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All Textbook Solutions for Calculus Volume 3
The position vector for a particle is r(t)=ti+t2j+t3k . The graph is shown here: 81. Find the speed of the particle at time t=2sec .The position vector for a particle is r(t)=ti+t2j+t3k . The graph is shown here: 82. Find the acceleration at time t=2sec .A particle travels along the path of a helix with the equation r(t)=cos(t)i+sin(t)j+tk . See the graph presented here: Find the following: 83. Velocity 0f the particle at any timeA particle travels along the path of a helix with the equation r(t)=cos(t)i+sin(t)j+tk . See the graph presented here: Find the following: 84. Speed 0f the particle at any timeA particle travels along the path of a helix with the equation r(t)=cos(t)i+sin(t)j+tk . See the graph presented here: Find the following: 85. Acceleration of the particle at any timeA particle travels along the path of a helix with the equation r(t)=cos(t)i+sin(t)j+tk . See the graph presented here: Find the following: 86 Find the unit tangent vector for the helix.A particle travels along the path of an ellipse with the equation r(t)=costi+2sintj+0k . Find the following: 87. Velocity of the particleA particle travels along the path of an ellipse with the equation r(t)=costi+2sintj+0k . Find the following: 88. Speed of the particle at t=4 .A particle travels along the path of an ellipse with the equation r(t)=costi+2sintj+0k . Find the following: 89. Acceleration of the particle at t=4 .Given the vector-valued function r(t)=tant,sect,0 (graph is shown here), find the following: 90. Velocity.Given the vector-valued function r(t)=tant,sect,0 (graph is shown here), find the following: 91. Speed.Given the vector-valued function r(t)=tant,sect,0 (graph is shown here), find the following: 92. AccelerationFind the minimum speed of a particle traveling along the curve r(t)=t+cost,tsintt[0,2] .Given r(t)=ti+2sintj+2costk and u(t)=1ti+2sintj+2costk , find the following: 94. r(t)u(t)Given r(t)=ti+2sintj+2costk and u(t)=1ti+2sintj+2costk , find the following: 95. ddt(r(t)u(t))Now, use the product rule for the derivative of the cross product of two vectors and show this result is the same as the answer for the preceding problem.Find the unit tangent vector T(t) for the following vector-valued functions. 97. r(t)=t,1t . The graph is shown here:Find the unit tangent vector T(t) for the following vector-valued functions. 98. r(t)=tcost,tsintFind the unit tangent vector T(t) for the following vector-valued functions. 99. r(t)=t+1,2t+1,2t+2Evaluate the following integrals: 100. ( e ti+sintj+ 1 2t1k)Evaluate the following integrals: 101. 01r(t)dt , where r(t)=t3,1t+1,etFind the arc length of the curve on the given interval. 102. r(t)=t2i+14tj,0t7 . This portion of the graph is shown here:Find the arc length of the curve on the given interval. 103. r(t)=t2i+(2t2+1)j,1t3Find the arc length of the curve on the given interval. 104. r(t)=2sint,5t,2cost,0t . This portion of the graph is shown here:Find the arc length of the curve on the given interval. 105. r(t)=t2+1,4t3+3,1t0r(t)=etcost,etsint over the interval [0,2] . Here is the portion of the graph on the indicated interval:Find the length of one turn of the helix given by r(t)=12costi+12sintj+34tk.Find the arc length of the vector-valued function r(t)=ti+4tj+3tk over s [0,1] .A particle travels in a circle with the equation of motion r(t)=3costi+3sintj+0k . Find the distance traveled around the Circle by the particle.Set up an integral to find the circumference of the ellipse with the equation r(t)=costi+2sintj+0k .Find the length of the curve r(t)=2t,et,et over the interval 0t1 . The graph is shown here:Find the length of the curve r(t)=2sint,5t,2cost for t[10,10] .The position function for a particle is r(t)=acos(t)i+bsin(t)j. Find the unit tangent vector and the unit normal vector at t=0 .Given r(t)=acos(t)i+bsin(t)j , find the binormal vector B(0) .Given r(t)=2et,etcost,etsint , determine the tangent vector T(t) .Given r(t)=2et,etcost,etsint , determine the unit tangent vector T(t) evaluated at t=0 .Given r(t)=2et,etcost,etsint , find the unit normal vector N(t) evaluated at t=0 , N(0) .Given r(t)=2et,etcost,etsint , find the unit normal vector evaluated at t=0 .Given r(t)=ti+t2j+tk . find the unit tangent vector T(t) . The graph is shown here:Find the unit tangent vector T(t) and unit normal vector N(t) at t=0 for the plane curve r(t)=t34t,5t22 . The graph is shown here:Find the unit tangent vector T(t) for r(t)=3ti+5t2j+2tkFind the principal normal vector to the curve r(t)=6cost,6sint at the paint determined by t=/3 .Find T(t) for the curve r(t)=(t34t)i+(5t22)j .Find N(t) for the curve r(t)=(t34t)i+(5t22)j .Find the unit normal vector N(t) for r(t)=2sint,5t,2cost .Find the unit tangent vector T(t) for r(t)=2sint,5t,2cost .Find the arc length function s(t) for the line segment given by r(t)=33t,4t . Write r as a parameter of s.Parameterize the helix r(t)=costi+sintj+tk using the arc-length parameter s, from t=0 .Parameterize the curve using the arc-length parameter s, at the point at which t=0 for r(t)=etsinti+etcostj .Find the curvature of the curve r(t)=5costi+4sintj at t=/3 . (Note: The graph is an ellipse.)Find the x-coordinate at which the curvature of the curve y=1/x is a maximum value.Find the curvature of the curve r(t)=5costi+5sintj . Does the curvature depend upon the parameter t ?Find the curvature k for the curve y=x14x2 at the point x=2 .Find the curvature k for the curve y=13x3 at the point x=1 .Find the curvature k of the curve r(t)=ti+6t2j+4tk . The graph is shown here:Find the mature of r(t)=2sint,5t,2cost .Find the curvature of r(t)=2ti+etj+etk at point P(0,1,1) .At what point does the curve y=ex have maximum curvature?What happens to the curvature as x on for the curve y=ex ?Find the point of maximum curvature on the curve y=lnx .Find the equations of the normal plane and the osculating plane of the curve r(t)=2sin(3t),t,2cos(3t) at point (0,,2) .Find equations of the osculating circles of the ellipse 4y2+9x2=36 at the points (2,0) and (0,3) .Find the equation for the osculating plane at point t=/4 on the curve r(t)=cos(2t)i+sin(2t)j+t .Find the radius of curvature of 6y=x3 at the point (2,43) .Find the curvature at each point (x,y) on the hyperbola r(t)=acosh(t),bsinh(t) .Calculate the mature of the circular helix r(t)=rsin(t)i+rcos(t)j+tk .Find the radius of curvature of y=ln(x+1) at point (2,ln3) .Find the radius of curvature of the hyperbola xy=1 at point (1,1) .A particle moves along the plane curve C described by r(t)=ti+t2j . Solve the following problems. 149. Find the length of the curve over the interval [0,2] .A particle moves along the plane curve C described by r(t)=ti+t2j . Solve the following problems. 150. Find the curvature of the plane curve at t=0,1,2 .A particle moves along the plane curve C described by r(t)=ti+t2j . Solve the following problems. 151. Describe the curvature as t increases from t=0 to t=2 .The surface of a large cup is formed by revolving the graph of the function y=0.25x1.6 from x=0 to x=5 about the y-axis (measured in centimeters). 152. [T] Use technology to graph the surface.The surface of a large cup is formed by revolving the graph of the function y=0.25x1.6 from x=0 to x=5 about the y-axis (measured in centimeters). 