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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

In Exercises sketch a graph of the polar equation.r = - 3 cos 2θ
In Exercises sketch a graph of the polar equation.r = cos 5θ
In Exercises find the area of the region.One petal of r = 3 cos 5θ
In Exercises find the area of the region.One petal of r = 2 sin 6θ
In Exercises find the area of the region.Interior of r = 2 + cos θ
In Exercises find the area of the region.Interior of r = 5(1 - sin θ)
In Exercises find the area of the region.Interior of r² = 4 sin 2θ
In Exercises find the points of intersection of the graphs of the equations. r = 1 = - cos 0 r = 1 + sin 0
In Exercises find the area of the region.Common interior of r = 4 cos θ and r = 2
In Exercises use a graphing utility to graph the polar equation. Find the area of the given region analytically.Inner loop of r = 3 - 6 cos θ
In Exercises find the points of intersection of the graphs of the equations. r = 1 + sin 0 r = 3 sin 0
In Exercises use a graphing utility to graph the polar equation. Find the area of the given region analytically.Inner loop of r = 2 + 4 sin θ
In Exercises use a graphing utility to graph the polar equation. Find the area of the given region analytically.Between the loops of r = 3 - 6 cos θ
In Exercises use a graphing utility to graph the polar equation. Find the area of the given region analytically.Between the loops of r = 2 + 4 sin θ
In Exercises find the length of the curve over the given interval. Polar Equation r = 5 cos 0 Interval = 0 Σπ 7/7 ≤0= 2
In Exercises find the length of the curve over the given interval. Polar Equation r = 3(1 - cos 0) Interval 0 ≤ 0 ≤ T
In Exercises find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. 2 1 + cos 0
In Exercises write an integral that represents the area of the surface formed by revolving the curve about the given line. Use the integration capabilities of a graphing utility to approximate the
In Exercises find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. r = 6 3 + 2 cos 0
In Exercises write an integral that represents the area of the surface formed by revolving the curve about the given line. Use the integration capabilities of a graphing utility to approximate the
In Exercises find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. r 6 1 - sin 0
In Exercises find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. || 4 2-3 sin 0
In Exercises find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. 8 2 - 5 cos 0
In Exercises find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) Conic Parabola Eccentricity e =
In Exercises find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. r 4 5 - 3 sin 0
In Exercises find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) Conic Ellipse Eccentricity 3 4 Directrix y
In Exercises find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.)
In Exercises find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) Conic Ellipse Vertex or Vertices (5, 0),
In Exercises find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) Conic Parabola Vertex or Vertices 2,
In Exercises find a polar equation for the conic with its focus at the pole. (For convenience, the equation for the directrix is given in rectangular form.) Conic Hyperbola Vertex or Vertices (1, 0),
In Exercises plot the point in polar coordinates and find the corresponding rectangular coordinates for the point. -2, 5 п 3
In Exercises find the area of the region.Interior of r = 1 - sin θ
In Exercises find a set of parametric equations for the rectangular equation that satisfies the given condition. y = x², t = 4 at the point (4, 16)
In Exercises convert the polar equation to rectangular form and sketch its graph.r = 4
In Exercises use the result of Exercise 104 to find the angle ψ between the radial and tangent lines to the graph for the indicated value of θ. Use a graphing utility to graph the polar equation,
In Exercises use the result of Exercise 104 to find the angle ψ between the radial and tangent lines to the graph for the indicated value of θ. Use a graphing utility to graph the polar equation,
In Exercises use the result of Exercise 104 to find the angle ψ between the radial and tangent lines to the graph for the indicated value of θ. Use a graphing utility to graph the polar equation,
In Exercises use the result of Exercise 104 to find the angle ψ between the radial and tangent lines to the graph for the indicated value of θ. Use a graphing utility to graph the polar 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.The polar equations r = sin 2θ, r = -sin 2θ, and r = sin
In Exercises use the result of Exercise 104 to find the angle ψ between the radial and tangent lines to the graph for the indicated value of θ. Use a graphing utility to graph the polar 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.If x > 0, then the point (x, y) on the rectangular coordinate
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 (r, θ₁) and (r, θ₂) represent the same point on the polar
Sketch the graph of each equation.(a)(b) r = 1 - sin 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 (r₁, θ₁) and (r₂, θ2) represent the same point on the
In Exercises use the result of Exercise 104 to find the angle ψ between the radial and tangent lines to the graph for the indicated value of θ. Use a graphing utility to graph the polar equation,
Prove that the tangent of the angle ψ(0 ≤ ψ ≤ π/2) between the radial line and the tangent line at the point (r, θ) on the graph of r = ƒ(θ) (see figure) is given by tan = dr/de R|N 2 Polar
Sketch the graph of r = 4 sin θ over each interval.(a)(b)(c) 0 < 0 < < 2
Identify each special polar graph and write its equation.(a)(b)(c)(d) π 元|2 1 1 2 -0
Write an equation for the rose curve r = 2 sin 2θ after it has been rotated by the given amount. Verify the results by using a graphing utility to graph the rotated rose curve for (a) θ =
Write an equation for the limaçon r = 2 - sin θ after it hasbeen rotated by the given amount. Use a graphing utility tograph the rotated limaçon for (a) θ = π/4(b) θ = π/2(c) θ = π(d)
In Exercises use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Name of Graph Hyperbolic spiral Polar Equation r = 2/0 Asymptote y = 2
The polar form of an equation of a curve is r = ƒ(sin θ). Show that the form becomes(a) r = ƒ(-cos θ) if the curve is rotated counterclockwise π/2 radians about the pole.(b) r = ƒ(-sin θ) if
In Exercises use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Name of Graph Conchoid Polar Equation r = 2 - sec 0 Asymptote x = -1
Verify that if the curve whose polar equation is r = ƒ(θ) is rotated about the pole through an angle ∅,then an equation for the rotated curve is r = ƒ(θ - ∅).
