Calculus: Early Transcendentals, Enhanced Etext
12th Edition
ISBN: 9781119777984
Author: Howard Anton; Irl C. Bivens; Stephen Davis
Publisher: Wiley Global Education US
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Textbook Question
Chapter 10.4, Problem 21ES
Find an equation for the ellipse that satisfies the given conditions.
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Consider the following system of equations, Ax=b :
x+2y+3z - w = 2
2x4z2w = 3
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-9x-2y+13z7w = -14
a. Find the solution to the system. Write it as a parametric equation. You can use a
computer to do the row reduction.
b. What is a geometric description of the solution? Explain how you know.
c. Write the solution in vector form?
d. What is the solution to the homogeneous system, Ax=0?
2. Find a matrix A with the following qualities
a. A is 3 x 3.
b. The matrix A is not lower triangular and is not upper triangular.
c. At least one value in each row is not a 1, 2,-1, -2, or 0
d. A is invertible.
Chapter 10 Solutions
Calculus: Early Transcendentals, Enhanced Etext
Ch. 10.1 - Prob. 1QCECh. 10.1 - The graph of the curve described by the parametric...Ch. 10.1 - Suppose that a parametric curve C is given by the...Ch. 10.1 - To find dy/dx directly from the parametric...Ch. 10.1 - Let L be the length of the curve x=lnt,y=sint1t An...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...Ch. 10.1 - Prob. 7ESCh. 10.1 - Sketch the curve by eliminating the parameter, and...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...Ch. 10.1 - Sketch the curve by eliminating the parameter, and...
Ch. 10.1 - Find parametric equations for the curve, and check...Ch. 10.1 - (a) Use a graphing utility to generate the...Ch. 10.1 - Prob. 21ESCh. 10.1 - In each part, match the parametric equation with...Ch. 10.1 - (a) Identify the orientation of the curves in...Ch. 10.1 - (a) Suppose that the line segment from the point...Ch. 10.1 - (a) Show that the line segment joining the points...Ch. 10.1 - (a) By eliminating the parameter, show that if a...Ch. 10.1 - Use a graphing utility and parametric equations to...Ch. 10.1 - Use a graphing utility and parametric equations to...Ch. 10.1 - Use a graphing utility and parametric equations to...Ch. 10.1 - Determine whether the statement is true or false....Ch. 10.1 - Determine whether the statement is true or false....Ch. 10.1 - Determine whether the statement is true or false....Ch. 10.1 - Prob. 36ESCh. 10.1 - Parametric curves can be defined piecewise by...Ch. 10.1 - Find parametric equations for the rectangle in the...Ch. 10.1 - (a) Find parametric equations for the ellipse that...Ch. 10.1 - We will show later in the text that if a...Ch. 10.1 - (a) Find the slope of the tangent line to the...Ch. 10.1 - For the parametric curve in Exercise 41, make a...Ch. 10.1 - Find dy/dxandd2y/dx2 at the given point without...Ch. 10.1 - Find dy/dxandd2y/dx2 at the given point without...Ch. 10.1 - Find at the given point without eliminating the...Ch. 10.1 - Find at the given point without eliminating the...Ch. 10.1 - Find dy/dxandd2y/dx2 at the given point without...Ch. 10.1 - (a) Find the equation of the tangent line to the...Ch. 10.1 - Find all values of t at which the parametric curve...Ch. 10.1 - In the and-1850s the French physicist Jules...Ch. 10.1 - The prolate cycloid x=2cost,y=2tsintt crosses...Ch. 10.1 - Show that the curve x=t2,y=t34t intersects itself...Ch. 10.1 - Show that the curve with parametric equations...Ch. 10.1 - Prob. 59ESCh. 10.1 - Verify that the cycloid