of the concentrate per gallon is running into the tank at the rate of r1, gallons per minute. The solution in the tank is
In Exercises use the differential equation for electric circuits given byIn this equation, I is the current, R is the resistance, L is the inductance, and E is the electromotive force (voltage).Use the result of Exercise 33 to find the equation for the current when I(0) = 0, E0 = 120 volts, R = 600
In Exercises find the general solution of the differential equation. dy dx x³ 2y²
In Exercises consider a tank that at time t = 0 contains v0 gallons of a solution of which, by weight, q0 pounds is soluble concentrate. Another solution containing q1 pounds of the concentrate per gallon is running into the tank at the rate of r1, gallons per minute. The solution in the tank is
In Exercises use the differential equation for electric circuits given byIn this equation, I is the current, R is the resistance, L is the inductance, and E is the electromotive force (voltage).Solve the differential equation for the current given a constant voltage E0. dl L + RI = E. dt
The sales S (in thousands of units) of a new product after it has been on the market for years is given by(a) Find S as a function of when 5000 units have been sold after 1 year and the saturation point for the market is 30,000 units (that is (b) How many units will have been sold after 5 years? S
In Exercises find the general solution of the differential equation.y' - ey sin x = 0
In Exercises find the general solution of the differential equation. dy dx || 5x y
Glucose is added intravenously to the bloodstream at the rat of units per minute, and the body removes glucose from the bloodstream at a rate proportional to the amoun present. Assume Q(t) that is the amount of glucose in the bloodstream at time t.(a) Determine the differential equation describing
In Exercises find the general solution of the differential equation.y' - 16xy = 0
In Exercises consider an eight-pound object dropped from a height of 5000 feet, where the air resistance is proportional to the velocity.Use the result of Exercise 31 to write the position of the object as a function of time. Approximate the velocity of the object when it reaches ground level.Data
In Exercises consider an eight-pound object dropped from a height of 5000 feet, where the air resistance is proportional to the velocity.Write the velocity of the object as a function of time when the velocity after 5 seconds is approximately -101 feet per second. What is the limiting value of the
In Exercises use the result of Exercise 26.Data from in Exercise 26A large corporation starts at time t = 0 to invest part of its receipts continuously at a rate of P dollars per year in a fund for future corporate expansion. Assume that the fund earns r percent interest per year compounded
In Exercises use the result of Exercise 26.Data from in Exercise 26A large corporation starts at time t = 0 to invest part of its receipts continuously at a rate of P dollars per year in a fund for future corporate expansion. Assume that the fund earns r percent interest per year compounded
Find the balance in an account when $1000 is deposited for 8 years at an interest rate of 4% compounded continuously.
In Exercises find the exponential function y = Cekt that passes through the two points. 5 4 3 نیا 2 1 y (1,4) + 1 2 . -3 (4,1) 4 5
Learning Curve The management at a certain factory has found that the maximum number of units a worker can produce in a day is 75. The rate of increase in the number of units N produced with respect to time t in days by a new employee is proportional to 75 - N.(a) Determine the differential
A population grows continuously at the rate of 1.85%. How long will it take the population to double?
A large corporation starts at time t = 0 to invest part of its receipts continuously at a rate of P dollars per year in a fund for future corporate expansion. Assume that the fund earns r percent interest per year compounded continuously. So, the rate of growth of the amount A in the fund is given
Radioactive radium has a half-life of approximately 1599 years. The initial quantity is 15 grams. How much remains after 750 years?
When predicting population growth, demographers must consider birth and death rates as well as the net change caused by the difference between the rates of immigration and emigration. Let P be the population at time t and let N be the net increase per unit time resulting from the difference between
Under ideal conditions, air pressure decreases continuously with the height above sea level at a rate proportional to the pressure at that height. The barometer reads 30 inches at sea level and 15 inches at 18,000 feet. Find the barometric pressure at 35,000 feet.
In Exercises find the exponential function y = Cekt that passes through the two points. 5 4 3 2 1 (4, 5) 2 3 4 5
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation 2xy' y = x³ x - - Initial Condition y(4) = 2
In Exercises find the exponential function y = Cekt that passes through the two points. 5 4 3 2 1 -(0,5) 12 3 (5, 1) 4 5
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation y' + (2x - 1)y = 0 Initial Condition y(1) = 2
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation x dy = (x + y + 2) dx Initial Condition y(1) = 10
In Exercises find the exponential function y = Cekt that passes through the two points. 5 4 3 2 y - (0, ²) 12 (5,5), + 3 4 5 t
In Exercises find the particular solution of the differential equation that satisfies the initial condition. Differential Equation y' + y sec x = sec x Initial Condition y(0) = 4
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

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