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mathematics
calculus 10th edition

Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions

In Exercises the initial and terminal points of a vector v are give(a) Sketch the given directed line segment(b) Write the vector in component form(c) Write the vector as the linear combination of the standard unit vectors i and j(d) Sketch the vector with its initial point at the origin. Initial
What is the x-coordinate of any point in the yz-plane?
In Exercises the initial and terminal points of a vector v are give(a) Sketch the given directed line segment(b) Write the vector in component form(c) Write the vector as the linear combination of the standard unit vectors i and j(d) Sketch the vector with its initial point at the origin.
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).Z = 6
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).Y = 2
In Exercises the initial and terminal points of a vector v are give(a) Sketch the given directed line segment(b) Write the vector in component form(c) Write the vector as the linear combination of the standard unit vectors i and j(d) Sketch the vector with its initial point at the origin.
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s). X = - 3
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).Y < 0
In Exercises the initial and terminal points of a vector v are give(a) Sketch the given directed line segment(b) Write the vector in component form(c) Write the vector as the linear combination of the standard unit vectors i and j(d) Sketch the vector with its initial point at the origin. Initial
In Exercises sketch each scalar multiple of v.(a) 2v(b) -3v(c) 7/2v(d) 2/3v V = (3,5)
In Exercises find(a) 2/3u, (b) 3v, (c) v - u, (d) 2u + 5v. u = (4, 9), v = (2,-5)
In Exercises find(a) 2/3u, (b) 3v, (c) v - u, (d) 2u + 5v. u = (-3, -8), v = (8, 25)
In Exercises sketch each scalar multiple of v.(a) 4v(b) -1/2v(c) 0v(d) -6v v = (-2, 3)
In Exercises use the figure to sketch a graph of the vector.-u y n - X
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).|y| ≤ 3
In Exercises use the figure to sketch a graph of the vector.2u y n - X
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).|x| > 4
In Exercises use the figure to sketch a graph of the vector.-v y n - X
In Exercises use the figure to sketch a graph of the vector.1/2v y n - X
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).xy > 0, z = -3
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).xy < 0, z = 4
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).xyz < 0
In Exercises use the figure to sketch a graph of the vector.u - v y n - X
In Exercises use the figure to sketch a graph of the vector.u + 2v y n - X
In Exercises determine the location of a point (x, y, z) that satisfies the condition(s).xyz > 0
In Exercises find the lengths of the sides of the triangle with the indicated vertices, and determine whether the triangle is a right triangle, an isosceles triangle, or neither. (0, 0, 4), (2, 6, 7), (6, 4, -8)
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 = 300 -12 + 6 sin 0
In Exercises use a graphing utility to graph the polar equation when (a) e = 1(b) e = 0.5(c) e = 1.5 Identify the conic. r = 2e 1 + e cos 0
In Exercises match the equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)4x² - y² = 4 -2 4 2 -2 -4 y 2 4 x
In Exercises match the equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)4x² + y² = 4 -2 4 2 -2 -4 y 2 4 x
In Exercises use a graphing utility to graph the polar equation when (a) e = 1(b) e = 0.5(c) e = 1.5 Identify the conic. || 2e 1 - e cos 0
In Exercises match the equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)y² = - 4x -2 4 2 -2 -4 y 2 4 x
In Exercises use a graphing utility to graph the polar equation when (a) e = 1(b) e = 0.5(c) e = 1.5 Identify the conic. = 2e 1 - e sin 0
In Exercises match the equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)y² - 4x² = 4 -2 4 2 -2 -4 y 2 4 x
Consider the parabola x² = 4y and the focal chord y = 3/4x + 1. (a) Sketch the graph of the parabola and the focal chord.(b) Show that the tangent lines to the parabola at the endpoints of the focal chord intersect at right angles.(c) Show that the tangent lines to the parabola at the
Prove Theorem 10.2, Reflective Property of a Parabola, as shown in the figure.Data from in Theorem 10.2 THEOREM 10.2 Reflective Property of a Parabola Let P be a point on a parabola. The tangent line to the parabola at point P makes equal angles with the following two lines. 1. The line passing
The curve given by the parametric equationsis called a strophoid.(a) Find a rectangular equation of the strophoid.(b) Find a polar equation of the strophoid.(c) Sketch a graph of the strophoid.(d) Find the equations of the two tangent lines at the origin.(e) Find the points on the graph at which
In Exercises use a graphing utility to graph the polar equation when (a) e = 1(b) e = 0.5(c) e = 1.5 Identify the conic. = 2e 1 + e sin 0
In Exercises match the polar equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) N * 21 RIN
Consider the parabola x2 = 4py and one of its focal chords.(a) Show that the tangent lines to the parabola at the endpoints of the focal chord intersect at right angles.(b) Show that the tangent lines to the parabola at the endpoints of the focal chord intersect on the directrix of the parabola.
In Exercises match the equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)x² + 4y² = 4 -2 4 2 -2 -4 y 2 4 x
In Exercises match the equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f)x² = 4y -2 4 2 -2 -4 y 2 4 x
Consider the cornu spiral given by(a) Use a graphing utility to graph the spiral over the interval(b) Show that the cornu spiral is symmetric with respect to the origin.(c) Find the length of the cornu spiral fromWhat is the length of the spiral from x(t) = [cos(1²) du COS 2 du and
Find a rectangular equation of the portion of the cycloid given by the parametric equations x = a(θ - sin θ) and y = a(1- cos θ), 0 ≤ θ ≤ π, as shown in the figure. 2a 0 an X
In Exercises match the polar equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) N * 21 RIN
In Exercises match the polar equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) N * 21 RIN
In Exercises match the polar equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) N * 21 RIN
