Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus (MindTap Course List)
Find equations of the osculating circles of the ellipse 9x2+4y2=36 at the points (2,0) and (0,3). Use a graphing calculator or computer to graph the ellipse and both osculating circles on the same screen.Find equations of the osculating circles of the parabola y=12x2 at the points (0,0) and (1,12). Graph both osculating circles and the parabola on the same screen.53EIs there a point on the curve in Exercise 53 where the osculating plane is parallel to the plane x+y+z=1? Note: You will need a CAS for differentiating, for simplifying, and for computing a cross product.Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x=y2 and z=x2 at the point (1,1,1).56EShow that at every point on the curve r(t)=etcost,etsint,et the angle between the unit tangent vector and the z-axis is the same. Then show that the same result holds true for the unit normal and binormal vectors.58E59E60Ea Show that dB/ds is perpendicular to B. b Show that dB/ds is perpendicular to T. c Deduce from parts a and b that dB/ds=(s)N for some number (s) called the torsion of the curve. The torsion measures the degree of twisting of a curve. d Show that for a plane curve the torsion is (s)=0.62EUse the Frenet-Serret formulas to prove each of the following. Primes denote derivatives with respect to t. Start as in the proof of Theorem 10. a r=sT+(s)2N b rr=(s)3B c r=[s2(s)3]T+[3ss+(s)2]N+(s)3B d =(rr)r|rr|2Show that the circular helix r(t)=acost,asint,bt, where a and b are positive constants, has constant curvature and constant torsion. Use the result of Exercise 63d.65E66E67E68EThe table gives coordinates of a particle moving through space along a smooth curve. a Find the average velocities over the time intervals 0, 1, 0.5, 1, 1, 2, and 1, 1.5. b Estimate the velocity and speed of the particle at t=1. t x y z 0 2.7 9.8 3.7 0.5 3.5 7.2 3.3 1.0 4.5 6.0 3.0 1.5 5.9 6.4 2.8 2.0 7.3 7.8 2.7The figure shows the path of a particle that moves with position vector r(t) at time t. a Draw a vector that represents the average velocity of the particle over the time interval 2t2.4. b Draw a vector that represents the average velocity over the time interval 1.5t2. c Write an expression for the velocity vector v(2). d Draw an approximation to the vector v(2) and estimate the speed of the particle at t=2.3E4EFind the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of t. r(t)=3costi+2sintj,t=/36E7E8E9E10E11E12EFind the velocity, acceleration, and speed of a particle with the given position function. r(t)=et(costi+sintj+tk)Find the velocity, acceleration, and speed of a particle with the given position function. r(t)=t2,sinttcost,cost+tsint,t015E16Ea Find the position vector of a particle that has the given acceleration and the specified initial velocity and position. b Use a computer to graph the path of the particle. a(t)=2ti+sintj+cos2tk,v(0)=i,r(0)=j18EThe position function of a particle is given by r(t)=t2,5t,t216t. When is the speed a minimum?20EA force with magnitude 20 N acts directly upward from the xy-plane on an object with mass 4 kg. The object starts at the origin with initial velocity v(0)=ij. Find its position function and its speed at time t.Show that if a particle moves with constant speed, then the velocity and acceleration vectors are orthogonal.A projectile is fired with an initial speed of 200 m/s and angle of elevation 60. Find a the range of the projectile, b the maximum height reached, and c the speed at impact.24E25EA projectile is fired from a tank with initial speed 400 m/s. Find two angles of elevation that can be used to hit a target 3000 m away.A rifle is fired with angle of elevation 36. What is the muzzle speed if the maximum height of the bullet is 1600 ft?A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle 50 above the horizontal. Is it a home run? In other words, does the ball clear the fence?A medieval city has the shape of a square and is protected by walls with length 500 m and height 15 m. You are the commander of an attacking army and the closest you can get to the wall is 100 m. Your plan is to set fire to the city by catapulting heated rocks over the wall with an initial speed of 80 m/s. At what range of angles should you tell your men to set the catapult? Assume the path of the rocks is perpendicular to the wall.Show that a projectile reaches three-quarters of its maximum height in half the time needed to reach its maximum height.A ball is thrown eastward into the air from the origin in the direction of the positive x-axis. The initial velocity is 50i+80k, with speed