153. Find the mature k of the generating curve as a function of x .The surface of a large cup is formed by revolving the graph of the function y=0.25x1.6 from x=0 to x=5 about the y-axis (measured in centimeters). 154. [T] Use technology to graph the curvature function.How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. Find the velocity function v(t) of the car. Show that vis tangent to the circular me. This means that, without a force to keep the car on the curve, the car will shoot off of it.How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. 2.Show that the speed of the car is R . Use this to show that (24)/|v|=(2)/ .How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. 3.Find the acceleration a. Show that this vector points toward the center of the circle and that |a|=R2 .How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. 4. The force required to produce this circular motion is called the centripetal forte, and it is denoted Fcent . This force points toward the center of the circle (not toward the ground). Show that |Fcent|=(m|v|2)R .How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (Figure 3.21). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that f=N for some positive constant . The constant is called the coefficient of friction. Figure 3.21 The car has three forces acting on it: gravity (denoted by mg), the friction force f, and the force exerted by the road N. Let vmax denote the maximum speed the car can attain through the curve without skidding. In other words, vmax is the fastest speed at which the car can navigate the turn. When the car is navigate at this speed, the magnitude of the centripetal force is |Fcent|=mvmax2R The next three questions deal with developing a formula that relates the speed vmax to the banking angle . 5. Show that Ncos=mg+fsin . Conclude that N=(mg)/(cossin) .How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (Figure 3.21). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that f=N for some positive constant . The constant is called the coefficient of friction. Figure 3.21 The car has three forces acting on it: gravity (denoted by mg), the friction force f, and the force exerted by the road N. Let vmax denote the maximum speed the car can attain through the curve without skidding. In other words, vmax is the fastest speed at which the car can navigate the turn. When the car is navigate at this speed, the magnitude of the centripetal force is |Fcent|=mvmax2R The next three questions deal with developing a formula that relates the speed vmax to the banking angle . 6. The centripetal force is the sum of the forces in the horizontal direction, since the centripetal force points toward the center of the circular curve. Show that fcent=Nsin+fcos . Conclude thatfcent=sin+coscossinmg.How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (Figure 3.21). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that f=N for some positive constant . The constant is called the coefficient of friction. Figure 3.21 The car has three forces acting on it: gravity (denoted by mg), the friction force f, and the force exerted by the road N. Let vmax denote the maximum speed the car can attain through the curve without skidding. In other words, vmax is the fastest speed at which the car can navigate the turn. When the car is navigate at this speed, the magnitude of the centripetal force is |Fcent|=mvmax2R The next three questions deal with developing a formula that relates the speed vmax to the banking angle . 7. Show that vmax2=((sin+cos)/(cossin))gR . Conclude that the maximum speed does not actually depend on the mass of the car. Now that we have a formula relating the maximum speed of the car and the banking angle, we are in a position to answer the questions like the one posed at the beginning of the project. The Bristol Motor Speedway is a NASCAR short track in Bristol, Tennessee. The track has the approximate shape shown in Figure 3.22. Each end of the track is approximately semicircular, so when cars make turns they are traveling along an approximately circular curve. If a car takes the inside track and speeds along the bottom of turn 1, the car travels along a semicircle of radius approximately 211ft with a banking angle of 24 . If the car decides to take the outside track and speeds along the top of turn 1, then the car travels along a semicircle with a banking angle of 28°. (The track has variable angle banking.) Figure 3.22 At the Bristol Motor Speedway, Bristol, Tennessee (a), the turns have an inner radius of about 211ft and a width of 40ft (b). (credit: part (c) photo by Raniel Diaz, Flickr) The coefficient of friction for a normal fire in Elly conditions is approximately 0.7 . Therefore, we assume the coefficient for a NASCAR tire in dry conditions is approximately 0.93 .How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (Figure 3.21). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that f=N for some positive constant . The constant is called the coefficient of friction. Figure 3.21 The car has three forces acting on it: gravity (denoted by mg), the friction force f, and the force exerted by the road N. Let vmax denote the maximum speed the car can attain through the curve without skidding. In other words, vmax is the fastest speed at which the car can navigate the turn. When the car is navigate at this speed, the magnitude of the centripetal force is |Fcent|=mvmax2R The next three questions deal with developing a formula that relates the speed vmax to the banking angle . Now that we have a formula relating the maximum speed of the car and the banking angle, we are in a position to answer the questions like the one posed at the beginning of the project. The Bristol Motor Speedway is a NASCAR short track in Bristol, Tennessee. The track has the approximate shape shown in Figure 3.22. Each end of the track is approximately semicircular, so when cars make turns they are traveling along an approximately circular curve. If a car takes the inside track and speeds along the bottom of turn 1, the car travels along a semicircle of radius approximately 211ft with a banking angle of 24 . If the car decides to take the outside track and speeds along the top of turn 1, then the car travels along a semicircle with a banking angle of 28°. (The track has variable angle banking.) Figure 3.22 At the Bristol Motor Speedway, Bristol, Tennessee (a), the turns have an inner radius of about 211ft and a width of 40ft (b). (credit: part (c) photo by Raniel Diaz, Flickr) The coefficient of friction for a normal fire in Elly conditions is approximately 0.7 . Therefore, we assume the coefficient for a NASCAR tire in dry conditions is approximately 0.93 . Before answering the following questions, note that it is easier to do computations in terms of feet and seconds, and then convert the answers to miles per hour as a final step. 8. In dry conditions, how fast can the car travel through the bottom of the turn without skidding?How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (Figure 3.21). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that f=N for some positive constant . The constant is called the coefficient of friction. Figure 3.21 The car has three forces acting on it: gravity (denoted by mg), the friction force f, and the force exerted by the road N. Let vmax denote the maximum speed the car can attain through the curve without skidding. In other words, vmax is the fastest speed at which the car can navigate the turn. When the car is navigate at this speed, the magnitude of the centripetal force is |Fcent|=mvmax2R The next three questions deal with developing a formula that relates the speed vmax to the banking angle . Now that we have a formula relating the maximum speed of the car and the banking angle, we are in a position to answer the questions like the one posed at the beginning of the project. The Bristol Motor Speedway is a NASCAR short track in Bristol, Tennessee. The track has the approximate shape shown in Figure 3.22. Each end of the track is approximately semicircular, so when cars make turns they are traveling along an approximately circular curve. If a car takes the inside track and speeds along the bottom of turn 1, the car travels along a semicircle of radius approximately 211ft with a banking angle of 24 . If the car decides to take the outside track and speeds along the top of turn 1, then the car travels along a semicircle with a banking angle of 28°. (The track has variable angle banking.) Figure 3.22 At the Bristol Motor Speedway, Bristol, Tennessee (a), the turns have an inner radius of about 211ft and a width of 40ft (b). (credit: part (c) photo by Raniel Diaz, Flickr) The coefficient of friction for a normal fire in Elly conditions is approximately 0.7 . Therefore, we assume the coefficient for a NASCAR tire in dry conditions is approximately 0.93 . Before answering the following questions, note that it is easier to do computations in terms of feet and seconds, and then convert the answers to miles per hour as a final step. 9. In dry conditions, how fast can the car travel through the top of the turn without skidding?How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (Figure 3.21). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that f=N for some positive constant . The constant is called the coefficient of friction. Figure 3.21 The car has three forces acting on it: gravity (denoted by mg), the friction force f, and the force exerted by the road N. Let vmax denote the maximum speed the car can attain through the curve without skidding. In other words, vmax is the fastest speed at which the car can navigate the turn. When the car is navigate at this speed, the magnitude of the centripetal force is |Fcent|=mvmax2R The next three questions deal with developing a formula that relates the speed vmax to the banking angle . Now that we have a formula relating the maximum speed of the car and the banking angle, we are in a position to answer the questions like the one posed at the beginning of the project. The Bristol Motor Speedway is a NASCAR short track in Bristol, Tennessee. The track has the approximate shape shown in Figure 3.22. Each end of the track is approximately semicircular, so when cars make turns they are traveling along an approximately circular curve. If a car takes the inside track and speeds along the bottom of turn 1, the car travels along a semicircle of radius approximately 211ft with a banking angle of 24 . If the car decides to take the outside track and speeds along the top of turn 1, then the car travels along a semicircle with a banking angle of 28°. (The track has variable angle banking.) Figure 3.22 At the Bristol Motor Speedway, Bristol, Tennessee (a), the turns have an inner radius of about 211ft and a width of 40ft (b). (credit: part (c) photo by Raniel Diaz, Flickr) The coefficient of friction for a normal fire in Elly conditions is approximately 0.7 . Therefore, we assume the coefficient for a NASCAR tire in dry conditions is approximately 0.93 . Before answering the following questions, note that it is easier to do computations in terms of feet and seconds, and then convert the answers to miles per hour as a final step. 10. In wet conditions, the coefficient of friction can become as low as 0.1 . If this is the case, how fast can the car travel through the bottom of the turn without skidding?How fast can a racecar travel through a circular tum without skidding and hitting the wall? The answer could depend on several factors: The weight of the car; The friction between the tires and the road; The radius of the circle; The “steepness” of the turn. In this project we investigate this question for NASCAR racecars at the Bristol Motor Speedway in Tennessee. Before considering this track in particular, we use vector functions to develop the mathematics and physics necessary for answering questions such as this. A car of mass m moves with constant angular speed to around a circular curve of radius R (Figure 3.20). The curve is banked at an angle . If the height of the car off the ground is h, then the position of the car at time t is given by the function r(t)Rcos(t),Rsin(t),h . Figure 3.20 Views of a lace ear moving around a track. As the car moves around the curve, three forces act on it: gravity, the force exerted by the road (this force is perpendicular to the ground), and the friction force (Figure 3.21). Because describing the frictional force generated by the tires and the road is complex, we use a standard approximation for the frictional force. Assume that f=N for some positive constant . The constant is called the coefficient of friction. Figure 3.21 The car has three forces acting on it: gravity (denoted by mg), the friction force f, and the force exerted by the road N. Let vmax denote the maximum speed the car can attain through the curve without skidding. In other words, vmax is the fastest speed at which the car can navigate the turn. When the car is navigate at this speed, the magnitude of the centripetal force is |Fcent|=mvmax2R The next three questions deal with developing a formula that relates the speed vmax to the banking angle . Now that we have a formula relating the maximum speed of the car and the banking angle, we are in a position to answer the questions like the one posed at the beginning of the project. The Bristol Motor Speedway is a NASCAR short track in Bristol, Tennessee. The track has the approximate shape shown in Figure 3.22. Each end of the track is approximately semicircular, so when cars make turns they are traveling along an approximately circular curve. If a car takes the inside track and speeds along the bottom of turn 1, the car travels along a semicircle of radius approximately 211ft with a banking angle of 24 . If the car decides to take the outside track and speeds along the top of turn 1, then the car travels along a semicircle with a banking angle of 28°. (The track has variable angle banking.) Figure 3.22 At the Bristol Motor Speedway, Bristol, Tennessee (a), the turns have an inner radius of about 211ft and a width of 40ft (b). (credit: part (c) photo by Raniel Diaz, Flickr) The coefficient of friction for a normal fire in Elly conditions is approximately 0.7 . Therefore, we assume the coefficient for a NASCAR tire in dry conditions is approximately 0.93 . Before answering the following questions, note that it is easier to do computations in terms of feet and seconds, and then convert the answers to miles per hour as a final step. 11. Suppose the measured speed of a car going along the outside edge of the turn is 105mph . Estimate the coefficient of friction for the car’s tires.Given r(t)=(3t22)i+(2tsin(t))j , find the velocity of a particle moving along this curve.Given r(t)=(3t22)i+(2tsin(t))j , find the acceleration vector of a particle moving along the curve in the preceding exercise.Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter t. 157. r(t)=3cost,sint,t2Given the following position functions, find the velocity, acceleration, and speed in terms of the parameter t. 158. r(t)=eti+t2j+tantkGiven the following position functions, find the velocity, acceleration, and speed in terms of the parameter t. 159. r(t)=2costj+3sintk . The graph is shown here:Find the velocity, acceleration, and speed of a particle with the given position function. 160. r(t)=t21,tFind the velocity, acceleration, and speed of a particle with the given position function. 161. r(t)=et,etFind the velocity, acceleration, and speed of a particle with the given position function. 162. r(t)=sint,t,cost . The graph is shown here:The position function of an object is given by r(t)=t2,5t,t216t . At what lime is the speed a minimum?Let r(t)=rcosh(t)i+rsinh(wt)j . Find the velocity and acceleration vectors and show that the acceleration is proportional to r(t) .Consider the motion of a point on the circumference of a rolling circle. As the circle rolls, it generates the cycloid r(t)=(tsin(t))i+(1cos(t))j , where to is the angular velocity of the circle and b is the radius of the circle: 165. Find the equations for the velocity, acceleration, and speed of the particle at any time.A person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector r(t)=(3cost)i+(3sint)j+t2k . The path is similar to that of a helix, although it is not a helix. The graph is shown here: Find the following quantities: 166. The velocity and acceleration vectorsA person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector r(t)=(3cost)i+(3sint)j+t2k . The path is similar to that of a helix, although it is not a helix. The graph is shown here: Find the following quantities: 167. The glider’s speed at any timeA person on a hang glider is spiraling upward as a result of the rapidly rising air on a path having position vector r(t)=(3cost)i+(3sint)j+t2k . The path is similar to that of a helix, although it is not a helix. The graph is shown here: Find the following quantities: 168. The times, if any, at which the glider’s acceleration is orthogonal to its velocityGiven that r(t)=e5tsint,e5tcost,4e5t is the position vector of a moving particle, find the following quantifies: 169. The velocity of the particleGiven that r(t)=e5tsint,e5tcost,4e5t is the position vector of a moving particle, find the following quantifies: 170. The speed of the particleGiven that r(t)=e5tsint,e5tcost,4e5t is the position vector of a moving particle, find the following quantifies: 171. The acceleration of the particleGiven that r(t)=e5tsint,e5tcost,4e5t is the position vector of a moving particle, find the following quantifies: 172. Find the maximum speed of a point on the circumference of an automobile fire of radius 1ft when the automobile is traveling at 55mph .A projectile is shot in the air from ground level with an initial velocity of 500m/sec at an angle of 60 with the horizontal. The graph is shown here: 173. At what lime does the projectile reach maximum height?A projectile is shot in the air from ground level with an initial velocity of 500m/sec at an angle of 60 with the horizontal. The graph is shown here: 174. What is the approximate maximum height of the projectile?A projectile is shot in the air from ground level with an initial velocity of 500m/sec at an angle of 60 with the horizontal. The graph is shown here: 175. At what lime is the maximum range of the projectile attained?A projectile is shot in the air from ground level with an initial velocity of 500m/sec at an angle of 60 with the horizontal. The graph is shown here: 176. What is the maximum range?A projectile is shot in the air from ground level with an initial velocity of 500m/sec at an angle of 60 with the horizontal. The graph is shown here: 177. 1What is the total flight lime of the projectile?A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 178. Determine the maximum height of the projectile.A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 179. Determine the range of the projectile.A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 180. A golf ball is hit in a horizontal direction off the top edge of a building that is 100ft tall. How fast must the ball he launched to land 450ft away?A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 181. A projectile is fired from ground level at an angle of 8 with the horizontal. The projectile is to have a range of 50m . Find the minimum velocity necessary to achieve this range.A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 182. Prove that an object moving in a straight line at a constant speed has an acceleration of zero.A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 183. The acceleration of an object is given by a(t)=tj+tk . The velocity at t=1sec is v(1)=5j and the position of the object at t=1sec is r(1)=0i+0j+0k . Find the object’s position at any time.A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 184. Find r(t) given that a(t)=32j , v(0)=6003i+600j , and r(0)=0 .A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 185. Find the tangential and normal components of acceleration for r(t)=acos(t)i+bsin(t)j at t=0 .A projectile is fired at a height of 1.5m above the ground with an initial velocity of 100m/sec and at an angle of 30 above the horizontal. Use this information to answer the following questions: 186. Given r(t)=t2i+2tj and t=1 . find the tangential and normal components of acceleration.For each of the following problems, find the tangential and normal components of acceleration. 187. r(t)=etcost,etsint,et . The graph is shown here:For each of the following problems, find the tangential and normal components of acceleration. 188. r(t)=cos(2t),sin(2t),1For each of the following problems, find the tangential and normal components of acceleration. 189. r(t)=2t,t2,t33For each of the following problems, find the tangential and normal components of acceleration. 190. r(t)=23(1+t)3/2,23(1t)3/2,2tFor each of the following problems, find the tangential and normal components of acceleration. 191. r(t)=6t,3t2,2t3For each of the following problems, find the tangential and normal components of acceleration. 192. r(t)=t2i+t2j+t3kFor each of the following problems, find the tangential and normal components of acceleration. 193. r(t)=3cos(2t)i+3sin(2t)jFor each of the following problems, find the tangential and normal components of acceleration. 194. Find the position vector-valued function r(t) , given that a(t)=i+etj,v(0)=2j , and r(0)=2i .For each of the following problems, find the tangential and normal components of acceleration. 