described by Formula (10)...Ch. 10.1 - (a) What is the slope of the tangent line at time...Ch. 10.1 - Suppose that a bee follows the trajectory...Ch. 10.1 - Consider the family of curves described by the...Ch. 10.1 - (a) Use a graphing utility to study how the curves...Ch. 10.1 - Find the exact arc length of the curve over the...Ch. 10.1 - Find the exact arc length of the curve over the...Ch. 10.1 - Find the exact arc length of the curve over the...Ch. 10.1 - Find the exact arc length of the curve over the...Ch. 10.1 - Find the exact arc length of the curve over the...Ch. 10.1 - Prob. 71ESCh. 10.1 - Use the parametric equations in Formula (10) to...Ch. 10.1 - Prob. 73ESCh. 10.1 - (a) If a thread is unwound from a fixed circle...Ch. 10.1 - If ftandgt are continuous functions, and if no...Ch. 10.1 - If ftandgt are continuous functions, and if no...Ch. 10.1 - If ftandgt are continuous functions, and if no...Ch. 10.1 - If ftandgt are continuous functions, and if no...Ch. 10.1 - If ftandgt are continuous functions, and if no...Ch. 10.2 - (a) Rectangular coordinates of a point x,y may be...Ch. 10.2 - Find the rectangular coordinates of the points...Ch. 10.2 - Prob. 3QCECh. 10.2 - Prob. 4QCECh. 10.2 - Prob. 1ESCh. 10.2 - Plot the points in polar coordinates....Ch. 10.2 - Find the rectangular coordinates of the points...Ch. 10.2 - Find the rectangular coordinates of the points...Ch. 10.2 - In each part, a point is given in rectangular...Ch. 10.2 - In each part, find polar coordinates satisfying...Ch. 10.2 - Use a calculating utility, where needed, to...Ch. 10.2 - Express the given equations in polar coordinates....Ch. 10.2 - A graph is given in a rectangular r-coordinate...Ch. 10.2 - A graph is given in a rectangular r-coordinate...Ch. 10.2 - A graph is given in a rectangular r-coordinate...Ch. 10.2 - A graph is given in a rectangular r-coordinate...Ch. 10.2 - Prob. 17ESCh. 10.2 - Prob. 19ESCh. 10.2 - Prob. 20ESCh. 10.2 - Prob. 21ESCh. 10.2 - Prob. 22ESCh. 10.2 - Sketch the curve in polar coordinates. r=3Ch. 10.2 - Prob. 24ESCh. 10.2 - Sketch the curve in polar coordinates. r2=2cosCh. 10.2 - Prob. 28ESCh. 10.2 - Prob. 29ESCh. 10.2 - Sketch the curve in polar coordinates. r=1+2sinCh. 10.2 - Sketch the curve in polar coordinates. r=1cosCh. 10.2 - Sketch the curve in polar coordinates. r=3sinCh. 10.2 - Sketch the curve in polar coordinates. r=3+4cosCh. 10.2 - Prob. 35ESCh. 10.2 - Prob. 37ESCh. 10.2 - Sketch the curve in polar coordinates. r2=cos2Ch. 10.2 - Sketch the curve in polar coordinates. r2=16sin2Ch. 10.2 - Sketch the curve in polar coordinates. r=40Ch. 10.2 - Sketch the curve in polar coordinates. r=40Ch. 10.2 - Sketch the curve in polar coordinates. r=2cos2Ch. 10.2 - Sketch the curve in polar coordinates. r=3sin2Ch. 10.2 - Sketch the curve in polar coordinates. r=9sin4Ch. 10.2 - Sketch the curve in polar coordinates. r=2cos3Ch. 10.2 - Determine whether the statement is true or false....Ch. 10.2 - Determine whether the statement is true or false....Ch. 10.2 - Prob. 49ESCh. 10.2 - Prob. 50ESCh. 10.2 - Prob. 53ESCh. 10.2 - Prob. 54ESCh. 10.2 - Prob. 55ESCh. 10.2 - The accompanying figure shows the graph of the...Ch. 10.2 - The accompanying figure shows the Archimedean...Ch. 10.2 - Find equations for the two families of circles in...Ch. 10.2 - (a) Show that if a varies, then the polar equation...Ch. 10.2 - Prob. 60ESCh. 10.2 - A polar graph of r=f is given over the stated...Ch. 10.2 - A polar graph of r=f is given over the stated...Ch. 10.2 - Use the polar graph from the indicated exercise to...Ch. 10.2 - Use