Use a graphing utility to graph the polar equation r = 6[1 + cos(θ - ∅)] for (a) ∅ = 0 (b) ∅ = π/4 (c) ∅ = π/2 Use the graphs to describe the effect of the angle ∅.
In Exercises use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Name of Graph Strophoid Polar Equation r = 2 cos 20 sec 0 Asymptote x = -2
In Exercises use a graphing utility to graph the equation and show that the given line is an asymptote of the graph. Name of Graph Conchoid Polar Equation r = 2 + csc 0 Asymptote y = 1
How are the slopes of tangent lines determined in polar coordinates? What are tangent lines at the pole and how are they determined?
Give the equations for the coordinate conversion from rectangular to polar coordinates and vice versa.
Describe the differences between the rectangular coordinate system and the polar coordinate system.
In Exercises sketch a graph of the polar equation. r 6 2 sin 3 cos 0
In Exercises sketch a graph of the polar equation. Ө
In Exercises sketch a graph of the polar equation.r² = 4 sin θ
In Exercises sketch a graph of the polar equation.r² = 4 cos 2θ
In Exercises sketch a graph of the polar equation.r = 2θ
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.Use the formula for the arc length of a curve in parametric form
A curve called the folium of Descartes can be represented by the parametric equations(a) Convert the parametric equations to polar form.(b) Sketch the graph of the polar equation from part (a).(c)
The curve represented by the equation r = aebθ, where a and b are constants, is called a logarithmic spiral. The figure shows the graph of r = eθ/6, -2π ≤ θ ≤ 2π. Find the area of the shaded
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 ƒ(θ) = g(θ) for θ = 0, π/2, and 3π/2, then the graphs of
Area The larger circle in the figure is the graph of r = 1. Find the polar equation of the smaller circle such that the shaded regions are equal. RIN 2
In Exercises sketch a graph of the polar equation.r = 3 csc θ
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 f(θ) > 0 for all θ and g(θ) < 0 for all θ, then the
In Exercises sketch a graph of the polar equation.r = 5 - 4 sin θ
In Exercises sketch a graph of the polar equation.r = 3 - 2 cos θ
Find the area of the circle given byCheck your result by converting the polar equation to rectangular form, then using the formula for the area of a circle. r = sin 0 + cos 0.
In Exercises sketch a graph of the polar equation.r = 1 + sin θ
In Exercises sketch a graph of the polar equation.r = 4(1 + cos θ)
The curve represented by the equation r = aθ, where a is a constant, is called the spiral of Archimedes.(a) Use a graphing utility to graph r = θ, where θ ≥ 0. What happens to the graph of r =
Consider the circle r = 3 sin θ.(a) Find the area of the circle.(b) Complete the table giving the areas A of the sectors of the circle between θ = 0 and the values of θ in the table.(c) Use the
In Exercises sketch a graph of the polar equation.r = 1
Consider the circle r = 8 cos θ. (a) Find the area of the circle. (b) Complete the table giving the areas A of the sectors of the circle between θ = 0 and the values of θ in the
In Exercises sketch a graph of the polar equation.r = 8
What conic section does the polar equation r = a sin θ + b cos θ represent?
Which graph, traced out only once, has a larger arc length? Explain your reasoning.(a)(b) Na + -0
In Exercises sketch a graph of the polar equation and find the tangents at the pole.r = 3 cos 2θ
In Exercises sketch a graph of the polar equation and find the tangents at the pole.r = 3 sin 2θ
Find the surface area of the torus generated by revolving the circle given by r = a about the line r = b sec θ, where 0 < a < b.
In Exercises sketch a graph of the polar equation and find the tangents at the pole.r = -sin 5θ
Find the surface area of the torus generated by revolving the circle given by r = 2 about the line r = 5 sec θ.
In Exercises sketch a graph of the polar equation and find the tangents at the pole.r = 4 cos 3θ
In Exercises sketch a graph of the polar equation and find the tangents at the pole.r = 3(1 - cos θ)
For each polar equation, sketch its graph, determine the interval that traces the graph only once, and find the area of the region bounded by the graph using a geometric formula and integration.(a) r
In Exercises use the integration capabilities of a graphing utility to approximate, to two decimal places, the area of the surface formed by revolving the curve about the polar axis. r = 4 cos 20,0
In Exercises sketch a graph of the polar equation and find the tangents at the pole.r = 2(1 - sin θ)
In Exercises find the area of the surface formed by revolving the curve about the given line. Polar Equation r = a(1 + cos 0) Interval 0 ≤ 0 ≤ T Axis of Revolution Polar axis
Give the integral formulas for the area of the surface of revolution formed when the graph of r = ƒ(θ) is revolved about(a) The polar axis.(b) The line θ = π/2.
In Exercises sketch a graph of the polar equation and find the tangents at the pole.r = 5 cos θ
Explain why finding points of intersection of polar graphs may require further analysis beyond solving two equations simultaneously.
In Exercises sketch a graph of the polar equation and find the tangents at the pole.r = 5 sin θ
In Exercises use the integration capabilities of a graphing utility to approximate, to two decimal places, the area of the surface formed by revolving the curve about the polar axis.r = θ, 0 ≤ θ

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