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All Textbook Solutions for Calculus (MindTap Course List)
38 Describe and sketch the surface. x2+z2=138 Describe and sketch the surface. 4x2+y2=438 Describe and sketch the surface. z=1y238 Describe and sketch the surface. y=z238 Describe and sketch the surface. xy=138 Describe and sketch the surface. z=sinya Find and identify the traces of the quadric surface x2+y2z2=1 and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1. b If we change the equation in part a to x2y2+z2=1, how is the graph affected? c What if we change the equation in part a to x2+y2+2yz2=0?a Find and identify the traces of the quadric surface x2y2+z2=1 and explain why the graph looks like the graph of the hyperboloid of one sheet in Table 1. b If the equation in part a is changed to x2y2z2=1, what happens to the graph? Sketch the new graph.1120 Use traces to sketch and identify the surface. x=y2+4z21120 Use traces to sketch and identify the surface. 4x2+9y2+9z2=361120 Use traces to sketch and identify the surface. x2=4y2+z21120 Use traces to sketch and identify the surface. z24x2y2=41120 Use traces to sketch and identify the surface. 9y2+4z2=x2+361120 Use traces to sketch and identify the surface. 3x2+y+3z2=01120 Use traces to sketch and identify the surface. x29+y225+z44=11120 Use traces to sketch and identify the surface. 3x2y2+3z2=01120 Use traces to sketch and identify the surface. y=z2x21120 Use traces to sketch and identify the surface. x=y2z22128 Match the equation with reasons for your choice. its graph labeled I-VIII. Give reasons for your choice. x2+4y2+9z2=12128 Match the equation with reasons for your choice. its graph labeled I-VIII. Give reasons for your choice. 9x2+4y2+z2=12128 Match the equation with reasons for your choice. its graph labeled I-VIII. Give reasons for your choice. x2y2+z2=124E25E2128 Match the equation with reasons for your choice. its graph labeled I-VIII. Give reasons for your choice. y2=x2+2z22128 Match the equation with reasons for your choice. its graph labeled I-VIII. Give reasons for your choice. x2+2z2=128E2930 Sketch and identify a quadric surface that could have the traces shown. Tracesinx=kTracesiny=k30E31E3138 Reduce the equation to one of the standard forms, classify the surface, and sketch it. 4x2y+2z2=033E34E35E3138 Reduce the equation to one of the standard forms, classify the surface, and sketch it. x2y2z24x2z+3=037E38E39E40E41E42E43E44EFind an equation for the surface obtained by rotating the curve y=x about the x-axis.46E47EFind an equation for the surface consisting of all points P for which the distance from P to the x-axis is twice the distance from P to the yz-plane. Identify the surface.Traditionally, the earths surface has been modeled as a sphere, but the World Geodetic System of 1984 WGS-84 uses an ellipsoid as a more accurate model. It places the center of the earth at the origin and the north pole on the positive z-axis. The distance from the center to the poles is 6356.523 km and the distance to a point on the equator is 6378.137 km. a Find an equation of the earths surface as used by WGS-84. b Curves of equal latitude are traces in the planes z=k What is the shape of these curves? c Meridians curves of equal longitude are traces in planes of the form y=mx What is the shape of these meridians?A cooling tower for a nuclear reactor is to be constructed in the shape of a hyperboloid of one sheet see the photo on page 879. The diameter at the base is 280 m and the minimum diameter, 500 m above the base, is 200 m. Find an equation for the tower.Show that if the point (a,b,c) lies on the hyperbolic paraboloid z=y2x2, then the lines with parametric equations x=a+t,y=b+t,z=c+2(ba)t and x=a+t,y=b+t,z=c2(ba)t both lie entirely on this paraboloid. This shows that the hyperbolic paraboloid is what is called a ruled surface; that is, it can be generated by the motion of a straight line. In fact, this exercise shows that through each point on the hyperbolic paraboloid there are two generating lines. The only other quadric surfaces that are ruled surfaces are cylinders, cones, and hyperboloids of one sheet.Show that the curve of intersection of the surfaces x2+2y2z2+3x=1 and 2x2+4y22z25y=0 lies in a plane.53EWhat is the difference between a