measured in feet per second. The spin of the ball results in a southward acceleration of 4ft/s2, so the acceleration vector is a=4j32k. Where does the ball land and with what speed?32EWater traveling along a straight portion of a river normally flows fastest in the middle, and the speed slows to almost zero at the banks. Consider a long straight stretch of river flowing north, with parallel banks 40 m apart. If the maximum water speed is 3 m/s, we can use a quadratic function as a basic model for the rate of water flow x units from the west bank: f(x)=3400x(40x). a A boat proceeds at a constant speed of 5 m/s from a point A on the west bank while maintaining a heading perpendicular to the bank. How far down the river on the opposite bank will the boat touch shore? Graph the path of the boat. b Suppose we would like to pilot the boat to land at the point B on the east bank directly opposite A. If we maintain a constant speed of 5 m/s and a constant heading, find the angle at which the boat should head. Then graph the actual path the boat follows. Does the path seem realistic?34E35E36E37E38E39E40EFind the tangential and normal components of the acceleration vector at the given point. r(t)=lnti+(t2+3t)j+4tk,(0,4,4)42EThe magnitude of the acceleration vector a is 10 cm/s2. Use the figure to estimate the tangential and normal components of a.44EThe position function of a spaceship is r(t)=(3+t)i+(2+lnt)j+(74t2+1)k and the coordinates of a space station are (6,4,9). The captain wants the spaceship to coast into the space station. When should the engines be turned off?46E1CC2CC3CC4CC5CC6CC7CC8CC9CC1TFQ2TFQ3TFQ4TFQ5TFQ6TFQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If T(t) is the unit tangent vector of a smooth curve, then the curvature is =|dT/dt|.8TFQ9TFQ10TFQ11TFQ12TFQ13TFQ14TFQ1E2E3E4E5E6E7E8E9E10EFor the curve given by r(t)=sin3t,cos3t,sin2t, 0t/2, find athe unit tangent vector, bthe unit normal vector, cthe unit binormal vector, and dthe curvature.Find the curvature of the ellipse x=3cost,y=4sint at the points (3,0) and (0,4).Find the curvature of the curve y=x4 at the point (1,1).Find an equation of the osculating circle of the curve y=x4x2 at the origin. Graph both the curve and its osculating circle.15EThe figure shows the curve C traced by a particle with position vector rt at time t. aDraw a vector that represents the average velocity of the particle over the time interval 3t3.2. bWrite an expression for the velocity v3. cWrite an expression for the unit tangent vector T3 and draw it.A particle moves with position function r(t)=tlnti+tj+etk. Find the velocity, speed, and acceleration of the particle.18EA particle starts at the origin with initial velocity ij+3k. Its acceleration is a(t)=6ti+12t2j6tk. Find its position function.20EA projectile is launched with an initial speed of 40 m/s from the floor of a tunnel whose height is 30 m. What angle of elevation should be used to achieve the maximum possible horizontal range of the projectile? What is the maximum range?22E23EIn designing transfer curves to connect sections of straight railroad tracks, its important to realize that the acceleration of the train should be continuous so that the reactive force exerted by the train on the track is also continuous. Because of the formulas for the components of acceleration in Section 13.4, this will be the case if the curvature varies continuously. a A logical candidate for a transfer curve to join existing tracks given by y=1 for x0 and y=2x for x1/2 might be the function f(x)=1x2,0x1/2, whose graph is the arc of the circle shown in the figure. It looks reasonable at first glance. Show that the function F(x)={1ifx01x2if0x1/22xifx1/2} is continuous and has continuous slope, but does not have continuous curvature. Therefore f is not an appropriate transfer curve. b Find a fifth-degree polynomial to serve as a transfer curve between the following straight line segments: y=0 for x0 and y=x for x1. Could this be done with a fourth-degree polynomial? Use a graphing calculator or computer to sketch the graph of the connected function and check to see that it looks like the one in the figure.A particle P moves with constant angular speed around a circle whose center is at the origin and whose radius is R. The particle is said to be in uniform circular motion. Assume that the motion is counterclockwise and that the particle is at the point (R,0) when t=0. The position vector at time t0 is r(t)=Rcosti+Rsintj. a Find the velocity vector v and show that vr=0. Conclude that v is tangent to the circle and points in the direction of the motion. b