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mathematics
calculus 6th edition

Questions and Answers of Calculus 6th edition

Solve the initial-value problem.xy' – y = x lnx, y(1)=2
The table gives the midyear population of Spain, in thousands, from 1955 to 2000.Use a graphing calculator to fit both an exponential function and a logistic function to these data. Graph the data
Solve the differential equation.x dy/dx – 4y = x4ex
Find the solution of the differential equation that satisfies the given initial condition.dy/dx = y cos x/1 + y2, y (0) = 1
Solve the differential equation.(1 + t)du/dt +u = 1 + t, t > 0
Find the solution of the differential equation that satisfies the given initial condition.x cos x = (2y + e3y)y', y(0) = 0
Solve the differential equation.t ln t dr/dt + r = tet
Solve the initial-value problem.y' = x + y, y(0) = 2
Solve the initial-value problem.t dy/dt + 2y = t3, t > 0, y(1) = 0
Find the solution of the differential equation that satisfies the given initial condition.xy' + y = y2, y(1) = –1
Solve the initial-value problem.dv/dt – 2tv = 3t2et3, v(0) = 5
Solve the initial-value problem.2xy' + y = 6x, x > 0, y(4) = 20
Solve the initial-value problem.xy' = y + x2 sin x, y(π) = 0
Find the function f such that f'(x) = f(x)(1 – f(x)) and f(0) = 1/2
Solve the initial-value problem.(x2 + 1)dy/dx + 3x(y – 1) = 0, y(0) = 2
Solve the second-order equation xy" + 2y' = 12x2 by making the substitution u = y'.
Find the area of the region that is bounded by the given curve and lies in the specified sector.r = θ2, 0 ≤ 0 ≤ π/4
Find dy/dx.x = t sin t, y = t2 + t
Find the area of the region that is bounded by the given curve and lies in the specified sector.kr = eθ/2, π ≤ θ ≤ 2π
Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.x = cos θ, y = sec θ, 0 ≤ θ ≤ π/2
Find the area of the region that is bounded by the given curve and lies in the specified sector.r = sin θ, π/3 ≤ θ ≤ 2π/3
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.x = t4 + 1, y = t3 + t; t = –1
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.x = t – t–1, y = 1+ t2; t = 1
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.x = e√t, y – t – In t2; t = 1
Find the area of the region that is bounded by the given curve and lies in the specified sector.r = √sin θ, 0 ≤ θ ≤ π
Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.x = cos θ + sin 2θ, y = sin θ + cos 2θ: θ = 0
Sketch the polar curve.r = 1 – cos θ
Sketch the curve and find the area that it encloses.r = 3 cos θ
Sketch the curve and find the area that it encloses.r = 3(1 + cos θ)
Find the vertices and foci of the ellipse and sketch its graph.x2/9 + y2/5 = 1
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = sin θ,
Sketch the curve and find the area that it encloses.r2 = 4 cos 2θ
Find the vertices and foci of the ellipse and sketch its graph.x2/64 + y2/100 = 1
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = 4
Sketch the curve and find the area that it encloses.r = 2 – sin θ
Find the distance between the points with polar coordinates (2, π/3) and (4, 2π/3).
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = sin t,
Sketch the curve and find the area that it encloses.r = 2 cos 3θ
Find the vertices and foci of the ellipse and sketch its graph.4x2 + 25y2 = 25
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = et –
Sketch the curve and find the area that it encloses.r = 2 + 2cos 2θ
Identify the curve by finding a Cartesian equation for the curve.r = 2
Sketch the polar curve.r = 3/1 + 2 sin θ
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = e2t, y
Graph the curve and find the area that it encloses.r = 1 + 2 sin 6θ
Find the vertices and foci of the ellipse and sketch its graph.x2 + 3y2 + 2x – 12y + 10 = 0
Identify the curve by finding a Cartesian equation for the curve.r cos θ = 1
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = ln t. y
Graph the curve and find the area that it encloses.r = 2 sin θ + 3 sin 9θ