In Exercises identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.16x² + 16y2 - 16x + 24y - 3 = 0
Let a and b be positive constants. Find the area of the region in the first quadrant bounded by the graph of the polar equation r ab (a sin 0 + b cos 0)' TT ²-5050
In Exercises identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.y² - 12y - 8x + 20 = 0
Use a graphing utility to graph the curve shown below. The curve is given byOver what interval must θ vary to produce the curve? 0 r = ecos - 2 cos 40+ sin5. 12
Consider the right triangle shown in the figure.(a) Show that the area of the triangle is(b) Show that(c) Use part (b) to derive the formula for the derivative of the tangent function. A(a) a = 1200*² sec² 0 do.
In Exercises match the polar equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) N * 21 RIN
In Exercises identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.3x² - 2y² + 24x + 12y + 24 = 0
A particle is moving along the path described by the parametric equations x = 1/t and y = (sin t)/t, for 1 ≤ t < ∞, as shown in the figure. Find the length of this path. -1 y
In Exercises match the polar equation with the correct graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).](a)(b)(c)(d)(e)(f) N * 21 RIN
In Exercises identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.5x² + y² - 20x + 19 = 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. r = 1 1- cos 0
In Exercises identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.3x² + 2y² - 12x +12y + 29 = 0
Determine the polar equation of the set of all points (r, θ), the product of whose distancesfrom the points (1, 0) and (-1, 0) is equal to 1, as shown in thefigure.
Four dogs are located at the corners of a square with sides of length d. The dogs all move counterclockwise at the same speed directly toward the next dog, as shown in the figure. Find the polar equation of a dog's path as it spirals toward the center of the square. d d d d
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 identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.12x² - 12y² - 12x + 24y - 45 = 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. = || 3 2 + 6 sin 0
In Exercises find an equation of the parabola. Vertex: (0, 2) Directrix: x = -3
In Exercises identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.x² - 6x - 8y + 1 = 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. r = 4 1 + cos 0
In Exercises find an equation of the parabola. Vertex: (2, 6) Focus: (2, 4)
In Exercises identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.9x² + 25y² + 18x - 100y - 116 = 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. r = 10 5 + 4 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. 5 -1 + 2 cos 0
In Exercises find an equation of the ellipse. Center: (0, 0) Major axis: vertical Points on the ellipse: (1, 2), (2, 0)
In Exercises find an equation of the ellipse. Center: (0, 0) Focus: (5, 0) Vertex: (7,0)
Consider a circle of radius a tangent to the y-axis and the line x = 2a, as shown in the figure. Let A be the point where the segment OB intersects the circle. The cissoid of Diocles consists of all points P such that OP = AB.(a) Find a polar equation of the cissoid.(b) Find a set of parametric
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 2 + cos 0
In Exercises find an equation of the ellipse. Vertices: (3, 1), (3, 7) Eccentricity:
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. || -6 3 + 7 sin 0
Use a graphing utility to graph the polar equation r = cos 5θ + n cos θ for 0 ≤ θ < π and for the integers n = - 5 to n = 5. What values of n produce the "heart" portion of the curve? What values of n produce the "bell" portion? (This curve, created by Michael W. Chamberlin, appeared in
In Exercises find an equation of the ellipse. Foci: (0, +7) Major axis length: 20
In Exercises find an equation of the hyperbola. Vertices: (0,8) Asymptotes: y = ±2x
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. 1 1 + sin 0
In Exercises find an equation of the hyperbola. Vertices: (±2, 0) Asymptotes: y = ±32x
In Exercises use a graphing utility to graph the polar equation. Identify the graph and find its eccentricity. || - 15 2 + 8 sin 0
In Exercises find an equation of the hyperbola. Center: (0, 0) Vertex: (0, 3) Focus: (0, 6)
In Exercises use a graphing utility to graph the polar equation. Identify the graph and find its eccentricity. r = 3 - 4 + 2 sin 0
In Exercises find an equation of the hyperbola. Vertices: (+7, -1) Foci: (+9, -1)
In Exercises use a graphing utility to graph the polar equation. Identify the graph and find its eccentricity. r = 6 6 + 7 cos 0
In Exercises use a graphing utility to graph the polar equation. Identify the graph and find its eccentricity. - 10 1 - cos 0
In Exercises use a graphing utility to graph the conic. Describe how the graph differs from the graph in the indicated exercise. 4 1 + cos(0 - π/3)
A cross section of a large parabolic antenna is modeled by the graph ofThe receiving and transmitting equipment is positioned at the focus.(a) Find the coordinates of the focus.(b) Find the surface area of the antenna. y = x² 200' -100≤x≤ 100.
In Exercises use a graphing utility to graph the conic. Describe how the graph differs from the graph in the indicated exercise. r = 10 5 + 4 sin(0π/4)
Consider the ellipse(a) Find the area of the region bounded by the ellipse.(b) Find the volume of the solid generated by revolving the region about its major axis. x² + = 1. 25 9
In Exercises use a graphing utility to graph the conic. Describe how the graph differs from the graph in the indicated exercise. r = 6 2 + cos(0 + π/6)
Write the equation for the ellipse rotated π/6 radian clockwise from the ellipse r = 8 8 + 5 cos 0
In Exercises sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter.x = 1 + 8t, y = 3 - 4t
In Exercises use a graphing utility to graph the conic. Describe how the graph differs from the graph in the indicated exercise. || -6 3 + 7 sin(0 + 2π/3)
In Exercises sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter.x = t - 6, y = t²
Write the equation for the parabola rotated π/6 radian counterclockwise from the parabola r = 9 1 + sin 0
In Exercises sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter.x = et - 1, y = e³t
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 = 1 Directrix x = -3

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