Show that the speed |v| of the particle is the constant R. The period T of the particle is the time required for one complete revolution. Conclude that T=2R|v|=2 c Find the acceleration vector a. Show that it is proportional to r and that it points toward the origin. An acceleration with this property is called a centripetal acceleration. Show that the magnitude of the acceleration vector is |a|=R2. d Suppose that the particle has mass m. Show that the magnitude of the force F that is required to produce this motion, called a centripetal force, is |F|=m|v|2RA circular curve of radius R on a highway is banked at an angle so that a car can safely traverse the curve without skidding when there is no friction between the road and the tires. The loss of friction could occur, for example, if the road is covered with a film of water or ice. The rated speed vR of the curve is the maximum speed that a car can attain without skidding. Suppose a car of mass m is traversing the curve at the rated speed vR. Two forces are acting on the car: the vertical force, mg, due to the weight of the car, and a force F exerted by, and normal to, the road see the figure. The vertical component of F balances the weight of the car, so that |F|cos=mg. The horizontal component of F produces a centripetal force on the car so that, by Newtons Second Law and part d of Problem 1, |F|sin=mvR2R a Show that vR2=Rgtan. b Find the rated speed of a circular curve with radius 400 ft that is banked at an angle of 12. c Suppose the design engineers want to keep the banking at 12, but wish to increase the rated speed by 50. What should the radius of the curve be?A projectile is fired from the origin with angle of elevation and initial speed v0. Assuming that air resistance is negligible and that the only force acting on the projectile is gravity, g, we showed in Example 13.4.5 that the position vector of the projectile is r(t)=(v0cos)ti+[(v0sin)t12gt2]j We also showed that the maximum horizontal distance of the projectile is achieved when =45 and in this case the range is R=v02/g. a At what angle should the projectile be fired to achieve maximum height and what is the maximum height? b Fix the initial speed v0 and consider the parabola x2+2RyR2=0, whose graph is shown in the figure at the left. Show that the projectile can hit any target inside or on the boundary of the region bounded by the parabola and the x-axis, and that it cant hit any target outside this region. c Suppose that the gun is elevated to an angle of inclination in order to aim at a target that is suspended at a height h directly over a point D units downrange see the figure below. The target is released at the instant the gun is fired. Show that the projectile always hits the target, regardless of the value v0, provided the projectile does not hit the ground before D.a A projectile is fired from the origin down an inclined plane that makes an angle with the horizontal. The angle of elevation of the gun and the initial speed of the projectile are and 0, respectively. Find the position vector of the projectile and the parametric equations of the path of the projectile as functions of the time t. Ignore air resistance. b Show that the angle of elevation that will maximize the downhill range is the angle halfway between the plane and the vertical. c Suppose the projectile is fired up an inclined plane whose angle of inclination is . Show that, in order to maximize the uphill range, the projectile should be fired in the direction halfway between the plane and the vertical. d In a paper presented in 1686, Edmond Halley summarized the laws of gravity and projectile motion and applied them to gunnery. One problem he posed involved firing a projectile to hit a target a distance R up an inclined plane. Show that the angle at which the projectile should be fired to hit the target but use the least amount of energy is the same as the angle in part c. Use the fact that the energy needed to fire the projectile is proportional to the square of the initial speed, so minimizing the energy is equivalent to minimizing the initial speed.A ball rolls off a table with a speed of 2 ft/s. The table is 3.5 ft high. a Determine the point at which the ball hits the floor and find its speed at the instant of impact. b Find the angle between the path of the ball and the vertical line drawn through the point of impact see the figure. c Suppose the ball rebounds from the floor at the same angle with which it hits the floor, but loses 20 of its speed due to energy absorbed by the ball on impact. Where does the ball strike the floor on the second