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mathematics
calculus

Questions and Answers of Calculus

The region bounded by the hyperbola b2x2 - a2y2 = a2b2 and a vertical line through a focus is revolved about the x-axis. Find the volume of the resulting solid.
If the ellipse of Problem 56 is revolved about the x-axis, find the volume of the resulting solid. In Problem 56 Find the volume of the solid obtained by revolving the ellipse b2x2 + a2y2 = a2b2
Find the dimensions of the rectangle having the greatest possible area that can be inscribed in the ellipse b2x2 + a2y2 = a2b2. Assume that the sides of the rectangle are parallel to the axes of the
Show that the point of contact of any tangent line to a hyperbola is midway between the points in which the tangent intersects the asymptotes.
Find the point in the first quadrant where the two hyper-bolas 25x2 - 9y2 = 225 and - 25x2 + 18y2 = 450 intersect.
Find the points of intersection of x2 + 4y2 = 20 and x + 2y = 6.
Sketch a design for a reflecting telescope that uses a parabola and an ellipse rather than a parabola and a hyperbola as described in the text and shown in Figure 17.
A ball placed at a focus of an elliptical billiard table is shot with tremendous force so that it continues to bounce off the cushions indefinitely. Describe its ultimate path?
Show that an ellipse and a hyperbola with the same foci intersect at right angles.
Describe a string apparatus for constructing a hyperbola. (There are several possibilities.)
Sound travels at u feet per second and a rifle bullet at v > u feet per second. The sound of the firing of a rifle and the impact of the bullet hitting the target were heard simultaneously, If the
Listeners A(-8, 0), 8(8, 0), and C(8, 10) recorded the exact times at which they heard an explosion. If B and C heard the explosion at the same time and A heard it 12 seconds later, where was the
Show that(√x2 - a2 - x) → as x → ∞.Rationalize the numerator.
For an ellipse, let p and q be the distances from a focus to the two vertices. Show that b = √pq, with 2b being the minor diameter.
The wheel in Figure 20 is rotating at 1 radian per second so that Q has coordinates (a cos t, a sin t). Find the coordinates (x, y) of R at time t and show that it is traveling in an elliptical path.
Let P be a point on a ladder of length a + b, P being a units from the top end. As the ladder slides with its top end on the y-axis and its bottom end on the x-axis, P traces out a curve. Find the
Show that a line through a focus of a hyperbola and perpendicular to an asymptote intersects that asymptote on the directrix nearest the focus.
If a horizontal hyperbola and a vertical hyperbola have the same asymptotes, show that their eccentricities e and E satisfy e-2 + E-2 = 1.
Let C be the curve of intersection of a right circular cylinder and a plane making an angle ϕ (0 < ϕ < π/2) with the axis of the cylinder. Show that C is an ellipse.
Using the same axes, draw the conics y = ±(ax2 + 1)1/2 for -2 ≤ x ≤ 2 and -2 ≤ y ≤ 2 using a = -2, -1, -0.5, -0.1, 0, 0.1, 0.6, 1. Make a conjecture about how the shape of the figure depends
In Problem a-c, sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). a. x2/16 + y2/4 = 1 b. x2/16 - y2/4 = 1 c. -x2/9 + y2/4 = 1
In Problems 1-3, name the conic or limiting form represented by the given equation. Usually you will need to use the process of completing the square 1. x2 + y2 - 2x + 2y + 1 = 0 2. x2 + y2 + 6x - 2y
In Problem a-c, sketch the graph of the given equation.a.b. (x + 3)2/4 + (y - 4)2 = 25 c.
Find the focus and directrix of the parabola 2y2 - 4y - 10x = 0
Determine the distance between the vertices of -9x2 + 18x + 4y2 + 24y = 9
Find the foci of the ellipse 16(x - 1)2 + 25(y + 2)2 = 400
Find the focus and directrix of the paralola x2 - 6x + 4y + 3 = 0
In Problems a to c, find the equation of the given conic. a. Horizontal ellipse with center (5, 1), major diameter 10, minor diameter 8 b. Hyperbola with center (2, -1), vertex at (4, -1), and focus
In Problems 1-2, eliminate the cross-product term by a suitable rotation of axes and then, if necessary, translate axes (complete the squares) to put the equation in standard form, Finally, graph the
A curve C goes through the three points, (-1, 2), (0, 0), and (3, 6). Find an equation for C if C is (a) A vertical parabola; (b) A horizontal parabola; (c) A circle.
