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All Textbook Solutions for Calculus (MindTap Course List)
20E21EPopulations of birds and insects are modeled by the equations dxdt=0.4x0.002xydydt=0.2y+0.000008xy a Which of the variables, x or y, represents the bird population and which represents the insect population? Explain. b Find the equilibrium solutions and explain their significance. c Find an expression for dy/dx. d The direction field for the differential equation in part c is shown. Use it to sketch the phase trajectory corresponding to initial populations of 100 birds and 40, 000 insects. Then use the phase trajectory to describe how both populations change. e Use part d to make rough sketches of the bird and insect populations as functions of time. How are these graphs related to each other?23E24EFind all functions f such that f is continuous and [f(x)]2=100+0x{[f(t)]2+[f(t)]2}dt for all real x2P3PFind all functions f that satisfy the equation (f(x)dx)(1f(x)dx)=15PA subtangent is a portion of the x-axis that lies directly beneath the segment of a tangent line from the point of contact to the x-axis. Find the curves that pass through the point c, 1 and whose subtangents all have length c.A peach pie is taken out of the oven at 5:00 PM. At that time it is piping hot, 100C. At 5:10 PM its temperature is 80C; at 5:20 PM it is 65C. What is the temperature of the room?Snow began to fall during the morning of February 2 and continued steadily into the afternoon. At noon a snowplow began removing snow from a road at a constant rate. The plow traveled 6 km from noon to 1 PM but only 3 km from 1 PM to 2 PM. When did the snow begin to fall? Hints: To get started, let t be the time measured in hours after noon; let x(t) be the distance traveled by the plow at time t; then the speed of the plow is dx/dt. Let b be the number of hours before noon that it began to snow. Find an expression for the height of the snow at time t. Then use the given information that the rate of removal R in m3/h is constant.A dog sees a rabbit running in a straight line across an open field and gives chase. In a rectangular coordinate system as shown in the figure, assume: iThe rabbit is at the origin and the dog is at the point L, 0 at the instant the dog first sees the rabbit. iiThe rabbit runs up the y-axis and the dog always runs straight for the rabbit. iiiThe dog runs at the same speed as the rabbit. a Show that the dogs path is the graph of the function y=f(x), where y satisfies the differential equation xd2ydx2=1+(dydx)2 b Determine the solution of the equation in part a that satisfies the initial conditions y=y=0 when x=L. Hint: Let z=dy/dx in the differential equation and solve the resulting first-order equation to find z; then integrate z to find y. c Does the dog ever catch the rabbit?a Suppose that the dog in Problem 9 runs twice as fast as the rabbit. Find a differential equation for the path of the dog. Then solve it to find the point where the dog catches the rabbit. b Suppose the dog runs half as fast as the rabbit. How close does the dog get to the rabbit? What are their positions when they are closest?A planning engineer for a new alum plant must present some estimates to his company regarding the capacity of a silo designed to contain bauxite ore until it is processed into alum. The ore resembles pink talcum powder and is poured from a conveyor at the top of the silo. The silo is a cylinder 100 ft high with a radius of 200 ft. The conveyor carries ore at a rate of 60,000ft3/h and the ore maintains a conical shape whose radius is 1.5 times its height. aIf, at a certain time t, the pile is 60 ft high, how long will it take for the pile to reach the top of the silo? bManagement wants to know how much room will be left in the floor area of the silo when the pile is 60 ft high. How fast is the floor area of the pile growing at that height? cSuppose a loader starts removing the ore at the rate of 20,000ft3/h when the height of the pile reaches 90 ft. Suppose, also, that the pile continues to maintain its shape. How long will it take for the pile to reach the top of the silo under these conditions?12P13P14P15Pa An outfielder fields a baseball 280 ft away from home plate and throws it directly to the catcher with an initial velocity of 100 ft/s. Assume that the velocity v(t) of the ball after t seconds satisfies the differential equation dv/dt=110v because of air resistance. How long does it take for the ball to reach home plate? Ignore any vertical motion of the