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mathematics
calculus early transcendentals

Calculus Early Transcendentals 2nd edition William L. Briggs, Lyle Cochran, Bernard Gillett - Solutions

Write the equation (x - 4)2 + y2 = 16 in polar coordinates and state values of θ that produce the entire graph of the circle.
Consider the equation r = 4/(sin θ + cos θ).a. Convert the equation to Cartesian coordinates and identify the curve it describes.b. Graph the curve and indicate the points that correspond to θ = 0, π/2, and 2π.c. Give an interval in θ on which the entire curve is generated.
Write the equationr2 + r(2 sin θ - 6 cos θ) = 0in Cartesian coordinates and identify the corresponding curve.
Jake responds to Liz with a graph that shows that his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?a. r = θ, for θ ≥ 0 b. r = 1/2 + sin θ c. r2 = cos 2θ
Liz wants to show her love for Jake by passing him a valentine on her graphing calculator. Sketch each of the following curves and determine which one Liz should use to get a heart-shaped curve.a. r = 5 cos θ b. r = 1 - sin θ c. r = cos 3θ
Match equations a–f with graphs A–F.a. r = 3 sin 4θ b. r2 = 4 cos θc. r = 2 - 3 sin θ d. r = 1 + 2 cos θe. r = 3 cos 3θ f. r = e-θ/6 УА х (A) (B) y. УА х (C) (D) y. (E) (F)
Sketch the following sets of points.{(r, θ): 0 ≤ r ≤ 4, -π/2 ≤ θ ≤ -π/3}
Sketch the following sets of points.{(r, θ): 4 ≤ r2 ≤ 9}
Find an equation of the line tangent to the cycloid x = t - sin t, y = 1 - cos t at the points corresponding to t = π/6 and t = 2π/3.
Write parametric equations for the following curves. Solutions are not unique.The segment of the curve f(x) = x3 + 2x from (0, 0) to (2, 12)
Write parametric equations for the following curves. Solutions are not unique.The line segment from P(-1, 0) to Q(1, 1) and the line segment from Q to P
Write parametric equations for the following curves. Solutions are not unique.The line y - 3 = 4(x + 2)
Write parametric equations for the following curves. Solutions are not unique.The right side of the ellipse x2/9 + y2/4 = 1, generated counterclockwise
Write parametric equations for the following curves. Solutions are not unique.The upper half of the ellipse x2/9 + y2/4 = 1, generated counterclockwise
Write parametric equations for the following curves. Solutions are not unique.The circle x2 + y2 = 9, generated clockwise
Find a description of the following curve in polar coordinates and describe the curve.x = (1 + cos t) cos t, y = (1 + cos t) sin t + 6; 0 ≤ t ≤ 2π
Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π
Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.x = 4 cos t - 1, y = 4 sin t + 2; 0 ≤ t ≤ 2π
Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.x = 4 cos t, y = 3 sin t; 0 ≤ t ≤ 2π
What is the relationship among a, b, c, and d such that the equations x = a cos t + b sin t, y = c cos t + d sin t describe a circle? What is the radius of the circle?
a. Plot the following curves, indicating the positive orientation.b. Eliminate the parameter to obtain an equation in x and y.c. Identify or briefly describe the curve.d. Evaluate dy/dx at the specified point.x = ln t, y = 8 ln t2, for 1 ≤ t ≤ e2; (1, 16)
a. Plot the following curves, indicating the positive orientation.b. Eliminate the parameter to obtain an equation in x and y.c. Identify or briefly describe the curve.d. Evaluate dy/dx at the specified point.x = 10 sin 2t, y = 16 cos 2t, for 0 ≤ t ≤ π; (5√3, 8)
a. Plot the following curves, indicating the positive orientation.b. Eliminate the parameter to obtain an equation in x and y.c. Identify or briefly describe the curve.d. Evaluate dy/dx at the specified point.x = et, y = 3e-2t, for -∞ < t < ∞; (1, 3)
a. Plot the following curves, indicating the positive orientation.b. Eliminate the parameter to obtain an equation in x and y.c. Identify or briefly describe the curve.d. Evaluate dy/dx at the specified point.x = t2 + 4, y = 6 - t, for -∞ < t < ∞; (5, 5)
Determine whether the following statements are true and give an explanation or counterexample.a. A set of parametric equations for a given curve is always unique.b. The equations x = et, y = 2et, for -∞ < t < ∞, describe a line passing through the origin with slope 2.c. The polar
Consider the parametric equationsx = a cos t + b sin t, y = c cos t + d sin t,where a, b, c, and d are real numbers.a. Show that (apart from a set of special cases) the equations describe an ellipse of the form Ax2 + Bxy + Cy2 = K, where A, B, C, and K are constants.b. Show that (apart from a set
Let H be the hyperbola x2 - y2 = 1 and let S be the 2-by-2 square bisected by the asymptotes of H. Let R be the anvil-shaped region bounded by the hyperbola and the horizontal lines y = ±p (see figure).a. For what value of p is the area of R equal to the area of S?b. For what value of p is the
Let H be the right branch of the hyperbola x2 - y2 = 1 and let ℓ be the line y = m(x - 2) that passes through the point (2, 0) with slope m, where -∞ < m < ∞. Let R be the region in the first quadrant bounded by H and ℓ (see figure). Let A(m) be the area of R. Note that for some
Show that the vertical distance between a hyperbola x2/a2 - y2/b2 = 1 and its asymptote y = bx/a approaches zero as x→∞, where 0 < b < a.
