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mathematics
calculus early transcendentals

Questions and Answers of Calculus Early Transcendentals

Find an equation of the following parabolas, assuming the vertex is at the origin.A parabola symmetric about the x-axis that passes through the point (1, -4)
Find an equation of the following parabolas, assuming the vertex is at the origin.A parabola symmetric about the y-axis that passes through the point (2, -6)
Find an equation of the following parabolas, assuming the vertex is at the origin.A parabola with focus at (-4, 0)
Find an equation of the following parabolas, assuming the vertex is at the origin.A parabola with focus at (3, 0)
Find an equation of the following parabolas, assuming the vertex is at the origin.A parabola that opens downward with directrix y = 6
Find an equation of the following parabolas, assuming the vertex is at the origin.A parabola that opens to the right with directrix x = -4
Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work.12x = 5y2
Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work.8y = -3x2
Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work.4x = -y2
Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work.x = -y2/16
Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work.y2 = 20x
Sketch a graph of the following parabolas. Specify the location of the focus and the equation of the directrix. Use a graphing utility to check your work.x2 = 12y
How does the eccentricity determine the type of conic section?
What are the equations of the asymptotes of a standard hyperbola with vertices on the x-axis?
Give the equation in polar coordinates of a conic section with a focus at the origin, eccentricity e, and a directrix x = d, where d > 0.
Given vertices (±a, 0) and eccentricity e, what are the coordinates of the foci of an ellipse and a hyperbola?
What is the equation of the standard hyperbola with vertices at (0, ±a) and foci at (0, ±c)?
What is the equation of the standard ellipse with vertices at (±a, 0) and foci at (±c, 0)?
What is the equation of the standard parabola with its vertex at the origin that opens downward?
Sketch the three basic conic sections in standard position with vertices and foci on the y-axis.
Sketch the three basic conic sections in standard position with vertices and foci on the x-axis.
Give the property that defines all hyperbolas.
Give the property that defines all ellipses.
Give the property that defines all parabolas.
Let a curve be described by r = f(θ), where f(θ) > 0 on its domain. Referring to the figure of Exercise 62, a curve is isogonal provided the angle φ is constant for all θ. a. Prove that
Let a polar curve be described by r = f(θ) and let ℓ be the line tangent to the curve at the point P(x, y) = P(r, θ) (see figure).a. Explain why tan α = dy/dx.b. Explain why tan θ = y/x.c. Let
Consider the following sequence of problems related to grazing goats tied to a rope.A circular corral of unit radius is enclosed by a fence. A goat is outside the corral and tied to the fence with a
Consider the following sequence of problems related to grazing goats tied to a rope.A circular concrete slab of unit radius is surrounded by grass. A goat is tied to the edge of the slab with a rope
Consider the following sequence of problems related to grazing goats tied to a rope.A circular corral of unit radius is enclosed by a fence. A goat inside the corral is tied to the fence with a rope
A blood vessel with a circular cross section of constant radius R carries blood that flows parallel to the axis of the vessel with a velocity of v(r) = V(1 - r2/R2), where V is a constant and r is
Find the area of the regions bounded by the following curves.The limaçon r = 4 - 2 cos θ
Find the area of the regions bounded by the following curves.The limaçon r = 2 - 4 sin θ
Find the area of the regions bounded by the following curves.The lemniscate r2 = 6 sin 2θ
Find the area of the regions bounded by the following curves.The complete three-leaf rose r = 2 cos 3θ
Let Rn be the region bounded by the nth turn and the (n + 1)st turn of the spiral r = e-θ in the first and second quadrants, for θ ≥ 0 (see figure).a. Find the area An of Rn.b. Evaluatec.
Assume m is a positive integer.a. Even number of leaves: What is the relationship between the total area enclosed by the 4m-leaf rose r = cos (2mθ) and m?b. Odd number of leaves: What is the
Use a graphing utility to determine the first three points with θ ≥ 0 at which the spiral r = 2θ has a horizontal tangent line. Find the first three points with θ ≥ 0 at which the spiral r =
Find the areas of the following regions.The region common to the circle r = 3 cos θ and the cardioid r = 1 + cos θ
Find the areas of the following regions.The region inside the outer loop but outside the inner loop of the limaçon r = 3 - 6 sin θ
Find the areas of the following regions.The region inside the inner loop of the limaçon r = 2 + 4 cos θ.
Find the areas of the following regions.The region common to the circles r = 2 sin θ and r = 1
Explain why the point (-1, 3π/2) is on the polar graph of r = 1 + cos θ even though it does not satisfy the equation r = 1 + cos θ.
Determine whether the following statements are true and give an explanation or counterexample. a. The area of the region bounded by the polar graph of r = f(θ) on the interval [α, β] isb. The
In Exercises 37–40, you found the intersection points of pairs of curves. Find the area of the entire region that lies within both of the following pairs of curves.r = 1 and r = √2 cos 2θ
In Exercises 37–40, you found the intersection points of pairs of curves. Find the area of the entire region that lies within both of the following pairs of curves.r = 1 + sin θ and r = 1 + cos θ
