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mathematics
calculus 10th edition

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

(a) Show that the equation of an ellipse can be written as(b) Use a graphing utility to graph the ellipse(c) Use the results of part (b) to make a conjecture about the change in the shape of the ellipse as e approaches 0. (x - h)² a² + (y - k)² a²(1 - e²) = 1.
Find the minimum value of (u (n − 1)² + ( √2 − u² − 2)² /2 for 0 < u 0.
Show that the equation of the tangent lineat the point to x² ₂2 y² b² 1
A hyperbolic mirror (used in some telescopes) has the property that a light ray directed at the focus will be reflected to the other focus. The mirror in the figure has the equation (x²/36) - (y2/64) = 1. At whichpoint on the mirror will light from the point (0, 10) be reflectedto the other focus?
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 C is the circumference of the ellipse then 2πb ≤ C ≤ 2πα. b² 1, b < a
For a point P on an ellipse, let d be the distance from the center of the ellipse to the line tangent to the ellipse at P. Prove that (PF₁)(PF₂)d2 is constant as P varies on the ellipse, where PF1 and PF2 are the distances from P to the foci F1 and F₂ of the ellipse.
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.Every tangent line to a hyperbola intersects the hyperbola only at the point of tangency.
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 the asymptotes of the hyperbola (x²/a²) - (y²/b²) = intersect at right angles, then a = b.
Prove Theorem 10.4 by showing that the tangent line to an ellipse at a point p makes equal angles with lines through p and the foci (see figure).Data from in Theorem 10.4 2 + (-c, 0) Tangent line B P = (xo, Yo α (c, 0), X
LORAN (long distance radio navigation) for aircraft and ships uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is
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 D ≠ 0 or E ≠ 0, then the graph of y² - x² + Dx + Ey = 0is a hyperbola.
The area of the ellipse in the figure is twice the area of the circle. What is the length of the major axis? (-a, 0) (0, 10) (0, -10) (a,0) X
Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form: 6² || = 1.
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 point on a parabola closest to its focus is its vertex.
Use the integration capabilities of a graphing utility to approximate to two-decimal-place accuracy the elliptical integral representing the circumference of the ellipse x² y2 + = 1. 25 49
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.It is possible for a parabola to intersect its directrix.
In Exercises (a) Find the area of the region bounded by the ellipse(b) The volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid)(c) The volume and surface area of the solid generated by revolving the region about its minor axis (oblate
In Exercises (a) Find the area of the region bounded by the ellipse(b) The volume and surface area of the solid generated by revolving the region about its major axis (prolate spheroid)(c) The volume and surface area of the solid generated by revolving the region about its minor axis (oblate
Find an equation of the hyperbola such that for any point on the hyperbola, the difference between its distances from the points (2, 2) and (10, 2) is 6.
Consider a particle traveling clockwise on the elliptical pathThe particle leaves the orbit at the point (-8, 3) and travels in a straight line tangent to the ellipse. At what point will the particle cross the y-axis? 위험 x2 100 25 = 1.
Probably the most famous of all comets, Halley’s comet, has an elliptical orbit with the sun at one focus. Its maximum distance from the sun is approximately 35.29 AU (1 astronomical unit is approximately 92.956 × 106 miles), and its minimum distance is approximately 0.59 AU. Find the
A church window is bounded above by a parabola and below by the arc of a circle (see figure). Find the surface area of the window. 8 ft 8 ft Circle radius 4 ft LLLL
The apogee (the point in orbit farthest from Earth) and the perigee (the point in orbit closest to Earth) of an elliptical orbit of an Earth satellite are given by A and P. Show that the eccentricity of the orbit is e = A - P A + P
A cable of a suspension bridge is suspended (in the shape of a parabola) between two towers that are 120 meters apart and 20 meters above the roadway (see figure). The cable touches the roadway midway between the towers.(a) Find an equation for the parabolic shape of the cable.(b) Find the length
A satellite signal receiving dish is formed by revolving the parabola given by x² = 20y about the y-axis. The radius of the dish is r feet. Verify that the surface area of the dish is given by 2[:/1+() X X 10 2 dx 7[100 + r²)³/² - 1000]. 15
On November 20, 1975, the United States launched the research satellite Explorer 55. Its low and high points above the surface of Earth were 96 miles and 1865 miles. Find the eccentricity of its elliptical orbit.