ball. b The manager of the team wonders whether the ball will reach home plate sooner if it is relayed by an infielder. The shortstop can position himself directly between the outfielder and home plate, catch the ball thrown by the outfielder, turn, and throw the ball to the catcher with an initial velocity of 105 ft/s. The manager clocks the relay time of the shortstop catching, turning, throwing at half a second. How far from home plate should the shortstop position himself to minimize the total time for the ball to reach home plate? Should the manager encourage a direct throw or a relayed throw? What if the shortstop can throw at 115 ft/s? c For what throwing velocity of the shortstop does a relayed throw take the same time as a direct throw?14 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. x=1t2, y=2tt2, 1t214 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. x=t3+t, y=t2+2, 2t214 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. x=t+sint, y=cost, t14 Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. x=et+t, y=ett, 2t25E6E7E8E9E10E1118 a Eliminate the parameter to find a Cartesian equation of the curve. b Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x=sin12,y=cos12,1118 a Eliminate the parameter to find a Cartesian equation of the curve. b Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x=12cos,y=2sin,01118 a Eliminate the parameter to find a Cartesian equation of the curve. b Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x=sint,y=csct,0t/21118 a Eliminate the parameter to find a Cartesian equation of the curve. b Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x=et,y=e2t1118 a Eliminate the parameter to find a Cartesian equation of the curve. b Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x=t2,y=lnt1118 a Eliminate the parameter to find a Cartesian equation of the curve. b Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x=t+1,y=t11118 a Eliminate the parameter to find a Cartesian equation of the curve. b Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x=sinht,y=cosht1118 a Eliminate the parameter to find a Cartesian equation of the curve. b Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x=tan2,y=sec,/2/21922 Describe the motion of a particle with position (x,y) as t varies in the given interval. x=5+2cost,y=3+2sint,1t21922 Describe the motion of a particle with position (x,y) as t varies in the given interval. x=2+sint,y=1+3cost,/2t21922 Describe the motion of a particle with position (x,y) as t varies in the given interval. x=5sint,y=2cost,t522E23E24E2527 Use the graphs of x=f(t) and y=g(t) to sketch the parametric curve x=f(t),y=g(t). Indicate with arrows the direction in which the curve is traced as t increases.2527 Use the graphs of x=f(t) and y=g(t) to sketch the parametric curve x=f(t),y=g(t). Indicate with arrows the direction in which the curve is traced as t increases.27EMatch the parametric equations with the graphs labeled IVI. Give reasons for your choices. Do not use a graphing device. a x=t4t+1,y=t2 b x=t22t,y=t c x=sin2t,y=sin(t+sin2t) d x=cos5t,y=sin2t e x=t+sin4t,y=t2+cos3t f x=sin2t4+t2,y=cos2t4+t229E30Ea Show that the parametric equations x=x1+(x2x1)ty=y1+(y2y1)t where 0t1, describe the line segment that joins the points and P1(x1,y1) and P2(x2y2). b Find parametric equations to represent the line segment from (2,7) to (3,1).Use a graphing device and the result of Exercise 31a to draw the triangle with vertices A(1,1), B(4,2), and C(1,5).33Ea Find parametric equations for the ellipse x2/a2+y2/b2=1. Hint: Modify the equations of the circle in Example 2. b Use these parametric equations to graph the ellipse when a=3 and b=1,2,4, and 8. c How does the shape of the ellipse change as b varies?35E36E37E3738 Compare the curves represented by the parametric equations. How do they differ? a x=t,y=t2 b x=cost,y=sec2t c x=et,y=e2t39E40E41EIf a and b are fixed numbers, find parametric equations for the curve that consists of all possible positions of the point P in the figure, using the angle as the parameter. The line segment AB is tangent to the larger circle.A curve, called a witch of Maria Agnesi, consists of all possible positions of the point P in the figure. Show that parametric equations for this curve can be written as x=2acoty=2asin2 Sketch the curve.a Find parametric equations for the set of all points P as shown in the figure such that |OP|=|AB|. This curve is called the cissoid of Diocles after the Greek scholar Diocles, who introduced the cissoid as a graphical