The ends of an elastic string with a knot at K(x, y) are attached to a fixed point A(a, b) and a point P on the rim of a wheel of radius r centered at (0,0), As the wheel turns. K traces a curve C.
Name the conic y2 = Lx + Kx2 according to the value of K and then show that in every case |L| is the length of the latus rectum of the conic. Assume that L ≠ 0.
Show that the equations of the parabola and hyperbola with vertex (a, 0) and focus (c, 0), c > a > 0, can be written as as = y2 = 4(c - a) (x - a) and y2 = (b2/a2) (x2 - a2), respectively. Then use
The graph of x cos α x + y sin α = d is a line. Show that the perpendicular distance from the origin to this line is |d| by making a rotation of axes through the angle a.
Transform the equation x1/2 + y l/2 = a1/2 by a rotation of axes through 45° and then square twice to eliminate radicals on variables. Identify the corresponding curve.
Solve the rotation formulas for u and v in terms of x and y.
Use the results of Problem 59 to find the uv-coordinates corresponding to (x. y) = (5, -3) after a rotation of axes through 60°. Solve the rotation formulas for u and v in terms of x and y.
Find the points of x2 + 14xy + 49y2 = 100 that are closest to the origin.
Recall that Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 transforms to au2 + buv + ev2 + du + ev + f = 0 under a rotation of axes. Find formulas for a and c, and show that a + c = A + C.
Refer to Figure 6 and show that y = u sin Î¸ + v cos Î¸.
In each problem 1-2, a parametric representation of a curve is given. a. Graph the curve. b. Is the curve closed? Is it simple? c. Obtain the Cartesian equation of the curve by eliminating the
In Problems a-b, find dy/dx and d2y/dx2 without eliminating the parameter. a. x = 3τ2, y = 4τ3; τ ≠ 0 b. x = 6s2, y = -2s3; s ≠ 0
In Problem 1-3, find the equation of the tangent line to the given curve at the given value of t without eliminating the parameter. Make a sketch. 1. x = t2, y = t3; t = 2 2. x = 3t, y = 8t3; t = -
In Problems 1 to 3, find the length of the parametric curve defined over the given interval. 1. x = 2t - 1, y = 3t - 4; 0 ≤ t ≤ 3 2. x = 2 - t, y = 2t - 3; - 3 ≤ t ≤ 3 2. x = t, y = t3/2; 0
Find the length of the curve with the given parametric equations(a) x = sin Î¸, y = cos Î¸ for Î¸ ‰¤ Î¸ ‰¤ 2Ï€(b) x =
Derive a formula for the surface area generated by the rotation of the curve x = F(t) y = G(t) for a ‰¤ t ‰¤ b about the y-axis for x ‰¤ 0, and show that the
A parametrization of a circle of radius 1 centered at (1, 0) in the xy-plane is given by x = 1 + cos t, y = sin t, for 0 ≤ t ≤ 2π. Find the surface area when this curve is revolved about the
Find the area of the surface generated by revolving the curve x = cos t, y = 3 + sin t, for 0 ≤ t ≤ 2π about the x-axis.
Find the area of the surface generated by revolving the curve x = 2 + cos t, y = 1 + sin t, for 0 ≤ t ≤ 2π about the x-axis.
Find the area of the surface generated by revolving the curve x = (2/3)t3/2, y = 2√t, for 0 ≤ t ≤ 2√3 about the y-axis.
Find the area of the surface generated by revolving the curve x = t + √7, y = t2/2 + √7, for - √7 ≤ t ≤ √7 about the y-axis.
Find the area of the surface generated by revolving the curve x = t2/2 + at. y = t + a. for -√a ≤ t ≤ √a about the x-axis.
Evaluate the integrals in Problems a and b. a. ∫01(x2 - 4y) dx, where x = t + 1, y = t3 + 4. b. ∫1√3 xy dy, where x = sec t, y = tan t.