195. The force on a particle is given by f(t)=(cost)i+(sint)j . The particle is located at point (c,0) at t=0 . The initial velocity of the particle is given by v(0)=v0j . Find the path of the particle of mass m. (Recall, F=ma .)All automobile that weighs 2700lb makes a turn on a flat mad while traveling at 56ft/sec . If the radius of the turn is 70ft , what is the required frictional force to keep the car from skidding?Using Kepler’s laws, it can be shown that v0=2GMr0 is the minimum speed needed when =0 so that an object will escape from the pull of a central force resulting from mass M. Use this result to find the minimum speed when =0 for a space capsule to escape from the gravitational pull of Earth if the probe is at an altitude of 300km above Earth’s surface.Find the lime in years it takes the dwarf planet Pluto to make one orbit about the Sun given that a=39.5A.U.Suppose that the position function for an object in three dimensions is given by the equation r(t)=tcos(t)i+tsin(t)j+3tk . 199. Show that the particle moves. on a circular cone.Suppose that the position function for an object in three dimensions is given by the equation r(t)=tcos(t)i+tsin(t)j+3tk . 200. Find the angle between the velocity and acceleration vectors when t=1.5 .Suppose that the position function for an object in three dimensions is given by the equation r(t)=tcos(t)i+tsin(t)j+3tk . 201. Find the tangential and normal components of acceleration when t=1.5 .True or False? Justify your answer with a proof or a counterexample. 202. A parametric equation that passes through points P and Q can be given by r(t)=t2,3t+1,t2 , where P(1,4,1) and Q(16,11,2) .True or False? Justify your answer with a proof or a counterexample. 203. ddt[u(t)u(t)]=2u'(t)u(t)True or False? Justify your answer with a proof or a counterexample. 204. The curvature of a circle of radius r is constant everywhere. Furthermore, the curvature is equal to 1/r .True or False? Justify your answer with a proof or a counterexample. 205. The speed of a particle with a position function r(t) is (r(t))/(|r(t)|) .Find the domains of the vector-valued functions. 206. r(t)=sin(t),ln(t),tFind the domains of the vector-valued functions. 207. r(t)=et,14t,sec(t)Sketch the tunes. for the following vector equations. Use a calculator if needed. 208. [T]r(t)=t2,t3Sketch the tunes. for the following vector equations. Use a calculator if needed. 209. [T]r(t)=sin(20t)et,cos(20t)et,etFind a vector function that describes the following curves. 210. Intersection of the cylinder x2+y2=4 with the plane x+z=6Find a vector function that describes the following curves. 211. Intersection 0f the cone z=x2+y2 and plane z=y4 .Find the derivatives of u(t),u(t),u(t)u(t) , u(t)u(t) and u(t)u(t) . Find the unit tangent vector. 212. u(t)=et,etFind the derivatives of u(t),u(t),u(t)u(t) , u(t)u(t) and u(t)u(t) . Find the unit tangent vector. 213. u(t)=t2,2t+6,4t512Evaluate the following integrals. 214. (tan( t)sec( t)it e 3tj)Evaluate the following integrals. 215. 14(t)dt , with u(t)=ln(t)t,1t,sin(t4)Find the length for the following curves. 216. r(t)=3(t),4cos(t),4sin(t) for 1t4Find the length for the following curves. 217. r(t)=2i+tj+3t2k for 0t1Reparametrize the following functions with respect to their arc length measured from t=0 in direction of increasing t . 218. r(t)=2ti+(4t5)j+(13t)kReparametrize the following functions with respect to their arc length measured from t=0 in direction of increasing t . 219. r(t)=cos(2t)i+8tjsin(2t)kFind the curvature for the following vector functions. 220. r(t)=(2sint)i+4tj+(2cost)kFind the curvature for the following vector functions. 221. r(t)=2eti+2etj+2tkFind the curvature for the following vector functions. 222. Find the unit tangent vector, the unit normal vector, and the binormal vector for r(t)=2costi+3tj+2sintk .Find the curvature for the following vector functions. 223. Find the tangential and normal acceleration components with the position vector r(t)=cost,sint,et .Find the curvature for the following vector functions. 224. A Ferris Wheel car is mating at a constant speed v and has a constant radius r . Find the tangential and normal acceleration of the Ferris wheel car.Find the curvature for the following vector functions. 225. The position of a particle is given by r(t)=t2,ln(t),sin(t) , where t is measured in seconds and r is measured in meters. Find the velocity, acceleration, and speed functions. What are the position, velocity, speed, and acceleration of the particle at 1sec ?The following problems consider launching a cannonball out of a cannon. The cannonball is shut out of the cannon with an angle and initial velocity v0 . The only force acting on the cannonball is gravity, so we begin with a constant acceleration a(t)=gj . 226. Find the velocity vector function v(t) .The following problems consider launching a cannonball out of a cannon. The cannonball is shut out of the cannon with an angle and initial velocity v0 . The only force acting on the cannonball is gravity, so we begin with a constant acceleration a(t)=gj . 227. Find the position vector r(t) and the parametric representation for the position.The following problems consider launching a cannonball out of a cannon. The cannonball is shut out of the cannon with an angle and initial velocity v0 . The only force acting on the cannonball is gravity, so we begin with a constant acceleration a(t)=gj . 228. At what angle do you need to fire the cannonball for the horizontal distance to be greatest? What is the total distance it would travel?For the following exercises, evaluate each function at the indicated values. 1. W(x,y)=4x2+y2.FindW(21),W(3,6).For the following exercises, evaluate each function at the indicated values. 2. W(x,y)=4x2+y2FindW(2+h,3+h).For the following exercises, evaluate each function at the indicated values. 3. The volume of a right circular cylinder is calculated by . . a function of two variables, V(x,y)=x2y . where x is the radius of the right circular cylinder and y represents the height of the cylinder. Evaluate V(2. 5) and explain what this means.For the following exercises, evaluate each function at the indicated values. 4. An oxygen tank is constructed of a right cylinder of height y and radius x with two hemispheres of radius x mounted on the top and bottom of the cylinder. Express the volume of the cylinder as a function of two variables, x and yq find V( 10. 2). and explain what this means.For the following exercises, find the domain of the function. 5. V(x,y)=4x2+y2For the following exercises, find the domain of the function. 6. f(x,y)=x2+y24For the following exercises, find the domain of the function. 7. f(x,y)=4ln(y2x)For the following exercises, find the domain of the function. 8. g(x,y)=164x2y2For the following exercises, find the domain of the function. 9. z(x,y)=y2x2For the following exercises, find the domain of the function. 10. f(x,y)=y+2x2Find the range of the functions. 11. g(x,y)=164x2y2Find the range of the functions. 12. V(x,y)=4x2+y2Find the range of the functions. 