Identify the curve by finding a Cartesian equation for the curve.r = 3 sin θ
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x sinht, y
Find the area of the region enclosed by one loop of the curve.r = sin 2θ
Identify the curve by finding a Cartesian equation for the curve.r = 2 sin θ + 2 cos θ
(a) Eliminate the parameter to find a Cartesian equation of the curve.(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.x = 2cosh
Find the area of the region enclosed by one loop of the curve.r = 4 sin 3θ
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.x2/144 – y2/25 = 1
Identify the curve by finding a Cartesian equation for the curve.r = csc θ
Find the area of the region enclosed by one loop of the curve.r = 3cos 5θ
Identify the curve by finding a Cartesian equation for the curve.r = tan θ sec θ
Find the area of the region enclosed by one loop of the curve.r = 2 sin 6θ
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.y2 – x2 = 4
Find a polar equation for the curve represented by the given Cartesian equation.x = 3
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.9x2 – 4y2 = 36
Find a polar equation for the curve represented by the given Cartesian equation.x2 + y2 = 9
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.4x2 – y2 – 24x – 4y + 28 = 0
Find the area of the region that lies inside the first curve and outside the second curve.r = 2 cos θ, r = 1
Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.y2 – 4x2 – 2y + 16x = 31
Find a polar equation for the curve represented by the given Cartesian equation.x + y = 9
Find a polar equation for the curve represented by the given Cartesian equation.x2 + y2 = cx
Identify the type of conic section whose equation is given and find the vertices and foci.x2 = y + 1
Find dy/dx and d2y/dx2. x = t + sin t, y = t – cos t
Find a polar equation for the curve represented by the given Cartesian equation.xy = 4
Identify the type of conic section whose equation is given and find the vertices and foci.x2 = y2 + 1
Find dy/dx and d2y/dx2.x = 1 + t2, y = t – t3
Find the area of the region that lies inside the first curve and outside the second curve.r = 2 + sinθ, r = 3 sinθ
Identify the type of conic section whose equation is given and find the vertices and foci.y2 – 8y = 6x – 16
Sketch the curve with the given polar equation.θ = –π/6
Sketch the curve with the given polar equation.r = sin θ
Find an equation for the conic that satisfies the given conditions.Parabola, vertex (0, 0), focus (0, –2)
Sketch the curve with the given polar equation.r = –3 cos θ
Find an equation for the conic that satisfies the given conditions.lParabola, vertex (1, 0), directrix x = –5
Sketch the curve with the given polar equation.r = 2(1 – sin θ), θ ≤ 0
Find the area of the region that lies inside both curves.r2 = sin 2θ, r2 = cos 2θ
Sketch the curve with the given polar equation.r = 1 – 3 cos θ
Find an equation for the conic that satisfies the given conditions.Parabola, focus (3,6), vertex (3,2)
Sketch the curve with the given polar equation.r = θ, θ ≥ 0
Find an equation for the conic that satisfies the given conditions.Parabola, vertex (2, 3), vertical axis, passing through (1,5)
Sketch the curve with the given polar equation.r = ln θ, θ ≥ 1
Find an equation for the conic that satisfies the given conditions.Parabola, horizontal axis, passing through (–1,0), (1, –1), and (3, 1)
Sketch the curve with the given polar equation.r = 4sin 3θ
Evaluate the integral. dx 1 + e*
Evaluate the integral. 1/2 10 xp x__soo
Evaluate the integral. *π/2 [*/² cos³x sin 2.x dx
Evaluate the integral. x³ + x² + 2x + 1 (x² + 1)(x² + 2) -dx
Evaluate the integral. +² X (3 + 4x - 4x²)3/2 dx
Evaluate the integral. √√3 | arctan(1/.x) d.x J1
Evaluate the integral. S √ √x² + 2x dx
Evaluate the integral. TT/3 [/³ tan³x sec¹x dx
Evaluate the integral. T t sin 3t dt
If 0 < a < b, find lim 1-0 1/t {f'² [bx + a[1 − x)}]}* dx} -

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