vector and a scalar?2CC3CC4CC5CC6CC7CC8CC9CC10CC11CC12CC13CC14CC15CC16CC17CCWhat are the traces of a surface? How do you find them?19CC1TFQ2TFQ3TFQ4TFQ5TFQ6TFQ7TFQ8TFQ9TFQ10TFQ11TFQ12TFQ13TFQ14TFQ15TFQ16TFQ17TFQ18TFQ19TFQ20TFQ21TFQ22TFQ1E2E3ECalculate the given quantity if a=i+j-2kb=3i-2j+kc=j-5k a 2a+3b b |b| c ab d (ab) e |bc| f a(bc) g cc h a(bc) i compab j projab k The angle between a and b correct to the nearest degree5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20EFind the point in which the line with parametric equations x=2t,y=1+3t,z=4t intersects the plane 2xy+z=2.22E23Ea Show that the planes x+yz=1 and 2x3y+4z=5 are neither parallel nor perpendicular. b Find, correct to the nearest degree, the angle between these planes.25Ea Find an equation of the plane that passes through the points A(2,1,1),B(1,1,10), and C(1,3,4). b Find symmetric equations for the line through B that is perpendicular to the plane in part a. c. A second plane passes through (2,0,4) and has normal vector 2,4,3. Show that the acute angle between the planes is approximately 43. d Find parametric equations for the line of intersection of the two planes.27E28E29E30E31E32E33E34E35E36E37E38EEach edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius r. The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus the balls are tightly packed in the box see the figure. Find r. If you have trouble with this problem, read about the problem-solving strategy entitled Use Analogy on page 98.Let B be a solid box with length L, width W, and height H. Let S be the set of all points that are a distance at most 1 from some point of B. Express the volume of S in terms of L, W, and H.Let L be the line of intersection of the planes cx+y+z=c and, xcy+cz=1 where c. is a real number. a Find symmetric equations for L. b As the number c varies, the line L sweeps out a surface S. Find an equation for the curve of intersection of S with the horizontal plane z=t the trace of S in the plane z=t. c Find the volume of the solid bounded by S and the planes z = 0 and z = 1.A plane is capable of flying at a speed of 180 km/h in still air. The pilot takes off from an airfield and heads due north according to the planes compass. After 30 minutes of flight time, the pilot notices that, due to the wind, the plane has actually traveled 80 km at an angle 5 east of north. a What is the wind velocity? b In what direction should the pilot have headed to reach the intended destination?5PFind an equation of the largest sphere that passes through the point (1,1,4) and is such that each of the points (x,y,z) inside the sphere satisfies the condition x2+y2+z2136+2(x+2y+3z)7PA solid has the following properties. When illuminated by rays parallel to the z-axis, its shadow is a circular disk. If the rays are parallel to the y-axis, its shadow is a square. If the rays are parallel to the x-axis, its shadow is an isosceles triangle. In Exercise 12.1.48 you were asked to describe and sketch an example of such a solid, but there are many such solids. Assume that the projection onto the xz-plane is a square whose sides have length 1. a What is the volume of the largest such solid? b Is there a smallest volume?1E2E3EFind the limit. limt1(t2tt1i+t+8j+sintlntk)5E6ESketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t)=sint,t8E9E10E11E12E13E14E15E16E17E18E19E20E21EMatch the parametric equations with the graphs labeled IIV. Give reasons for your choices. x=cost,y=sint,z=1/(1+t2)Match the parametric equations with the graphs labeled IIV. Give reasons for your choices. x=t,y=1/(1+t2),z=t2Match the parametric equations with the graphs labeled IIV. Give reasons for your choices. x=cost,y=sint,z=cos2t25E26E27E28E29E30E31E32E33E34E35EUse a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and view-points that reveal the true nature of the curve. r(t)=cos(8cost)sint,sin(8cost)sint,costUse a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and view-points that reveal the true nature of the curve. r(t)=cos2t,cos3t,cos4tGraph the curve with parametric equations x=sint,y=sin2t,z=cos4t. Explain its shape by graphing its projections onto the three coordinate planes.Graph the curve with parametric equations x=(1+cos16t)costy=(1+cos16t)sintz=1+cos16t Explain the appearance of the graph by