13. z=y2x2For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 14. z(x,y)=y2x2,c=1For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 15. z(x,y)=y2x2,c=4For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 16. g(x,y)=x2+y2;c=4,c=9For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 17. g(x,y)=4xy;c=0,4For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 18. f(x,y)=xy;c=1;c=1For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 19. h(x,y)=2xy;c=0,2,2For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 20. f(x,y)=x2yc=1,0,2For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 21. f(x,y)=xx+y;c=1,0,2For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 22. g(x,y)=x3y;c=1,0,2For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 23. g(x,y)=exy;c=12,3For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 24. f(x,y)=x2;c=4,9For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 25. f(x,y)=xyx;c=2,0,2For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 26. h(x,y)=ln(x2+y2);c=1,0,1For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 27. g(x,y)=ln(yx2)c=2,0,2For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 28. z=f(x,y)=x2+y2,c=3For the following exercises, find the level curves of each function at the indicated value of c to visualize the given function. 29. f(x,y)=y+2x2,c=anyconstantFor the following exercises, find the vertical traces of the functions at the indicated values of x and y, and plot the traces. 30. z=4xy;x=2For the following exercises, find the vertical traces of the functions at the indicated values of x and y, and plot the traces. 31. f(x,y)=3x+y3,x=1For the following exercises, find the vertical traces of the functions at the indicated values of x and y, and plot the traces. 32. z=cosx2+y2x=1Find the domain of the following functions. 33. z=1004x225y2Find the domain of the following functions. 34. z=ln(xy2)Find the domain of the following functions. 35. f(x,y,z)=1364x29y2z2Find the domain of the following functions. 36. f(x,y,z)=49x2y2z2Find the domain of the following functions. 37. f(x,y,z)=16x2y2z23Find the domain of the following functions. 38. f(x,y)=cosx2+y2For the following exercises, plot a graph of the function. 39. z=f(x,y)=x2+y2For the following exercises, plot a graph of the function. 40. z=x2+y2For the following exercises, plot a graph of the function. 41. Use technology to graph z=x2ySketch the following by finding the level curves. Verify the graph using technology. 42. f(x,y)=4x2y2Sketch the following by finding the level curves. Verify the graph using technology. 43. f(x,y)=2x2+y2Sketch the following by finding the level curves. Verify the graph using technology. 44. z=1+ex2y2Sketch the following by finding the level curves. Verify the graph using technology. 45. z=cosx2+y2Sketch the following by finding the level curves. Verify the graph using technology. 46. z=y2x2Sketch the following by finding the level curves. Verify the graph using technology. 47. Describe the contour lines for several values of c for z=x2+y22x2y .Find the level surface for the functions of three variables and describe it. 48. w=(x,y,z)=x2y+z,c=4Find the level surface for the functions of three variables and describe it. 49. w(x,y,z)=x2+y2+z2,c=9the level surface for the functions of three variables and describe it. 50. w(x,y,x)=x2+y2z2,c=4Find the level surface for the functions of three variables and describe it. 51. w(x,y,z)=x2+y2z2,c=4Find the level surface for the functions of three variables and describe it. 52. w(x,y,z)=9x24y2+36z2,c=0For the following exercises, find an equation of the level curve of f that contains the point P. 53. f(x,y)=14x2y2,P(0,1)For the following exercises, find an equation of the level curve of f that contains the point P. 54. g(x,y)=y2arctanx,P(1,2)For the following exercises, find an equation of the level curve of f that contains the point P. 55. g(x,y)=exy(x2+y2),P(1,0)The strength E of an electric field at point (x. y. z) resulting from an infinitely long charged wire lying along the y-axis is given by E(x,y,z)=k/x2+y2 where k is a positive constant. For simplicity, let k = 1 and find the equations of the level surfaces for E = 10 and E = 100.A thin plate made of iron is located in the xv-plane. The temperature T in degrees Celsius at a point P(x. y) is inversely proportional to the square of its distance from the origin. Express T as a function of x and yRefer to the preceding problem. Using the temperature function found there, determine the proportionality constant if the temperature at point P( 1, 2) is 50°C. Use this constant to determine the temperature at point Q(3. 4).Refer to the preceding problem. Find the level curves for T=40C and T = 100°C. and describe what the level curves represent.For the following exercises, find the limit of the function. 60. (x,y)(1,2)xlimFor the following exercises, find the limit of the function. 61. (x,y)(1,2)lim5x2yx2+y2For the following exercises, find the limit of the function. 62. Show that the (x,y)(0,0)lim5x2yx2+y2exists and is the same along the paths: y-axis and x-axis. and along y = x.For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 63. (x,y)(0,0)lim4x2+10y2+44x210y2+6For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 64. (x,y)(11,13)lim1xyFor the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 65. (x,y)(0,1)limy2sinxxFor the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 66. (x,y)(0,0)limsin(x8+y7xy+10)For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 67. (x,y)(/4,1)limytanxy+1For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 68. (x,y)(0,/4)limsecx+23xtanyFor the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 69. (x,y)(2,5)lim(1x5y)For the following exercises, evaluate the limits at the indicated values of x and v. If the limit does not exist, state this and explain why the limit does not exist. 70. (x,y)(2,5)lim(1x5y)For the following exercises, evaluate the limits at the indicated values of x and v. If the limit does not exist, state this and explain why the limit does not exist. 71. (xy)(4,4)limex2y2For the following exercises, evaluate the limits at the indicated values of x and v. If the limit does not exist, state this and explain why the limit does not exist. 72. (x,y)(0,0)lim9x2y2For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 73. (x,y)(1,2)lim(x2y3x3y2+3x+2y)For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 74. (x,y)()limxsin(x+y4)For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 75. (x,y)(0,0)limxy+1x2+y2+1For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 76. (x,y)(0,0)limx2+y2x2+y2+11For the following exercises, evaluate the limits at the indicated values of x and y. If the limit does not exist, state this and explain why the limit does not exist. 77. (x,y)(0,0)limln(x2+y2)For the following exercises, complete the statement. 78. A point ( x0,y0 ) in a plane region R is an interior point of R ifFor the following exercises, complete the statement. 79. A point (x0,y0)in a plane region R is called a boundary point of R if ___________For the following exercises, use algebraic techniques to evaluate the limit. 80. (x,y)(2,1)limxy1xy1For the following exercises, use algebraic techniques to evaluate the limit. 81. (x,y)(0,0)limx44y4x2+2y2For the following exercises, use algebraic techniques to evaluate the limit. 82. (xy)(0,0)limx3y3xyFor the following exercises, use algebraic techniques to evaluate the limit. 83. (xy)(0,0)limx2xyxyFor the following exercises, evaluate the limits of the functions of three variables. 84. (x,y,z)(1,2,3)limxz2y2zxyz1For the following exercises, evaluate the limits of the functions of three variables. 85. (x,y,z)(0,0,0)limx2y2z2x2+y2z2For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. 