method for constructing the edge of a cube whose volume is twice that of a given cube. b Use the geometric description of the curve to draw a rough sketch of the curve by hand. Check your work by using the parametric equations to graph the curve.Suppose that the position of one particle at time t is given by x1=3sinty1=2cost0t2 and the position of a second particle is given by x2=3+costy2=1+sint0t2 a Graph the paths of both particles. How many points of intersection are there? b Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points. c Describe what happens if the path of the second particle is given by x2=3+costy2=1+sint0t246E47EThe swallowtail catastrophe curves are defined by the parametric equations x=2ct4t3,y=ct2+3t4. Graph several of these curves. What features do the curves have in common? How do they change when c increases?49E50E51E52E12 Find dy/dx. x=t1+t,y=1+t2E3E4E36 Find and equation of the tangent to the curve at the point corresponding to the given value of the parameter. x=tcost,y=tsint;t=36 Find and equation of the tangent to the curve at the point corresponding to the given value of the parameter. x=etsint,y=e2t;t=07E8E9E10E11E12E13E14E15E16E17E18E1720 Find the points on the curve where the tangent is horizontal or vertical. If you have a graphing device, graph the curve to check your work. x=cos,y=cos320E21EUse a graph to estimate the coordinates of the lowest point and the leftmost point on the curve x=t42t,y=t+t4. Then find the exact coordinates.23E24E25E26E27Ea Find the slope of the tangent to the astroid x=acos3,y=asin3 terms of . Astroids are explored in the Laboratory Project on page 689. b At what points is the tangent horizontal or vertical? c At what points does the tangent have slope 1 or 1?29E30EUse the parametric equations of an ellipse, x=acos,y=bsin,02, to find the area that it encloses.32E33E34E35ELet R be the region enclosed by the loop of the curve in Example 1. a Find the area of R. b If R is rotated about the x-axis, find the volume of the resulting solid. c Find the centroid of R. A curve C is defined by the parametric equations x=t2,y=t33t.37E38E39E40E41E4144 Find the exact length of the curve. x=ett,y=4et/2,0t243E44E4546 Graph the curve and find its exact length. x=etcost,y=etsint,0t4546 Graph the curve and find its exact length. x=cost+ln(tan12t),y=sint,/4t3/447E48EUse Simpsons Rule with n=6 to estimate the length of the curve x=tet, y=t+et, 6t6.50E51E52EShow that the total length of the ellipse x=asin,y=bcos,ab0, is L=4a0/21e2sin2d where e is the eccentricity of the ellipse (e=c/a,wherec=a2b2).Find the total length of the astroid x=acos3,y=asin3,wherea0.55E56E57E5760 Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. x=sint,y=sin2t,0t/259E60E61E6163 Find the exact area of the surface obtained by rotating the given curve about the x-axis. x=2t2+1/t,y=8t,1t36163 Find the exact area of the surface obtained by rotating the given curve about the x-axis. x=acos3,y=asin3,0/264E65E66EIf f is continuous and f(t)0 for atb, show that the parametric curve x=f(t),y=g(t),atb, can be put in the form y=F(x). Hint: Show that f1 exists.68EThe curvature at a point P of a curve is defined as =|dds| Where is the angle of inclination of the tangent line at P, as shown in the figure. Thus the curvature is the absolute value of the rate of change of with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at P and will be studied in greater detail in Chapter 13. a For a parametric curve x=x(t),y=y(t), derive the formula =|xyxy|[x2+y2]3/2 where the dots indicate derivatives with respect to t so x=dx/dt. Hint: Use =tan1(dy/dx) and Formula 2 to find d/dt. Then use the Chain Rule to find d/ds. b By regarding a curve y=f(x) as the parametric curve x=x,y=f(x), with parameter x, show that the formula in part a becomes =|d2y/dx2|[1+(dy/dx)2]3/270E71E72EA string is wound around a circle and then unwound while being held taut. The curve traced by the point P at the end of the string is called the involute of the circle. If the circle has radius r and center O and the initial position of P is r, 0, and if the parameter is chosen as in the figure, show that parametric equations of the involute are x=r(cos+sin)y=r(sincos)A cow is tied to a silo with radius r by a rope just long enough to reach the opposite side of the silo. Find the grazing area available for the cow.1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E1520 Identify the curve by finding a Cartesian equation for the curve. r2cos2=11520 Identify the curve by finding a Cartesian