Find the area of the region between the curve x = e2t, y = e-t, and x-axis from t = 0 to t = ln 5. Make a sketch.
The path of a projectile fired from level ground with a speed of v0 feet per second at an angle a with the ground is given by the parametric equations x = (v0 cos α)t, y = - 16t2 + (v0 sin α)t a.
Modify the text discussion of the cycloid (and its accompanying diagram) to handle the case where the point P is b < a units from the center of the wheel. Show that the corresponding parametric
Follow the instructions of Problem 59 for the case b > a (a flanged wheel, as on a train), showing that you get the same parametric equations. Sketch the graph of these equations (called a prolate
Let a circle of radius b roll, without slipping, inside a fixed circle of radius a. a > b. A point P on the rolling circle traces out a curve called a hypocycloid. Find parametric equations of the
Show that if b = a/4 in Problem 61, the parametric equations of the hypocycloid may be simplified tox = Î± cos3 t, y = Î± sin3 tThis is called a hypocycloid of four cusps.
Consider the ellipse x2/a2 + y2/b2 = 1.a. Show that its perimeter isb The integral in part (a) is called an elliptic integral. It has been studied at great length, and it is known that the inte-grand
The parametric curve given by x = cos at and y = sin bt is known as a Lissajous figure. The x-coordinate oscillates a times between 1 and -1 as t goes from 0 to 2ΰ, while the y-coordinate oscillates
Plot the Lissajous figure defined by x = cos 2t, y = sin 7t, 0 ≤ t ≤ 2π. Explain why this is a closed curve even though its graph does not look closed.
Plot Lissajous figures for the following combinations of a and b for 0 ≤ t ≤ 2π: a. a = 1, b = 2 b. a = 4, b = 8 c. a = 5, b = 10 d. a = 2, b = 3 e. a = 6, b = 9 f. a = 12, b = 18
Plot the following parametric curves. Describe in words how the point moves around the curve in each case. a. x = cos(t2 - t), y = sin(t2 - t) b. x = cos(2t2 + 3t + 1), y = sin(2t2 + 3t + 1) c. x =
Using a computer algebra system, plot the following parametric curves for 0 ≤ t ≤ 2. Describe the shape of the curve in each case and the similarities and difference among all the curves. a. x =
Plot the graph of the hypocycloidFor appropriate values oft in each of the following cases: a. a = 4, b = 1 b. a = 3, b = 1 c. a = 5, = 2 d. a = 7, b = 4 Experiment with other positive integer values
Draw the graph of the epicycloidFor various values of a and b. What conjectures can you make
Draw the folium of Descartes x = 3t / (t3 + 1), y = 3t2 / (t3 + 1), then determine the values of t for which this graph is in each of the four quadrants.
Plot the points whose polar coordinates are (3, 1/3π), (1, 1/2π), (4, 1/3π), (0, π), (1, 4π), (3, 11/7π), (5/3, 1/2π), and (4, 0).
In each of Problems a to c, sketch the graph of the given Cartesian equation, and then find the polar equation for it. a. x - 3y + 2 = 0 b. x = 0 c. y = -2
In Problems a-c find the Cartesian equations of the graphs of the given polar equations. a. θ = 1/2 π b. r = 3 c. r cos θ + 3 = 0 d. r - 5 cos θ = 0
Plot the points whose polar coordinates are (3, 2π), (2, 1/2π), (4, -1/3π), (0, 0), (1, 54π), (3, -1 /6π), (1, 1/2π), and (3, - 3/2π).
In Problems 1-3, name the curve with the given polar equation. If it is a conic, give its eccentricity. Sketch the graph. 1. r = 6 2. θ = 2π / 3 3. r = 3 / sin θ
Show that the polar equation of the circle with center (c, a) and radius a is r2 + c2 - 2rc cos(θ - α) = a2.
Prove that r = a sin θ + b cos θ represents a circle and find its center and radius.
Find the length of the latus rectum for the general conic r = ed/[1 + e cos(θ - θ0)] in the terms of e and d.
Plot the points whose polar coordinates are (3, 9/4π), (-2, 1/2π), (-2, -1/3π), (-1, -1), (1, -7π), (3, - 1/6π), (-2, - 1/2π), and (3, - 33/2π).