On November 27, 1963, the United States launched the research satellite Explorer 18. Its low and high points above the surface of Earth were 119 miles and 123,000 miles. Find the eccentricity of its elliptical orbit.
A solar collector for heating water is constructed with a sheet of stainless steel that is formed into the shape of a parabola (see figure). The water will flow through a pipe that is located at the focus of the parabola. At what distance from the vertex is the pipe? INNAN 6 m I'm 1
In parts (a) –(d), describe in words how a plane could intersect with the double-napped cone to form the conic section (see figure).(a) Circle (b) Ellipse(c) Parabola (d) Hyperbola
Earth moves in an elliptical orbit with the sun at one of the foci. The length of half of the major axis is 149,598,000 kilometers, and the eccentricity is 0.0167. Find the minimum distance (perihelion) and the maximum distance (aphelion) of Earth from the sun.
A simply supported beam that is 16 meters long has a load concentrated at the center (see figure). The deflection of the beam at its center is 3 centimeters. Assume that the shape of the deflected beam is parabolic.(a) Find an equation of the parabola. (Assume that the origin is at the center of
Sketch the graphs of x² = 4py for p = 1/4, 1/2, 1, 3/2, and 2 on the same coordinate axes. Discuss the change in the graphs as p increases.
(a) Prove that if any two tangent lines to a parabola intersect at right angles, their point of intersection must lie on the directrix.(b) Demonstrate the result of part (a) by showing that the tangent lines to the parabola x² - 4x - 4y + 8 = 0 at the points (-2, 5) and (3, 5/4) intersect at right
(a) Prove that any two distinct tangent lines to a parabola intersect.(b) Demonstrate the result of part (a) by finding the point of intersection of the tangent lines to the parabolax² - 4x - 4y = 0 at the points (0, 0) and (6, 3).
Define the eccentricity of an ellipse. In your own words, describe how changes in the eccentricity affect the ellipse.
(a) Give the definition of a hyperbola.(b) Give the standard forms of a hyperbola with center at (h, k).(c) Write equations for the asymptotes of a hyperbola.
(a) Give the definition of an ellipse.(b) Give the standard form of an ellipse with center at (h, k).
(a) Give the definition of a parabola.(b) Give the standard forms of a parabola with vertex at (h, k).(c) In your own words, state the reflective property of a parabola.
In Exercises classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.9(x + 3)² = 36 - 4(x - 2)²
In Exercises classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.3(x − 1)² = 6 + 2(y + 1)²
In Exercises find an equation of the ellipse. Center: (1, 2) Major axis: vertical Points on the ellipse: (1,6), (3, 2)
In Exercises find equations for (a) The tangent lines and (b) The normal lines to the hyperbola for the given value of x. y² x² 2 4 1, x = 4
In Exercises find an equation of the hyperbola. Center: (0, 0) Vertex: (6, 0) Focus: (10, 0)
In Exercises classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.2x(x - y) = y(3 - y - 2x)
In Exercises classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.9x² + 9y² - 36x + 6y + 34 = 0
In Exercises find an equation of the hyperbola. Focus: (20, 0) Asymptotes: y = x ±
In Exercises find equations for (a) The tangent lines and (b) The normal lines to the hyperbola for the given value of x. x² - y² = 1₁ x = 6 1, 1, 9
In Exercises classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.y² - 4y = x + 5
In Exercises find an equation of the hyperbola. Vertices: (0, 2), (6,2) 2 3- y = 4 - 3x Asymptotes: y = x
In Exercises find an equation of the hyperbola. Vertices: (2, +3) Point on graph: (0,5)
In Exercises classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.25x² - 10x - 200y - 119 = 0
In Exercises classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.4x² - y² - 4x - 3 = 0
In Exercises find an equation of the hyperbola. Center: (0, 0) Vertex: (0, 2) Focus: (0,4)
In Exercises classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.x² + 4y² - 6x + 16y + 21 = 0
In Exercises find an equation of the hyperbola. Vertices: (2, +3) Foci: (2, +5)
In Exercises find an equation of the hyperbola. Vertices: (0, +4) Asymptotes: y = ±2x
In Exercises find an equation of the hyperbola. Vertices: (±1, 0) Asymptotes: y = ±5x
In Exercises find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. (y + 3)²(x - 5)² 225 64 1
In Exercises find an equation of the ellipse. Center: (0, 0) Major axis: horizontal Points on the ellipse: (3, 1), (4,0)
In Exercises find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid. x² 25 y² 16 || 1