86. (x,y)(0,0)limxy+y3x2+y2 a. Along the x-axis(y=0) b. Along the y-axis(x=0) c. Along the path y=2xFor the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. 87. Evaluate (x,y)(0,0)limxy+y3x2+y2 using the results of previous problem.For the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. 87. Evaluate (x,y)(0,0)limxy+y3x2+y2 using the results of previous problem. a. Along the x-axis(y=0) b. Along the y-axis(x=0) c. Along the path y=2xFor the following exercises, evaluate the limit of the function by determining the value the function approaches along the indicated paths. If the limit does not exist, explain why not. 89. Evaluate (x,y)(0,0)limx2yx4+y2 using the results of previous problem.Discuss the continuity of the following functions. Find the largest region in the xy-plane in which the following functions are continuous. 90. f(x,y)=sin(xy)Discuss the continuity of the following functions. Find the largest region in the xy-plane in which the following functions are continuous. 91. f(x,y)=ln(x+y)Discuss the continuity of the following functions. Find the largest region in the xy-plane in which the following functions are continuous. 92. f(x,y)=e3xyDiscuss the continuity of the following functions. Find the largest region in the xy-plane in which the following functions are continuous. 93. f(x,y)=1xyFor the following exercises, determine the region in which the function is continuous. Explain your answer. 94. f(x,y)=x2yx2+y2For the following exercises, determine the region in which the function is continuous. Explain your answer. 95. f(x,y)={x2y0x2+y2if(x,y)(0,0)}Show that the function approaches different values along two different paths.) 96. f(x,y)=sin(x2+y2)x2+y2Show that the function approaches different values along two different paths.) 97. Determine whether g(x,y)=x2y2x2y2 , is continuous at (0. 0).Create a plot using graphing software to determine where the limit does not exist. Determine the region of the coordinate plane in which f(x,y)=1x2y is continuous.Determine the region of the xy-plane in which the composite function g(x,y)=arctan(xy2x+y) is continuous. Use technology to support your conclusion.Determine the region of the xv-plane in which f(x. y) = In(x2+ y2- 1) as continuous. Use technology to support your conclusion. (Hint: Choose the range of values for x and y carefully!)At what points in space g(x,y,z)=x2+y22z2 continuous?At what points in space is g(x,y,z)=1x2+z21 continuous?Show that (x,y)(0,0)lim1x2+y2 does not exist at (0, 0) by plotting the graph of the function.[T] Evaluate (x,y)(0,0)limxy2x2+y2 by plotting the function using a CAS. Determine analytically the limit along the path x = y2.[T] a. Use a CAS to draw a contour map of z=9x2y2 b. What is the name of the geometric shape of the level curves? c. Give the general equation of the level curves. d. What is the maximum value of z? e. What is the domain of the function? f. What is the range of the function?True or False: If we evaluate (x,y)(0,0)limf(x) along several paths and each time the limit is I we can conclude that (x,y)(0,0)limf(x)=1Use polar coordinates to find (xy)(0,0)limsin x 2+ y 2 x 2+ y 2You can also find the limit using L’Hôpital’s rule.Use polar coordinates to find urn (xy)(0,0)limcos(x2+y2)Discuss the continuity of f(g(x. y)) where f(i) = 1/t and g(x. y) = 2x — 5y.Given f(x. v) = x2-4y. find h0limf(x+h,y)f(x,y)h.Given f(x,y)=x24y find h0limf(1+h,y)f(1,y)hLord Kelvin and the Age of Earth Figure 4.25 (a) William Thomson (Lord Kelvin). 1824-1907, was a British physicist and electrical engineer. (b) Kelvin used the heat diffusion equation to estimate the age of Earth (credit: modification of work by NASA). During the late 1800s, the scientists of the new field of geology were coming to the conclusion the Earth must be “millions and millions” of years old. At about the same time. Charles Darwin had published his treatise on evolution. Darwin’s view was that evolution needed many millions of years to take place, and he made a bold claim that the Weald chalk fields, where important fossils were found, were the result of 300 million years of erosion. At that time, eminent physicist William Thomson (lord Kelvin) used an important partial differential equation, known as the heat diffusion equation, to estimate the age of Earth by determining how long it would take Earth to cool from molten rock to what we had at tha time. His conclusion was a range of 20 to 4(X) million years, but most likely about 5() million years. For many decades, the proclamations of this irrefutable icon of science did not sit well with geologists or with Darwin. tJR ead Kelvin’s paper (http:Iiwww.openstaxcollege.orgIlI2O KelEarthAge) on estimating the age of the Earth. Kelvin made reasonable assumptions based on what was known in his time, but he also made several assumptions that turned out to be wrong. One incorrect assumption was that Earth is solid and that the cooling was therefore via conduction only, hence justifying the use of the diffusion equation. But the most serious error was a forgivable one—omission of the fact that Earth contains radioactive elements that continually supply heat beneath Earth’s mantle. The discoveiy of radioactivity came near the end of Kelvin’s life and he acknowledged that his calculation would have to be modified. Kelvin used the simple one-dimensional model applied only to Earth’s outer shell, and derived the age from gsaphs and the roughly known temperature gs-adietn near Earth’s surface. Let’s take a look at a more appropriate version of the diffusion equation in radial coordinates, which has the form Tt=K[2T2r+2rTr] (4.23) Here, T(r.t) is temperature as a function of r (measured from the center of Earth) and time i. K is the heat conductivity—for molten rock, in this case. ibe standard method of solving such a partial differential equation is by separation of variables, where we express the solution as the product of functions containing each variable separately. In this case, we would write the temperature as T(r,t)=R(r)f(t).Lord Kelvin and the Age of Earth Figure 4.25 (a) William Thomson (Lord Kelvin). 1824-1907, was a British physicist and electrical engineer. (b) Kelvin used the heat diffusion equation to estimate the age of Earth (credit: modification of work by NASA). During the late 1800s, the scientists of the new field of geology were coming to the conclusion the Earth must be “millions and millions” of years old. At about the same time. Charles Darwin had published his treatise on evolution. Darwin’s view was that evolution needed many millions of years to take place, and he made a bold claim that the Weald chalk fields, where important fossils were found, were the result of 300 million years of erosion. At that time, eminent physicist William Thomson (lord Kelvin) used an important partial differential equation, known as the heat diffusion equation, to estimate the age of Earth by determining how long it would take Earth to cool from molten rock to what we had at tha time. His conclusion was a range of 20 to 4(X) million years, but most likely about 5() million years. For many decades, the proclamations of this irrefutable icon of science did not sit well with geologists or with Darwin. tJR ead Kelvin’s paper (http:Iiwww.openstaxcollege.orgIlI2O KelEarthAge) on