Let r1 and r2 be the minimum and maximum distances (perihelion and aphelion, respectively) of the ellipse r = ed/[1 + e cos(θ - θ0)] from a focus. Show that a. r1 = ed/(1 + e), r2 = ed/(1 - e), b.
The perihelion and aphelion for the orbit of the asteroid Icarus are 17 and 183 million miles, respectively. What is the eccentricity of its elliptical orbit?
Earth's orbit around the sun is an ellipse of eccentricity 0.0167 and major diameter 185.8 million miles. Find its perihelion.
The path of a certain comet is a parabola with the sun at the focus. The angle between the axis of the parabola and a ray from the sun to the comet is 120° (measured from the point of the perihelion
The position of a comet with a highly eccentric elliptical orbit (e very near 1) is measured with respect to a fixed polar axis (sun is at a focus but the polar axis is not an axis of the ellipse) at
In order to graph a polar equation such as r = f(t) using a parametric equation grapher, you must replace this equation by x = f(t) cos t and y = f(t) sin t. These equations can be obtained by
Plot the points whose polar coordinates follow. For each point, give four other pairs of polar coordinates, two with positive r and two with negative r. a. (1, 1/2 π) b. (-1, 1/4 π) c. (√2, - 1/3
Find the Cartesian coordinates of the points Problem 5. Plot the points whose polar coordinates follow. For each point, give four other pairs of polar coordinates, two with positive r and two with
Find polar coordinates of the points whose Cartesian coordinates are given. a. (3 √3, 3) b. (-2 √3, 2) c. (- √2, - √2) d. (0, 0)
In Problems 1-3, sketch the graph of the given polar equation and verify its symmetry1. θ2 - π2 / 16 = 02. (r - 3) (θ - π / 4) = 03. r sin θ + 4 = 0
In Problems a to b, sketch the given curves and find their points of intersection. a. r = 6, r = 4 + 4 cos θ b. r = 1 - cos θ, r = 1 + cos θ
The conditions for symmetry given in the text are sufficient conditions, not necessary conditions. Give an example of a polar equation r = f (θ) whose graph is symmetric with respect to the y-axis,
Let a and b be fixed positive numbers and suppose that AP is part of the line that passes through (0, 0), with A on the line x = a and |AP| = b. Find both the polar equation and the rectangular
Let F and F' be fixed points with polar coordinates (a, 0) and (-a, 0), respectively. Show that the set of points P satisfying |PF| |PF'| = a2 is a lemniscate by finding its polar equation.
A line segment L of length 2a has its two end points on the x- and y-axes, respectively. The point P is on L and is such that OP is perpendicular to L. Show that the set of points P satisfying this
Find the polar equation for the curve described by the following Cartesian equations. a. y = 45 b. x2 + y2 = 36 c. x2 - y2 = 1 d. 4xy = 1 e. y = 3x + 2 f. 3x2 + 4y = 2 g. x2 + 2x + y2 - 4y - 25 = 0
Graph the curve r = cos(8θ/5) using the parametric graphing facility of a graphing calculator or computer. Notice that it is necessary to determine the proper domain for 0. Assuming that you start
Match the polar equations to the graphs labeled I-VIII in Figure 11, giving reasons for your choices.a. r = cos(Î¸/2)b. r = sec(3Î¸)c. r = 2 - 3 sin(5Î¸)d. r = 1 -
A Problems a to c, use a computer or graphing calculator to graph the given equation. Make sure that you choose a sufficiently large interval for the parameter so that the entire curve is drawn. a. r
In many cases, polar graphs are related to each other by rotation. We explore that concept here. a. How are the graphs of r = 1 + sin(θ - π/3) and r = 1 + sin(θ + π /3) related to the graph of r
Investigate the family of curves given by r = a + b cos(n(θ + ϕ)) where a, b, and ϕ are real numbers and n is a positive integer. As you answer the following questions, be sure that you graph a
Sketch the reciprocal spiral given by r = c/θ. For c > 0, does it unwind in the clockwise direction?
The following polar equations are represented by six graphs in Figure 12. Match each graph with its equation.a. r = sin 3Î¸ + sine2 2Î¸b. r = cos 2Î¸ + cos2

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