showing that it lies on a cone.40EShow that the curve with parametric equations x=t2,y=13t,z=1+t3 passes through the points (1,4,0) and (9,8,28) but not through the point (4,7,6).42E43E44E45E46ETry to sketch by hand the curve of intersection of the circular cylinder x2+y2=4 and the parabolic cylinder z=x2. Then find parametric equations for this curve and use these equations and a computer to graph the curve.Try to sketch by hand the curve of intersection of the parabolic cylinder y=x2 and the top half of the ellipsoid x2+4y2+4z2=16. Then find parametric equations for this curve and use these equations and a computer to graph the curve.If two objects travel through space along two different curves, its often important to know whether they will collide. Will a missile hit its moving target? Will two aircraft collide? The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions r1(t)=t2,7t12,t2r2(t)=4t3,t2,5t6 for t0. Do the particles collide?50Ea Graph the curve with parametric equations x=2726sin8t839sin18ty=2726cos8t+839cos18tz=14465sin5t b Show that the curves lies on the hyperboloid of one sheet 144x2+144y225z2=100.52E53E54E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29EFind parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. x=2cost,y=2sint,z=4cos2t;(3,1,2)31E32E33E34EEvaluate the integral. 02(tit3j+3t5k)dt36E37E38EEvaluate the integral. (sec2ti+t(t2+1)3j+t2lntk)dt40E41E42E43EProve Formula 3 of Theorem 3.Prove Formula 5 of Theorem 3.46E47EIf u and v are the vector functions in Exercise 47, use Formula 5 of Theorem 3 to find ddt[u(t)v(t)]49E50EIf r(t)=acost+bsint, where a and b are constant vectors, show that r(t)r(t)=ab.52E53EFind an expression for ddt[u(t)(v(t)w(t))].55E56E57E58EFind the length of the curve. r(t)=t,3cost,3sint,5t52E3E4EFind the length of the curve. r(t)=i+t2j+t3k,0t16E7EFind the length of the curve correct of four decimal places. Use a calculator to approximate the integral. r(t)=t,et,tet,1t39EGraph the curve with parametric equations x=sint,y=sin2t,z=sin3t. Find the total length of this curve correct to four decimal places.Let C be the curve of intersection of the parabolic cylinder x2=2y and the surface 3z=xy. Find the exact length of C from the origin to the point (6,18,36).Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x2+y2=4 and the plane x+y+z=2.a Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P, and b find the point 4 units along the curve in the direction of increasing t from P. r(t)=(5t)i+(4t3)j+3tk,P(4,1,3)a Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P, and b find the point 4 units along the curve in the direction of increasing t from P. r(t)=etsinti+etcostj+2etk,P(0,1,2)15EReparametrize the curve r(t)=(2t2+11)i+2tt2+1j with respect to arc length measured from the point (1,0) in the direction of increasing t. Express the reparametrization in its simplest form. What can you conclude about the curve?a Find the unit tangent and unit normal vectors T(t) and N(t). b Use Formula 9 to find the curvature. r(t)=t,3cost,3sint18E19E20EUse Theorem 10 to find the curvature. r(t)=t3j+t2kUse Theorem 10 to find the curvature. r(t)=ti+t2j+etk23EFind the curvature of r(t)=t2,lnt,tlnt at the point (1,0,0).Find the curvature of r(t)=t,t2,t3 at the point (1,1,1).Graph the curve with parametric equations x=cost,y=sint,z=sin5t and find the curvature at the point (1,0,0).Use Formula 11 to find the curvature. y=x428EUse Formula 11 to find the curvature. y=xex30E31EFind an equation of a parabola that has curvature 4 at the origin.a Is the curvature of the curve C shown in the figure greater at P or at Q? Explain. b Estimate the curvature at P and at Q by sketching the osculating circles at those points.34E35E36E37ETwo graphs, a and b, are shown. One is a curve y=f(x) and the other is the graph of its curvature function y=(x). Identify each curve and explain your choices.Two graphs, a and b, are shown. One is a curve y=f(x) and the other is the graph of its curvature function y=(x). Identify each curve and explain your choices.40E41E42E43E44E45E46E47E48EFind equations of the normal plane and osculating plane of the curve at the given point. x=sin2t,y=cos2t,z=4t;(0,1,2)Find equations of the normal plane and osculating plane of the curve at the given point. x=lnt,y=2t,z=t2;(0,2,1)