In Exercises 37–40, you found the intersection points of pairs of curves. Find the area of the entire region that lies within both of the following pairs of curves.r = 2 + 2 sin θ and r = 2 - 2
In Exercises 37–40, you found the intersection points of pairs of curves. Find the area of the entire region that lies within both of the following pairs of curves.r = 3 sin θ and r = 3 cos θ
Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.r = 1 and r = √2 cos 2θ
Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.r = 1 + sin θ and r = 1 + cos θ
Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.r = 2 + 2 sin θ and r = 2 - 2 sin θ
Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.r = 3 sin θ and r = 3 cos θ
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the lemniscate r2 = 2 sin 2θ and outside the circle r = 1.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the rose r = 4 sin 2θ and inside the circle r = 2
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the rose r = 4 cos 2θ and outside the circle r = 2.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside one leaf of the rose r = cos 5θ.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the right lobe of r = √cos 2θ and inside the circle r = 1/√12 in the first quadrant
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the curve r = √cos θ and inside the circle r = 1/√2 in the first quadrant.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the curve r = √cos θ and outside the circle r = 1/√2.
Make a sketch of the region and its bounding curves. Find the area of the region.The region outside the circle r = 1/2 and inside the circle r = cos θ.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the inner loop of r = cos θ - 1/2.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside one leaf of r = cos 3θ.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside all the leaves of the rose r = 3 sin 2θ.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the limaçon r = 2 + cos θ.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the cardioid r = 4 + 4 sin θ.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the circle r = 8 sin θ.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the right lobe of r = √cos 2θ.
Make a sketch of the region and its bounding curves. Find the area of the region.The region inside the curve r = √cos θ.
Find the points at which the following polar curves have a horizontal or a vertical tangent line.r = sec θ.
Find the points at which the following polar curves have a horizontal or a vertical tangent line.r = 1 - sin θ.
Find the points at which the following polar curves have a horizontal or a vertical tangent line.r = 3 + 6 sin θ.
Find the points at which the following polar curves have a horizontal or a vertical tangent line.r = sin 2θ.
Find the points at which the following polar curves have a horizontal or a vertical tangent line.r = 2 + 2 sin θ.
Find the points at which the following polar curves have a horizontal or a vertical tangent line.r = 4 cos θ.
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in
What integral must be evaluated to find the area of the region bounded by the polar graphs of r = f(θ) and r = g(θ) on the interval α ≤ θ ≤ β, where f(θ) ≥ g(θ) ≥ 0?
Explain why the slope of the line tangent to the polar graph of r = f(θ) is not dr/dθ.
How do you find the slope of the line tangent to the polar graph of r = f(θ) at a point?
Express the polar equation r = f(θ) in parametric form in Cartesian coordinates, where θ is the parameter.
Consider the polar curve r = cos (nθ/m), where n and m are integers.a. Graph the complete curve when n = 2 and m = 3.b. Graph the complete curve when n = 3 and m = 7.c. Find a general rule in terms
Without using a graphing utility, determine the symmetries (if any) of the curve r = 4 - sin (θ/2).
Find the equation in Cartesian coordinates of the lemniscate r2 = a2 cos 2θ, where a is a real number.
Show that the equation r = a cos θ + b sin θ, where a and b are real numbers, describes a circle. Find the center and radius of the circle.
Water flows in a shallow semicircular channel with inner and outer radii of 1 m and 2 m (see figure). At a point P(r, θ) in the channel, the flow is in the tangential direction (counterclockwise
A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and 3, respectively, and that Earth completes one orbit in one year while Mars takes two years. When t = 0,
Consider the curve r = f(θ) = cos aθ - 1.5, where a = (1 + 12π)1/(2π) ≈ 1.78933 (see figure).a. Show that f(0) = f(2π) and find the point on the curve that corresponds to θ = 0 and θ =
The butterfly curve of Example 8 is enhanced by adding a term:r = esin θ - 2 cos 4θ + sin5 (θ/12), for 0 ≤ u ≤ 24π.a. Graph the curve.b. Explain why the new term produces the observed effect.
Points at which the graphs of r = f(θ) and r = g(θ) intersect must be determined carefully. Solving f(θ) = g(θ) identifies some—but perhaps not all—intersection points. The reason is that the
Points at which the graphs of r = f(θ) and r = g(θ) intersect must be determined carefully. Solving f(θ) = g(θ) identifies some—but perhaps not all—intersection points. The reason is that the
Points at which the graphs of r = f(θ) and r = g(θ) intersect must be determined carefully. Solving f(θ) = g(θ) identifies some—but perhaps not all—intersection points. The reason is that the
Points at which the graphs of r = f(θ) and r = g(θ) intersect must be determined carefully. Solving f(θ) = g(θ) identifies some—but perhaps not all—intersection points. The reason is that the
Graph the following spirals. Indicate the direction in which the spiral is generated as θ increases, where θ > 0. Let a = 1 and a = -1.Hyperbolic spiral: r = a/θ

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