the polar graph from the indicated exercise to...Ch. 10.2 - Prob. 65ESCh. 10.2 - Use the result in Exercise 65 to find an equation...Ch. 10.2 - Prob. 67ESCh. 10.2 - Prob. 68ESCh. 10.2 - Find the highest point on the cardioid r=1+cos.Ch. 10.2 - Show that in a polar coordinate system the...Ch. 10.2 - Use the formula obtained in Exercise 71 to find...Ch. 10.2 - In the late seventeenth century the Italian...Ch. 10.2 - Prob. 76ESCh. 10.2 - Vertical and horizontal asymptotes of polar curves...Ch. 10.2 - Prob. 79ESCh. 10.3 - Prob. 1QCECh. 10.3 - Prob. 2QCECh. 10.3 - (a) To find the arc length L of the polar curve...Ch. 10.3 - Prob. 4QCECh. 10.3 - Find the area of the circle r=a by integration.Ch. 10.3 - Find the slope of the tangent line to the polar...Ch. 10.3 - Find the slope of the tangent line to the polar...Ch. 10.3 - Find the slope of the tangent line to the polar...Ch. 10.3 - Find the slope of the tangent line to the polar...Ch. 10.3 - Find the slope of the tangent line to the polar...Ch. 10.3 - Calculate the slopes of the tangent lines...Ch. 10.3 - Find polar coordinates of all points at which the...Ch. 10.3 - Find polar coordinates of all points at which the...Ch. 10.3 - Use a graphing utility to make a conjecture about...Ch. 10.3 - Prob. 13ESCh. 10.3 - Prob. 16ESCh. 10.3 - Prob. 17ESCh. 10.3 - Sketch the polar curve and find polar equations of...Ch. 10.3 - Use Formula (3) to calculate the arc length of the...Ch. 10.3 - Use Formula (3) to calculate the arc length of the...Ch. 10.3 - Use Formula (3) to calculate the arc length of the...Ch. 10.3 - Use Formula (3) to calculate the arc length of the...Ch. 10.3 - (a) Show that the arc length of one petal of the...Ch. 10.3 - Write down, but do not evaluate, an integral for...Ch. 10.3 - Find the area of the shaded region in Exercise...Ch. 10.3 - In each part, find the area of the circle by...Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Find the area of the shaded region.Ch. 10.3 - Find the area of the shaded region.Ch. 10.3 - Find the area of the shaded region.Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Find the area of the region described. The region...Ch. 10.3 - Determine whether the statement is true or false....Ch. 10.3 - Determine whether the statement is true or false....Ch. 10.3 - (a) Find the error: The area that is inside the...Ch. 10.3 - Find the area inside the curve r2=sin2.Ch. 10.3 - Prob. 53ESCh. 10.3 - As illustrated in the accompanying figure, suppose...Ch. 10.3 - (a) Show that the Folium of Descartes x33xy+y3=0...Ch. 10.3 - (a) (Figure 3.1.3a). What is the area that is...Ch. 10.3 - One of the most famous problems in Greek antiquity...Ch. 10.3 - Use Formula (9) of Section 10.1 to derive the arc...Ch. 10.3 - As illustrated in the accompanying figure, let Pr,...Ch. 10.3 - Use the formula for obtained in Exercise 61. (a)...Ch. 10.3 - Use the formula for obtained in Exercise 61. Show...Ch. 10.3 - (a) In the discussion associated with Exercises...Ch. 10.3 - Sketch the surface, and use the formulas in...Ch. 10.3 - Sketch the surface, and use the formulas in...Ch. 10.3 - Sketch the surface, and use the formulas in...Ch. 10.3 - Sketch the surface, and use the formulas in...Ch. 10.4 - Prob. 1QCECh. 