Show that an ellipse and a hyperbola that have the same two foci intersect at right angles.
A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.The length of the latus rectum of a hyperbola centered at the origin is 2b2/a = 2b√e2 - 1.
A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.The length of the latus rectum of an ellipse centered at the origin is 2b2/a = 2b√1 - e2.
A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.The length of the latus rectum of the parabola y2 = 4px or x2 = 4py is 4|p|.
A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.Let L be the latus rectum of the parabola y2 = 4px, for p > 0. Let F be the focus of the
A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties.The lines tangent to the endpoints of any focal chord of a parabola y2 = 4px intersect on the
Suppose that two hyperbolas with eccentricities e and E have perpendicular major axes and share a set of asymptotes. Show that e-2 + E-2 = 1.
Show that the polar equation of an ellipse or a hyperbola with one focus at the origin, major axis of length 2a on the x-axis, and eccentricity e is a(1 – e²) 1 + e cos 0
Show that the set of points equidistant from a circle and a line not passing through the circle is a parabola. Assume the circle, line, and parabola lie in the same plane.
Consider a hyperbola to be the set of points in a plane whose distances from two fixed points have a constant difference of 2a or -2a. Derive the equation of a hyperbola. Assume the two fixed points are on the x-axis equidistant from the origin.
Consider an ellipse to be the set of points in a plane whose distances from two fixed points have a constant sum 2a. Derive the equation of an ellipse. Assume the two fixed points are on the x-axis equidistant from the origin.
Completed in 1937, San Francisco’s Golden Gate Bridge is 2.7 km long and weighs about 890,000 tons. The length of the span between the two central towers is 1280 m; the towers themselves extend 152 m above the roadway. The cables that support the deck of the bridge between the two towers hang in
Consider the parabola y = x2/4p with its focus at F(0, p) (see figure). The goal is to show that the angle of incidence between the ray ℓ and the tangent line L (α in the figure) equals the angle of reflection between the line PF and L (β in the figure). If these two angles are equal, then the
The region bounded by the parabola y = ax2 and the horizontal line y = h is revolved about the y-axis to generate a solid bounded by a surface called a paraboloid (where a > 0 and h > 0). Show that the volume of the solid is 32 the volume of the cone with the same base and vertex.
Consider the region R bounded by the right branch of the hyperbola x2/a2 - y2/b2 = 1 and the vertical line through the right focus.a. What is the volume of the solid that is generated when R is revolved about the x-axis?b. What is the volume of the solid that is generated when R is revolved about
Consider the region R bounded by the right branch of the hyperbola x2/a2 - y2/b2 = 1 and the vertical line through the right focus.a. What is the area of R?b. Sketch a graph that shows how the area of R varies with the eccentricity e, for e > 1.
Suppose that the ellipse x2/a2 + y2/b2 = 1 is revolved about the x-axis. What is the volume of the solid enclosed by the ellipsoid that is generated? Is the volume different if the same ellipse is revolved about the y-axis?
Find an equation of the line tangent to the hyperbola x2/a2 - y2/b2 = 1 at the point (x0, y0).
Show that an equation of the line tangent to the ellipse x2/a2 + y2/b2 = 1 at the point (x0, y0) is ххо Уo = 1. b?
Let R be the region bounded by the upper half of the ellipse x2/2 + y2 = 1 and the parabola y = x2/√2.a. Find the area of R.b. Which is greater, the volume of the solid generated when R is revolved about the x-axis or the volume of the solid generated when R is revolved about the y-axis?
