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All Textbook Solutions for Multivariable Calculus
8E9E10E11E12E13E14ERectangular-to-Cylindrical ConversionIn Exercises 1522, find an equation in cylindrical coordinates for the surface represented by the rectangular equation. z=416E17E18ERectangular-to-Cylindrical ConversionIn Exercises 1522, find an equation in cylindrical coordinates for the surface represented by the rectangular equation. y=x220E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42ERectangular-to-Spherical ConversionIn Exercises 4350, find an equation in spherical coordinates for the surface represented by the rectangular equation. y=244E45E46E47E48E49E50ESpherical-to-Rectangular Conversion In Exercises 5158, find an equation in rectangular coordinates for the surface represented by the spherical equation, and sketch its graph. =152E53E54ESpherical-to-Rectangular Conversion In Exercises 5158, find an equation in rectangular coordinates for the surface represented by the spherical equation, and sketch its graph. =4cos56ESpherical-to-Rectangular Conversion In Exercises 5158, find an equation in rectangular coordinates for the surface represented by the spherical equation, and sketch its graph. =csc58E59E60E61E62E63E64E65E66E67E68E69E70E71E72E73EMatchingIn Exercises 7176, match the equation (written in terms of cylindrical or spherical coordinates) with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (d) (e) (f) =4MatchingIn Exercises 7176, match the equation (written in terms of cylindrical or spherical coordinates) with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) (c) (d) (e) (f) r2=z76E77E78E79E80E81E82EConverting a Rectangular EquationIn Exercises 7986, convert the rectangular equation to an equation in (a) cylindrical coordinates and (b) spherical coordinates. x2+y2=4y84E85E86ESketching a Solid In Exercises 8790, sketch the solid that has the given description in cylindrical coordinates. 0/2,0r2,0z488ESketching a SolidIn Exercises 8790, sketch the solid that has the given description in cylindrical coordinates. 02,0ra,rza90E91E92E93E94E95E96E97E98E99E100E101E102EIntersection of SurfaceIdentify the curve of intersection of the surfaces (in cylindrical coordinates) z=sin and r=1.104EDomain and Continuity In Exercises 1-4, (a) find the domain of r, and (b) determine the interval(s) on which the function is continuous. r(t)=tanti+j+tk2REDomain and Continuity In Exercises 1-4, (a) find the domain of r, and (b) determine the interval(s) on which the function is continuous. r(t)=t29ij+ln(t1)kDomain and Continuity In Exercises 1-4, (a) find the domain of r, and (b) determine the values (if any) of t for which the function is continuous. r(t)=(2t+1)i+t2j+tk5RE6RE7RE8RE9RE10RESketching a Curve In Exercises 9-12, sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t)=(t+1)i+(3t1)j+2tkSketching a Curve In Exercises 9-12, sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t)=2costi+tj+2sintk13RE14RERepresenting a Graph by a Vector-Valued Function In Exercises 15 und 16, sketch the space cane represented by the intersection of the surfaces. 1'hcn use the parameter x=t to find a sector-valued function for the space curve. x=x2+y2,y=2Representing a Graph by a Vector-Valued Function In Exercises 15 und 16, sketch the space cane represented by the intersection of the surfaces. 1'hcn use the parameter x=t to find a sector-valued function for the space curve. x2+z2=4,xy=017REFinding a Limit In Exercises 17 and 18, find the limit. limt(sin2tti+etj+4k)19RE20REHigher-Order Differentiation In Exercise 21 and 22, find (a) r'(t), (b) r'(t). (c) r(t) r(t), and (d) r(t)r(t). r(t)=2t3i+4tjt2k22RE23REFinding Intervals on Which a Curve is SmoothIn Exercises 23 and 24, find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t)=tt2i+tj+1+tk25RE26RE27RE28RE29RE30RE31RE32RE33RE34RE35RE36RE37RE38RE39RE40RE41REProjectile Motion In Exercises 41 and 42, use the model for projectile motion, assuming there is no air resistance and g=32 feet per second per second. Determine the maximum height and range of a projectile Bred at a height of 6 feet above the ground with an initial speed of 400 feet per second and an angle of 60' above the horizontalFinding the Unit Tangent Vector In Exercises 43 and 44, find the unit tangent sector to the curse at the specified value of the parameter. r(t)=6tit2j,t=244RE45RE46RE47RE48RE49RE50RE51RE52RE53REFinding Tangential and Normal Components of Acceleration In Exercises 53 and 54, Find the tangential and normal components of acceleration at the given time t for the space curve r(t). r(t)=t23i6tj+t2k,t=255RE56RE57RE58RE59RE60RE61RE62RE63RE64REFinding CurvatureIn Exercises 6366, find the curvature of the curve. r(t)=2ti+12t2j+t2kFinding CurvatureIn Exercises 6366, find the curvature of the curve. r(t)=2ti+5costj+5sintkFinding Curvature In Exercises 67 and 68, find the curvature of the curve at the point P. r(t)=12t2i+tj+13t3k,P(12,1,13)68RE69RE70RE71RE72RE73RECornu Spiral