10.4 - (a) The equation of the parabola with focus p,0...Ch. 10.4 - (a) Suppose that an ellipse has semimajor axis a...Ch. 10.4 - (a) Suppose that a hyperbola has semifocal axis a...Ch. 10.4 - Prob. 1ESCh. 10.4 - (a) Find the focus and directrix for each parabola...Ch. 10.4 - Find an equation for the parabola that satisfies...Ch. 10.4 - Find an equation for the parabola that satisfies...Ch. 10.4 - Find an equation for the parabola that satisfies...Ch. 10.4 - Find an equation for the ellipse that satisfies...Ch. 10.4 - Find an equation for the ellipse that satisfies...Ch. 10.4 - Find an equation for the ellipse that satisfies...Ch. 10.4 - Find an equation for a hyperbola that satisfies...Ch. 10.4 - Find an equation for a hyperbola that satisfies...Ch. 10.4 - Find an equation for a hyperbola that satisfies...Ch. 10.4 - Determine whether the statement is true or false....Ch. 10.4 - Determine whether the statement is true or false....Ch. 10.4 - Determine whether the statement is true or false....Ch. 10.4 - Determine whether the statement is true or false....Ch. 10.4 - (a) As illustrated in the accompanying figure, a...Ch. 10.4 - (a) Find an equation for the parabolic arch with...Ch. 10.4 - Show that the vertex is the closest point on a...Ch. 10.4 - For the parabolic reflector in the accompanying...Ch. 10.4 - (a) Show that the right and left branches of the...Ch. 10.4 - (a) Show that the right and left branches of the...Ch. 10.4 - An ant is walking on the is such a way that its...Ch. 10.4 - Suppose that the thumbtacks in Figure 10.4.3(b)...Ch. 10.4 - Find the equation of the hyperbola traced by a...Ch. 10.4 - Prob. 41ESCh. 10.4 - Suppose that the base of a solid is elliptical...Ch. 10.4 - Show that an ellipse with semimajor axis a and...Ch. 10.4 - Show that if a plane is not parallel to the axis...Ch. 10.4 - As illustrated in the accompanying figure, a...Ch. 10.4 - As illustrated in the accompanying figure, suppose...Ch. 10.4 - As illustrated in the accompanying figure, suppose...Ch. 10.4 - A nuclear cooling tower is to have a height of h...Ch. 10.4 - Let R be the region that is above the x-axis and...Ch. 10.4 - As illustrated in the accompanying figure, the...Ch. 10.4 - Prove: The line tangent to the parabola x2=4py at...Ch. 10.4 - Prove: The line tangent to the ellipse
at the...Ch. 10.4 - Prove: The line tangent to the hyperbola...Ch. 10.4 - Use the results in Exercises 52 and 53 to show...Ch. 10.4 - Find the coordinates of all points on the...Ch. 10.4 - A line tangent to the hyperbola 4x2y2=36...Ch. 10.4 - Consider the second-degree equation where A and C...Ch. 10.4 - In each part, use the result in Exercise 58 to...Ch. 10.4 - Derive the equation x2=4py. in Figure 10.4.6.Ch. 10.4 - Derive the equation given in Figure 10.4.14.
Ch. 10.4 - Derive the equation given in Figure 10.4.22.
Ch. 10.4 - Prove Theorem 10.4.4. [Hint: Choose coordinate...Ch. 10.4 - Given two intersecting lines, let be the line...Ch. 10.5 - Prob. 1QCECh. 10.5 - Prob. 2QCECh. 10.5 - In each part, determine a rotation angle that...Ch. 10.5 - Express 2x2+xy+2y2=1 in the xy-coordinate system...Ch. 10.5 - Let an xy-coordinate system be obtained by...Ch. 10.5 - Let an xy-coordinate system be obtained by...Ch. 10.5 - Rotate the coordinate axes to remove the xy-term....Ch. 10.5 - Rotate the coordinate axes to remove the xy-term....Ch. 10.5 - Rotate the coordinate axes to remove the xy-term....Ch. 10.5 - Rotate the coordinate axes to remove the xy-term....Ch. 10.5 - Rotate the coordinate axes to remove the xy-term....Ch. 10.5 - Let an xy-coordinate system be obtained by...Ch. 10.5 - Let an xy-coordinate system be obtained by...Ch. 10.5 - Prob. 16ESCh. 10.5 - Let an xy-coordinate system be obtained by...Ch. 10.5 - Let an xy-coordinate system be obtained by...Ch. 10.5 - Prob. 19ESCh. 10.5 - Show that the graph of the given equation is a...Ch. 10.5 - Show that the graph of the given equation