Suppose two circles, whose centers are at least 2a units apart (see figure), are centered at F1 and F2, respectively. The radius of one circle is 2a + r and the radius of the other circle is r, where r ≥ 0. Show that as r increases, the intersection point P of the two circles describes one branch
Modify Figure 10.57 to derive the polar equation of a conic section with a focus at the originin the following three cases.a. Vertical directrix at x = -d, where d > 0b. Horizontal directrix at y = d, where d > 0c. Horizontal directrix at y = -d, where d > 0Figure 10.57 УА Directrix |PF| $УА Directrix |PF| = e\PL| P(r, 0) (d, 0) F(0, 0) х r cos 0 ed 1 + e cos 0$
Find a polar equation for each conic section. Assume one focus is at the origin. У 2 - Directrix y = -2 (0. –) 3 (0, -4)
Find a polar equation for each conic section. Assume one focus is at the origin. 2+G.0) (-2, 0) Directrix
Find an equation of the line tangent to the following curves at the given point. :( .2 5 6, 4 1; 64
Find an equation of the line tangent to the following curves at the given point. 1 + sin 0 \3’6 $1 + sin 0 \3’6$
Find an equation of the line tangent to the following curves at the given point.x2 = -6y; (-6, -6)
Find an equation of the line tangent to the following curves at the given point.y2 = 8x; (8, -8)
Determine whether the following statements are true and give an explanation or counterexample.a. The hyperbola x2/4 - y2/9 = 1 has no y-intercept.b. On every ellipse, there are exactly two points at which the curve has slope s, where s is any real number. c. Given the directrices and foci of a
Use a graphing utility to graph the hyperbolas for e = 1.1, 1.3, 1.5, 1.7, and 2 on the same set of axes. Explain how the shapes of the curves vary as e changes. 1 +есos Ө
Use a graphing utility to graph the parabolas y2 = 4px, for p = -5, -2, -1, 1, 2, and 5 on the same set of axes. Explain how the shapes of the curves vary as p changes.
Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as u increases from 0 to 2π. 1 1 - 2 cos 0
Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as u increases from 0 to 2π. cos 0
Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as u increases from 0 to 2π. 1 + 2 cos 0
Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as u increases from 0 to 2π. 1 + sin 0
Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. 12 cos O 3 –
Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. 1 2 – 2 sin 0
Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. 3 + 2 sin 0
Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. 2 – cos 0
Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. 4 2 + cos 0
Graph the following conic sections, labeling the vertices, foci, directrices, and asymptotes (if they exist). Use a graphing utility to check your work. 4 1 + cos 0
Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.A hyperbola with vertices (0, ±4) and eccentricity 2
Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.A hyperbola with vertices (±1, 0) and eccentricity 3
Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.An ellipse with vertices (0, ±9) and eccentricity 1/4
Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.An ellipse with vertices (±9, 0) and eccentricity 1/3
Write an equation of the following hyperbolas. Focus (0, 10) (0, 6) 10 -10 -(0, –6) Focus (0, – 10)
Write an equation of the following hyperbolas. (-4, 0) (4, 0) + 2 Focus (5, 0) Focus (-5, 0)
Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work.A hyperbola with vertices (0, ±4) and asymptotes y = ±2x
Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work.A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2
Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work.A hyperbola with vertices (±1, 0) that passes through (5/3 , 8)
Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work.A hyperbola with vertices (±4, 0) and foci (±6, 0)
Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work.10x2 - 7y2 = 140
Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work.x2/3 - y2/5 = 1
Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work.25y2 - 4x2 = 100
Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work.4x2 - y2 = 16
Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work.y2/16 - x2/9 = 1
Sketch a graph of the following hyperbolas. Specify the coordinates of the vertices and foci, and find the equations of the asymptotes. Use a graphing utility to check your work.x2/4 - y2 = 1
Write an equation of the following ellipses. 4+ 4.
Write an equation of the following ellipses. 4 4
Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci.An ellipse with vertices 10, {102, passing through the point(√3/2, 5)
Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci.An ellipse with vertices (±5, 0), passing through the point (4, 3/5)
Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci.An ellipse with vertices (±6, 0) and foci (±4, 0)
Find an equation of the following ellipses, assuming the center is at the origin. Sketch a graph labeling the vertices and foci.An ellipse whose major axis is on the x-axis with length 8 and whose minor axis has length 6.
Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work.12x2 + 5y2 = 60
Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. y2 5
Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. y2 У x² 9.
Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. 16
Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. x² У 4
Sketch a graph of the following ellipses. Plot and label the coordinates of the vertices and foci, and find the lengths of the major and minor axes. Use a graphing utility to check your work. .2 + y² = 1
Write an equation of the following parabolas. УА Directrix y = 4 (0, 2) х -4
Write an equation of the following parabolas. УА (-1,0), х Directrix x = -2 4-

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