bounce?6PIf a projectile is fired with angle of elevation and initial speed , then parametric equations for its trajectory are x=(cos)t y=(sin)t12gt2 See Example 13.4.5. We know that the range horizontal distance traveled is maximized when =45. What value of maximizes the total distance traveled by the projectile? State your answer correct to the nearest degree.8P9PIn Example 2 we considered the function W=f(T,v), where W is the wind-chill index, T is the actual temperature, and v is the wind speed. A numerical representation is given in Table 1 on page 929. aWhat is the value of f(15,40)? What is its meaning? bDescribe in words the meaning of the question For what value of v is f(20,v)=30? Then answer the question. cDescribe in words the meaning of the question For what value of T is f(T,20)=49? Then answer the question. dWhat is the meaning of the function W=f(5,v)? Describe the behavior of this function. eWhat is the meaning of the function W=f(T,50)? Describe the behavior of this function.The temperature-humidity index I or humidex, for short is the perceived air temperature when the actual temperature is T and the relative humidity is h, so we can write I=f(T,h). The following table of values of I is an excerpt from a table compiled by the National Oceanic Actual temperature F h T 20 30 40 50 60 70 80 77 78 79 81 82 83 85 82 84 86 88 90 93 90 87 90 93 96 100 106 95 93 96 101 107 114 124 100 99 104 110 120 132 144 aWhat is the value of f(95,70)? What is its meaning? bFor what value of h is f(90,h)=100? cFor what value of T is f(T,50)=88? dWhat are the meanings of the functions I=f(80,h) and I=f(100,h)? Compare the behavior of these two functions of h.3EVerify for the Cobb-Douglas production function P(L,K)=1.01L0.75K0.25 discussed in Example 3 that the production will be doubled if both the amount of labor and the amount of capital are doubled. Determine whether this is also true for the general production function P(L,K)=bLK15E6E7EA company makes three sizes of cardboard boxes: small, medium, and large. It costs 2.50 to make a small box, 4.00 for a medium box, and 4.50 for a large box. Fixed costs are 8000. aExpress the cost of making x small boxes, y medium boxes, and z large boxes as a function of three variables: C=f(x,y,z). bFind f(3000,5000,4000) and interpret it. cWhat is the domain of f?9ELet F(x,y)=1+4y2. aEvaluate F(3,1). bFind and sketch the domain of F. cFind the range of F.Let f(x,y,z)=x+y+z+ln(4x2y2z2). aEvaluate f(1,1,1). bFind and describe the domain of f.Let g(x,y,z)=x3y2z10xyz. aEvaluate g(1,2,3). bFind and describe the domain of g.Find and sketch the domain of the function. f(x,y)=x2+y1Find and sketch the domain of the function. f(x,y)=x3y415E16EFind and sketch the domain of the function. g(x,y)=xyx+yFind and sketch the domain of the function. g(x,y)=ln(2x)1x2y2Find and sketch the domain of the function. f(x,y)=yx21x220EFind and sketch the domain of the function. f(x,y,z)=4x2+9y2+1z222E23E24E25E26ESketch the graph of the function. f(x,y)=sinx28E29ESketch the graph of the function. f(x,y)=4x2+y2Sketch the graph of the function. f(x,y)=44x2y2Match the function with its graph labeled IVI. Give reasons for your choices. a f(x,y)=11+x2+y2 b f(x,y)=11+x2y2 c f(x,y)=ln(x2+y2) d f(x,y)=cosx2+y2 e f(x,y)=|xy| f f(x,y)=cos(xy)33EShown is a contour map of atmospheric pressure in North America on August 12, 2008. On the level curves called isobars the pressure is indicated in millibars mb. aEstimate the pressure at C Chicago, N Nashville, S San Francisco, and V Vancouver. bAt which of these locations were the winds strongest?Level curves isothermals are shown for the typical water temperature in C in Long Lake Minnesota as a function of depth and time of year. Estimate the temperature in the lake on June 9 day 160 at a depth of 10 m and on June 29 day 180 at a depth of 5 m.Two contour maps are shown. One is for a function f whose graph is a cone. The other is for a function g whose graph is a paraboloid. Which is which, and why?37E38EThe body mass index BMI of a person is defined by B(m,h)=mh2 where m is the persons mass in kilograms and h is the height in meters. Draw the level curves B(m,h)=18.5, B(m,h)=25, B(m,h)=30, and B(m,h)=40. A rough guideline is that a person is underweight if the BMI is less than 18.5; optimal if the BMI lies between 18.5 and 25; overweight if the BMI lies between 25 and 30; and obese if the BMI exceeds 30. Shade the region corresponding to optimal BMI. Does someone who weighs 62 kg and is 152 cm tall fall into this category?40EA contour map of a function is shown. Use it to make a rough sketch of