equation for the curve. r2sin2=121E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50EShow that the curve r=sintan called a cissoid of Diocles has the line x=1 as a vertical asymptote. Show also that the curve lies entirely within the vertical strip 0x1. Use these facts to help sketch the cissoid.52Ea In Example 11 the graphs suggest that the limaon r=1+csin has an inner loop when |c|1. Prove that this is true, and find the values of that correspond to the inner loop. b From Figure 19 it appears that the limaon loses its dimple when c=12. Prove this.54E5560 Find the slope of the tangent line to the given polar curve at the point specified by the value of . r=2cos,=/356E57E58E59E60E61E6164 Find the points on the given curve where the tangent line is horizontal or vertical. r=1sin63E64E65EShow that the curves r=asin and r=acos intersect at right angles.67E68E69E70E71E72E73E74E75E76E77E78E14 Find the area of the region that is bounded by the given curve and lies in the specified sector. r=e/4, /214 Find the area of the region that is bounded by the given curve and lies in the specified sector. r=cos, 0/614 Find the area of the region that is bounded by the given curve and lies in the specified sector. r=sin+cos, 04E58 Find the area of the shaded region. r2=sin258 Find the area of the shaded region. r=2+cos58 Find the area of the shaded region. r=4+3sin58 Find the area of the shaded region. r=ln, 12912 Sketch the curve and find the area that it encloses. r=2sin912 Sketch the curve and find the area that it encloses. r=1sin912 Sketch the curve and find the area that it encloses. r=3+2cos912 Sketch the curve and find the area that it encloses. r=2cos13E14E15E16E1721 Find the area of the region enclosed by one loop of the curve. r=4cos318E1721 Find the area of the region enclosed by one loop of the curve. r=sin41721 Find the area of the region enclosed by one loop of the curve. r=2sin51721 Find the area of the region enclosed by one loop of the curve. r=1+2sin(innerloop)Find the area enclosed by the loop of the strophoid r=2cossec.23E2328 Find the area of the region that lies inside the first curve and outside the second curve. r=1sin, r=125E26E2328 Find the area of the region that lies inside the first curve and outside the second curve. r=3cos, r=1+cos28E2934 Find the area of the region that lies inside both curves. r=3sin, r=3cos2934 Find the area of the region that lies inside both curves. r=1+cos, r=1cos31E32E33E34E35EFind the area between a larger loop and enclosed small loop of the curve r=1+2cos3.37E3742 Find all points of intersection of the given curves. r=1+cos, r=1sin3742 Find all points of intersection of the given curves. r=2sin2, r=140E41E42E43EWhen recording live performances, sound engineers often use a microphone with a cardioid pickup pattern because it suppresses noise from the audience. Suppose the microphone is placed 4 m from the front of the stage as in the figure and the boundary of the optimal pickup region is given by the cardioid r=8+8sin, where r is measured in meters and the microphone is at the pole. The musicians want to know the area they will have on stage within the optimal pickup range of the microphone. Answer their question.45E4548 Find the exact length of the polar curve. r=5, 024548 Find the exact length of the polar curve. r=2, 048E4950 Find the exact length of the curve. Use a graph to determine the parameter interval. r=cos4(/4)50E51E52E53E54Ea Use Formula 10.2 to show that the area of the surface generated by rotating the polar curve r=f()ab where f is continuous and 0ab about the polar axis is S=ba2rsinr2+(drd)2d b Use the formula in part a to find the surface area generated by rotating lemniscate r2=cos2 about the polar axis.a Find a formula for the area of the surface generated by rotating the polar curve r=f(), ab where f is continuous and 0ab, about the line is =/2. b Find the surface area generated by rotating lemniscate r2=cos2 about the line =/2.18 Find the vertex, focus, and directrix of the parabola and sketch its graph. x2=6y2E3E4E18 Find the vertex, focus, and directrix of the parabola and sketch its graph. (x+2)2=8(y3)6E7E8E910 Find an equation of the parabola. Then find the focus and directrix.910 Find an equation of the parabola. Then find the focus and directrix.1116 Find the vertices and foci of the ellipse and sketch its graph. x22+y24=112E13E1116 Find the vertices and foci of the ellipse and sketch its graph. 100x2+36y2=22515E16E1718 Find an equation of the ellipse. Then find its foci.18E19E