In Exercises find an equation of the ellipse. Center: (0, 0) Focus: (5, 0) Vertex: (6, 0)
In Exercises find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.9x² - 4y² + 54x + 8y + 78 = 0
In Exercises find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.x² - 9y² + 2x - 54y - 80 = 0
In Exercises find an equation of the ellipse. Foci: (0, +9) Major axis length: 22
In Exercises find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.y² - 16x² + 64x - 208 = 0
In Exercises find a geometric power series, centered at 0, for the function. g(x) || 2 3 - x
In Exercises find an equation of the ellipse. Vertices: (3, 1), (3,9) Minor axis length: 6
In Exercises find the center, foci, and vertices of the hyperbola, and sketch its graph using asymptotes as an aid.9x2 - y² - 36x - бу + 18 = 0 -
In Exercises find an equation of the ellipse. Vertices: (0, 3), (8,3) 3 Eccentricity: 4
In Exercises show that the function represented by the power series is a solution of the differential equation. (-3) x2n 2" n! y = Î n=0 y" + 3xy' + 3y = 0
In Exercises use the Ratio Test or the Root Test to determine the convergence or divergence of the series. 18 n=1 n en²
In Exercises use the Ratio Test or the Root Test to determine the convergence or divergence of the series. n=1 4n n 7n 1,
In Exercises use the Ratio Test or the Root Test to determine the convergence or divergence of the series. S n=1 3n-1) 2n + 5 n
In Exercises find the center, foci, vertices, and eccentricity of the ellipse, and sketch its graph.16x² + 25y² - 64x + 150y + 279 = 0
In Exercises use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. n=2 (-1)" In n³ n
In Exercises use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. n=2 (-1)^n n² - 3
In Exercises use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. n=1 (-1)^(n + 1) n² + 1
In Exercises use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. n=1 (-1)" nº
In Exercises use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. (-1)"n n=4 n-3
In Exercises use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. n=1 1 3n - 5
In Exercises use the Alternating Series Test, if applicable, to determine the convergence or divergence of the series. Σ n=1 (−1)" vn n + 1
In Exercises use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. n=1 n+ 1 n(n + 2)
In Exercises use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. n=1 n √n³ + 3n
In Exercises use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. n=2 1 3√n-1
In Exercises use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. n=1 1.3.5 (2n-1) 2.4.6... (2n) -
In Exercises use the Direct Comparison Test or the Limit Comparison Test to determine the convergence or divergence of the series. n=1 1 n³ + 2n
In Exercises use the Integral Test or a p-series to determine the convergence or divergence of the series. 1 2 (-/---/-) n² n=1 n
In Exercises use the Integral Test or a p-series to determine the convergence or divergence of the series. n=1 1 n5/2
In Exercises use the Integral Test or a p-series to determine the convergence or divergence of the series. ∞o 1 5n n=1
In Exercises use the Integral Test or a p-series to determine the convergence or divergence of the series. n=1 In n
In Exercises use the Integral Test or a p-series to determine the convergence or divergence of the series. n=1 1 4/n³
In Exercises use geometric series or the nth-Term Test to determine the convergence or divergence of the series. n=2 (-1)^n In n
In Exercises use the Integral Test or a p-series to determine the convergence or divergence of the series. n=1 2 би + 1
In Exercises use geometric series or the nth-Term Test to determine the convergence or divergence of the series. ∞ (0.36) n=0
In Exercises use geometric series or the nth-Term Test to determine the convergence or divergence of the series. (1.67)n n=0
In Exercises use geometric series or the nth-Term Test to determine the convergence or divergence of the series. n=0 2n + 1 3n + 2
In Exercises find the intervals of convergence of (a) ƒ(x), (b) ƒ'(x),(c) ƒ"(x), (d) ∫ ƒ(x) dx. Include a check for convergence atthe endpoints of the interval. f(x) = = n=0 5 n
A deposit of $125 is made at the end of each month for 10 years in an account that pays 3.5% interest, compounded monthly. Determine the balance in the account at the end of 10 years.
In Exercises show that the function represented by the power series is a solution of the differential equation. x2n 4"(n!)² y = Σ (-1)". n=0 x²y" + xy + x²y = 0
A ball is dropped from a height of 8 meters. Each time it drops h meters, it rebounds 0.7h meters. Find the total distance traveled by the ball.

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