the graph of f.A contour map of a function is shown. Use it to make a rough sketch of the graph of f.A contour map of a function is shown. Use it to make a rough sketch of the graph of f.A contour map of a function is shown. Use it to make a rough sketch of the graph of f.45E46E47EDraw a contour map of the function showing several level curves. f(x,y)=ln(x2+4y2)49EDraw a contour map of the function showing several level curves. f(x,y)=yarctanx51EDraw a contour map of the function showing several level curves. f(x,y)=y/(x2+y2)Sketch both a contour map and a graph of the function and compare them. f(x,y)=x2+9y254E55E56E57EUse a computer to graph the function using various domains and viewpoints. Get a printout of one that, in your opinion, gives a good view. If your software also produces level curves, then plot some contour lines of the same function and compare with the graph. f(x,y)=xy3yx3(dogsaddle)59E60EMatch the function a with its graph labeled AF below and b with its contour map labeled IVI. Give reasons for your choices. z=sin(xy)Match the function a with its graph labeled AF below and b with its contour map labeled IVI. Give reasons for your choices. z=excosyMatch the function a with its graph labeled AF below and b with its contour map labeled IVI. Give reasons for your choices. z=sin(xy)Match the function a with its graph labeled AF below and b with its contour map labeled IVI. Give reasons for your choices. z=sinxsinyMatch the function a with its graph labeled AF below and b with its contour map labeled IVI. Give reasons for your choices. z=(1x2)(1y2)Match the function a with its graph labeled AF below and b with its contour map labeled IVI. Give reasons for your choices. z=xy1+x2+y267E68E69EDescribe the level surfaces of the function. f(x,y,z)=x2y2z271EDescribe how the graph of g is obtained from the graph of f. a g(x,y)=f(x2,y) b g(x,y)=f(x,y+2) c g(x,y)=f(x+3,y41)73E74EGraph the function using various domains and viewpoints. Comment on the limiting behavior of the function. What happens as both x and y become large? What happens as (x,y) approaches the origin? f(x,y)=x+yx2+y276E77EUse a computer to investigate the family of surfaces z=(ax2+by2)ex2y2 How does the shape of the graph depend on the numbers a and b?79E80E81E1EExplain why each function is continuous or discontinuous. aThe outdoor temperature as a function of longitude, latitude, and time bElevation height above sea level as a function of longitude, latitude, and time cThe cost of a taxi ride as a function of distance traveled and time3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25EFind h(x,y)=g(f(x,y)) and the set of points at which h is continuous. g(t)=t+lnt, f(x,y)=1xy1+x2y227E28E29E30E31E32E33EDetermine the set of points at which the function is continuous. G(x,y)=ln(1+xy)35E36E37E38E39E40E41E42E43E44E45E46E1EAt the beginning of this section we discussed the function I=f(T,H), where I is the heat index, T is the temperature, and H is the relative humidity. Use Table 1 to estimate fT(92,60), and fH(92,60). What are the practical interpretations of these values?3EThe wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h=f(v,t) are recorded in feet in the following table. Duration hours Wind speed knots vt 5 10 15 20 30 40 50 10 2 2 2 2 2 2 2 15 4 4 5 5 5 5 5 20 5 7 8 8 9 9 9 30 9 13 16 17 18 19 19 40 14 21 25 28 31 33 33 50 19 29 36 40 45 48 50 60 24 37 47 54 62 67 69 a What are the meanings of the partial derivatives h/v and h/t? b Estimate the values of fv(40,15) and ft(40,15). What are the practical interpretations of these values? c What appears to be the value of the following limit? limtht5E6E7E8E9EA contour map is given for a function f. Use it to estimate fx(2,1) and fy(2,1).If f(x,y)=164x2y2, find fx(1,2) and fy(1,2) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.If f(x,y)=4x24y2, find fx(1,0) and fy(1,0) and interpret these numbers as slopes. Illustrate with either hand-drawn sketches or computer plots.Find fx and fy and graph f, fx, and fy with domains and viewpoints that enable you to see the relationships between them. f(x,y)=x2y3Find fx and fy and graph f, fx, and fy with domains and viewpoints that enable you to see the relationships between them. f(x,y)=y1+x2y215E16E17E18E19E20E21E22E23EFind the first partial derivatives of the function. w=evu+v225E26E27E28E29E30E31E32E33E34EFind the first partial derivatives of the function. p=t4+u2cosv36E37E38E39E40E41E42E43E44E45E46E47E48EUse implicit differentiation to find z/x and z/y. ez=xyz50EFind z/y and z/y. a z=f(x)+g(y) b z=f(x+y)Find z/y and z/y. a z=f(x)g(y) b z=f(xy) c z=f(x/y)53E