is an...Ch. 10.5 - Show that the graph of the given equation is an...Ch. 10.5 - Show that the graph of the given equation is an...Ch. 10.5 - Show that the graph of the given equation is an...Ch. 10.5 - Show that the graph of the given equation is a...Ch. 10.5 - Show that the graph of the given equation is a...Ch. 10.5 - Show that the graph of the given equation is a...Ch. 10.5 - Show that the graph of the given equation is a...Ch. 10.5 - Show that the graph of the equation x+y=1 is a...Ch. 10.5 - Derive the expression for Bin9.Ch. 10.5 - Use (9) to prove that B24AC=B24AC for all values...Ch. 10.5 - Use (9) to prove that A+C=A+C for all values of .Ch. 10.5 - Prob. 36ESCh. 10.5 - Prove: If B0, then the graph of x2+Bxy+F=0 is a...Ch. 10.6 - Prob. 1QCECh. 10.6 - Prob. 2QCECh. 10.6 - If the distance from a vertex of an ellipse to the...Ch. 10.6 - If the distance from a vertex of a hyperbola to...Ch. 10.6 - Find the eccentricity and the distance from the...Ch. 10.6 - Find the eccentricity and the distance from the...Ch. 10.6 - Prob. 3ESCh. 10.6 - Prob. 5ESCh. 10.6 - Prob. 6ESCh. 10.6 - Prob. 7ESCh. 10.6 - Prob. 8ESCh. 10.6 - Find the distance from the pole to the vertices,...Ch. 10.6 - Find the distance from the pole to the vertices,...Ch. 10.6 - Prob. 11ESCh. 10.6 - Prob. 12ESCh. 10.6 - Prob. 13ESCh. 10.6 - Prove that a hyperbola is an equilateral hyperbola...Ch. 10.6 - (a) Show that the coordinates of the point P on...Ch. 10.6 - (a) Show that the eccentricity of an ellipse can...Ch. 10.6 - (a) Show that the eccentricity of a hyperbola can...Ch. 10.6 - Prob. 18ESCh. 10 - (a) Suppose that the equations x=ft,y=gt describe...Ch. 10 - (a) Find the slope of the tangent line to the...Ch. 10 - Find dy/dx and d2y/dx2att=2 for the parametric...Ch. 10 - Find all values of t at which a tangent line to...Ch. 10 - Find the exact arc length of the curve...Ch. 10 - In each part, find the rectangular coordinates of...Ch. 10 - Prob. 8RECh. 10 - In each part, use a calculating utility to...Ch. 10 - Prob. 10RECh. 10 - In each part, identify the curve by converting the...Ch. 10 - Prob. 13RECh. 10 - Prob. 14RECh. 10 - Prob. 15RECh. 10 - Sketch the curve in polar coordinates. r2=sin2Ch. 10 - Sketch the curve in polar coordinates. r=3cosCh. 10 - Prob. 18RECh. 10 - A parametric curve of the form...Ch. 10 - (a) Find the arc length of the polar curve...Ch. 10 - Find the area of the region that is enclosed by...Ch. 10 - Find the area of the region that is common to the...Ch. 10 - Find the area of the region that is inside the...Ch. 10 - Sketch the parabola, and label the focus, vertex,...Ch. 10 - Sketch the parabola, and label the focus, vertex,...Ch. 10 - Sketch the ellipse, and label the foci, the...Ch. 10 - Sketch the ellipse, and label the foci, the...Ch. 10 - Sketch the ellipse, and label the foci, the...Ch. 10 - Sketch the hyperbola, and label the vertices,...Ch. 10 - Find an equation for the conic described. A...Ch. 10 - Prob. 40RECh. 10 - Find an equation for the conic described. A...Ch. 10 - It can be shown in the accompanying figure that...Ch. 10 - Prob. 43RECh. 10 - Mickey Mantle is recognized as baseball's...Ch. 10 - Prob. 45RECh. 10 - Rotate the coordinate axes to remove the xy-term,...Ch. 10 - Rotate the coordinate axes to show that the graph...Ch. 10 - Prob. 49RECh. 10 - Prob. 50RECh. 10 - Prob. 51RECh. 10 - Use the parametric equations x=acost,y=bsint to...Ch. 10 - Use Simpson's rule or the numerical integration...Ch. 10 - (a) Calculate the eccentricity of the Earth's...
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