estimating the age of the Earth. Kelvin made reasonable assumptions based on what was known in his time, but he also made several assumptions that turned out to be wrong. One incorrect assumption was that Earth is solid and that the cooling was therefore via conduction only, hence justifying the use of the diffusion equation. But the most serious error was a forgivable one—omission of the fact that Earth contains radioactive elements that continually supply heat beneath Earth’s mantle. The discoveiy of radioactivity came near the end of Kelvin’s life and he acknowledged that his calculation would have to be modified. Kelvin used the simple one-dimensional model applied only to Earth’s outer shell, and derived the age from gsaphs and the roughly known temperature gs-adietn near Earth’s surface. Let’s take a look at a more appropriate version of the diffusion equation in radial coordinates, which has the form Tt=K[2T2r+2rTr] (4.23) Here, T(r.t) is temperature as a function of r (measured from the center of Earth) and time i. K is the heat conductivity—for molten rock, in this case. ibe standard method of solving such a partial differential equation is by separation of variables, where we express the solution as the product of functions containing each variable separately. In this case, we would write the temperature as T(r,t)=R(r)f(t).Lord Kelvin and the Age of Earth Figure 4.25 (a) William Thomson (Lord Kelvin). 1824-1907, was a British physicist and electrical engineer. (b) Kelvin used the heat diffusion equation to estimate the age of Earth (credit: modification of work by NASA). During the late 1800s, the scientists of the new field of geology were coming to the conclusion the Earth must be “millions and millions” of years old. At about the same time. Charles Darwin had published his treatise on evolution. Darwin’s view was that evolution needed many millions of years to take place, and he made a bold claim that the Weald chalk fields, where important fossils were found, were the result of 300 million years of erosion. At that time, eminent physicist William Thomson (lord Kelvin) used an important partial differential equation, known as the heat diffusion equation, to estimate the age of Earth by determining how long it would take Earth to cool from molten rock to what we had at tha time. His conclusion was a range of 20 to 4(X) million years, but most likely about 5() million years. For many decades, the proclamations of this irrefutable icon of science did not sit well with geologists or with Darwin. tJR ead Kelvin’s paper (http:Iiwww.openstaxcollege.orgIlI2O KelEarthAge) on estimating the age of the Earth. Kelvin made reasonable assumptions based on what was known in his time, but he also made several assumptions that turned out to be wrong. One incorrect assumption was that Earth is solid and that the cooling was therefore via conduction only, hence justifying the use of the diffusion equation. But the most serious error was a forgivable one—omission of the fact that Earth contains radioactive elements that continually supply heat beneath Earth’s mantle. The discoveiy of radioactivity came near the end of Kelvin’s life and he acknowledged that his calculation would have to be modified. Kelvin used the simple one-dimensional model applied only to Earth’s outer shell, and derived the age from gsaphs and the roughly known temperature gs-adietn near Earth’s surface. Let’s take a look at a more appropriate version of the diffusion equation in radial coordinates, which has the form Tt=K[2T2r+2rTr] (4.23) Here, T(r.t) is temperature as a function of r (measured from the center of Earth) and time i. K is the heat conductivity—for molten rock, in this case. ibe standard method of solving such a partial differential equation is by separation of variables, where we express the solution as the product of functions containing each variable separately. In this case, we would write the temperature as T(r,t)=R(r)f(t).1. Substitute this form into Equation 4.13 and, noting that f(t) is constant with respect to distance (r) and R(r) is constant with respect to time (I). show that 1fft=KR[2Rr2+2rRr]=2For the following exercises, calculate the partial derivative using the limit definitions only. 112. zxforz=x23xy+y2For the following exercises, calculate the partial derivative using the limit definitions only. 113. zxforz=x23xy+y2For the following exercises, calculate the sign of the partial derivative using the graph of the surface. 114. fx(1,1)For the following exercises, calculate the sign of the partial derivative using the graph of the surface. 115. fx(1,1)For the following exercises, calculate the sign of the partial derivative using the graph of the surface. 116. fy(1,1)For the following exercises, calculate the sign of the partial derivative using the graph of the surface. 117. fy(0,0)For the following exercises, calculate the partial derivatives. 118. zxforz=sin(3x)cos(3y)For the following exercises, calculate the partial derivatives. 119. zxforz=sin(3x)cos(3y)For the following exercises, calculate the partial derivatives. 120. zxandzyforz=x8e3yFor the following exercises, calculate the partial derivatives. 121. zxandzyforz=ln(x6+y4)For the following exercises, calculate the partial derivatives. 122. Find fy(x,y)forf(x,y)=exycos(x)sin(y)For the following exercises, calculate the partial derivatives. 123. Let Letz=exy.Findzxandzy .For the following exercises, calculate the partial derivatives. 124. Letz=ln(xy).FindzxandzyFor the following exercises, calculate the partial derivatives. 125. Let z=tan(2xy).Findzxandzy.For the following exercises, calculate the partial derivatives. 126. z=sin(2x3y).Findzxandzy.For the following exercises, calculate the partial derivatives. 127. Let f(x,y)=arctan(yx).Evaluatefx(2,2)andfy(2,2).For the following exercises, calculate the partial derivatives. 128. f(x,y)=xyxy,Findfx(2,2)andfy(2,2)Evaluate the partial derivatives at point P(O. 1). 129. Find zx ,at (0,1) for z=excos(y)Evaluate the partial derivatives at point P(O. 1). 130. Given f(x,y,z)=x3yz2 , find 2fxy and fz(1,1,1)Evaluate the partial derivatives at point P(O. 1). 131. Given 2fxyf(x,y,z)=2sin(x+y) find fx(0,2,4),fy(0,2,4) and fz(0,3,4)The area of a parallelogram with adjacent side lengths that are a and b. and in which the angle between these two sides is . is given by the function A(a. b. ) = ba sin( ). Find the rate of change of the area of the parallelogram with respect to the following: a. Side a b. Side b c. Angle 0Express the volume of a right circular cylinder as a function of two variables: a. its radius r and its height h. b. Show that the rate of change of the volume of the cylinder with respect to its radius is the product of its circumference multiplied by its height. c. Show that the rate of change of the volume of the cylinder with respect to its height is equal to the area of the circular base.Calulate wz for w=zsin(xy2+2z) .Find the indicated higher-order partial derivatives. 135. f, for c = ln(x — y)Find the indicated higher-order partial derivatives. 136. fyxforz=ln(xy)Find the indicated higher-order partial derivatives. 137. Let z=x2+3xy+2y2 . Find 2zx2 and 2zx2Find the indicated higher-order partial derivatives. 138. Given z=ex tan y, find 2zx2 and 2zy2