The cornu spiral is given by x(t)=0tcos(u22)duandy(t)=0tsin(u22)du The spiral shown in the figure was plotted over the interval t. (a) Find the arc length of this curve from t=0tot=a. (b) Find the curvature of the graph when t=a. (c) The cornu spiral was discovered by James Bernoulli. He found that the spiral has an amazing relationship between curvature and arc length. What is this relationship?2PS3PS4PSCycloid Consider one arch of the cycloid r()=(sin)i+(1cos)j,02 as shown in the figure. Let s() be the arc length from the highest point on the arch to the point (x(),y()), and let ()=1/K be the radius of curvature at the point (x(),y()). Show that s and are related by the equation s2+2=16. (This equation is called a natural equation for the curve.)6PS7PS8PSBinormal VectorIn Exercises 911, use the binormal vector defined by the equation B=TN. Find the unit tangent, principal unit normal, and binormal vectors for the helix r(t)=4costi+4sintj+3tk at t=/2. Sketch the helix together with these three mutually orthogonal unit vectors.10PS11PS12PS13PSFerris Wheel You want to toss an object to a friend who is riding a Ferris wheel (see figure). The following parametric equation give the path of the friend r1(t) and the path of the object r2(t). Distance is measured in meters, and time is measured in seconds. r1(t)=15(sint10)i+(1615cost10)jr2(t)=[ 228.03(tt0) ]i+[1+11.47(tt0)4.9(tt0)2]jCONCEPTS CHECK Vector-valued FunctionDescribe how you can use a vector-valued function to represent a curve.Continuity of a Vector-valued FunctionDescribe what it means for vector-valued function r(t) to be continuous at a point.3E4E5E6E7E8E9E10E11E12EWriting a Vector-Valued FunctionIn Exercises 1316, represent the line segment from P to Q by a vector-valued function and by a set of parametric equations. P(0,0,0),Q(5,2,2)14EWriting a Vector-Valued FunctionIn Exercises 1316, represent the line segment from P to Q by a vector-valued function and by a set of parametric equations. P(3,6,1),Q(1,9,8)16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32ESketching a Space Curve In Exercises 31-38, sketch the space curve represented by the vector- valued function and give the orientation of the curve. r(t)=2costi+2sintj+tk34E35E36E37E38E39E40E41E42E43E44E45E46ERepresenting a Graph by a Vector-Valued Function In Exercises 47-54, represent the plane curve by a vector-valued function. (There are many correct answers.) y=x+548ERepresenting a Graph by a Vector-Valued Function In Exercises 47-54, represent the plane curve by a vector-valued function. (There are many correct answers.) y=(x2)250E51E52ERepresenting a Graph by a Vector-Valued Function In Exercises 47-54, represent the plane curve by a vector-valued function. (There are many correct answers.) x216y24=154E55E56E57E58E59E60E61ERepresenting a Graph by Vector-Valued Function In exercises 55-62 sketch the space curve represented by the intersection of the surfaces a. Then represent the curve by a vector valued function using the given parameter. Surfaces Parameter x2+y2+z2=16,xy=4 x=t (first octant)63E64EFinding a Limit In Exercises 65-70, find the limit (if it exists). limt(ti+costj+sintk)66EFinding a Limit In Exercises 65-70, find the limit (if it exists). limt0(t2i+3tj+1costtk)68EFinding a Limit In Exercises 65-70, find the limit (if it exists). limt0(eti+sinttj+etk)70EContinuity of a Vector-Valued Function In Exercises 71-76, determine the interval(s) on which the vector valued function is continuous. r(t)=12t+1i+1tj72EContinuity of a Vector-Valued Function In Exercises 71-76, determine the interval(s) on which the vector valued function is continuous. r(t)=ti+arcsintj+(t1)k74EContinuity of a Vector-Valued Function In Exercises 71-76, determine the interval(s) on which the vector valued function is continuous. r(t)=ett2tant76E77E78E79E80E81E82E83E84E85E86E87E88E89E90ECONCEPT CHECK Derivative Describe the relationship between the graph of r(t0) and the curve represented by r(t).2E3E4EDifferentiation of Vector-Valued FunctionsIn Exercises 310, find rt(t), r(t0), and rt(t0) for the given value of t0. Then sketch the curve represented by the vector-valued function and sketch the vectors r(t0) and rt(t0). r(t)=costi+sintj,t0=26EDifferentiation of Vector-Valued FunctionsIn Exercises 310, find rt(t), r(t0), and rt(t0) for the given value of t0. Then sketch the curve represented by the vector-valued function and sketch the vectors r(t0) and rt(t0). r(t)=et,e2t,t0=08E9E10E11E12E13E14E15E16E17E18E19E20EHigher-Order DifferentiationIn Exercises 1922, find (a) r(t), (b) r(t), and (c) r(t)r(t). r(t)=4costi+4sintj22EHigher-Order DifferentiationIn Exercises 2326, find (a) r(t), (b) r(t), and (c) r(t)r(t), and (d) r(t)r(t). r(t)=12t2itj+16t3k24EHigher-Order DifferentiationIn Exercises 2326, find (a) r(t), (b) r(t), and (c) r(t)r(t), and (d) r(t)r(t). r(t)=cost+tsint,sinttcost,tHigher-Order DifferentiationIn Exercises 2326, find (a) r(t), (b) r(t), and (c) r(t)r(t), and (d) r(t)r(t). r(t)=et,t2,tant