1 Limits And Continuity 2 The Derivative 3 Topics In Differentiation 4 The Derivative In Graphing And Applications 5 Integration 6 Applications Of The Definite Integral In Geometry, Science, And Engineering 7 Principles Of Integral Evaluation 8 Mathematical Modeling With Differential Equations 9 Infinite Series 10 Parametric And Polar Curves; Conic Sections 11 Three-dimensional Space; Vectors 12 Vector-valued Functions 13 Partial Derivatives 14 Multiple Integrals 15 Topics In Vector Calculus expand_more
12.1 Introduction To Vector-valued Functions 12.2 Calculus Of Vector-valued Functions 12.3 Change Of Parameter; Arc Length 12.4 Unit Tangent, Normal, And Binormal Vectors 12.5 Curvature 12.6 Motion Along A Curve 12.7 Kepler’s Laws Of Planetary Motion Chapter Questions expand_more
Problem 1QCE: If rt is a smooth vector-valued function, then the integral abdrdtdt may be interpreted... Problem 2QCE: If r(s) is a smooth vector-valued function parametrized by arc length s, then drds= and the arc... Problem 3QCE: If rt is a smooth vector-valued function, then the arc length parameter s having rt0 as the... Problem 4QCE: Suppose that rt is a smooth vector-valued function of t with r1=3,3,1, and let r1t be defined by the... Problem 1ES: Determine whether rt is a smooth function of the parameter t. rt=t3i+3t22tj+t2k Problem 2ES: Determine whether rt is a smooth function of the parameter t. rt=cost2i+sint2j+etk Problem 3ES: Determine whether rt is a smooth function of the parameter t. rt=teti+t22tj+costk Problem 4ES: Determine whether rt is a smooth function of the parameter t. rt=sinti+2tlntj+t2tk Problem 5ES: Find the arc length of the parametric curve. x=cos3t,y=sin3t,z=2;0t/2 Problem 6ES: Find the arc length of the parametric curve. x=3cost,y=3sint,z=4t;0t Problem 7ES: Find the arc length of the parametric curve. x=et,y=et,z=2t;0t1 Problem 8ES: Find the arc length of the parametric curve. x=12t,y=131t3/2,z=131+t3/2;1t1 Problem 9ES: Find the arc length of the graph of rt. rt=t3i+tj+126t2k;1t3 Problem 10ES: Find the arc length of the graph of rt. rt=4+3ti+22tj+5+tk;3t4 Problem 11ES: Find the arc length of the graph of rt. rt=3costi+3sintj+tk;0t2 Problem 12ES: Find the arc length of the graph of rt. rt=t2i+cost+tsintj+sinttcostk;0t Problem 13ES: Calculate dr/d by the chain rule, and then check your result by expressing r in terms of and... Problem 14ES: Calculate dr/d by the chain rule, and then check your result by expressing r in terms of and... Problem 15ES: Calculate dr/d by the chain rule, and then check your result by expressing r in terms of and... Problem 16ES: Calculate dr/d by the chain rule, and then check your result by expressing r in terms of and... Problem 17ES Problem 18ES Problem 19ES Problem 20ES Problem 21ES: (a) Find the arc length parametrization of the line x=t,y=t that has the same orientation as the... Problem 22ES: Find arc length parametrizations of the lines in Exercise 21 that have the stated reference point... Problem 23ES Problem 24ES: (a) Find the arc length parametrization of the line x=5+3t,y=2t,z=5+t that has the same direction as... Problem 25ES: Find an arc length parametrization of the curve that has the same orientation as the given curve and... Problem 26ES: Find an arc length parametrization of the curve that has the same orientation as the given curve and... Problem 27ES: Find an arc length parametrization of the curve that has the same orientation as the given curve and... Problem 28ES: Find an arc length parametrization of the curve that has the same orientation as the given curve and... Problem 29ES: Find an arc length parametrization of the curve that has the same orientation as the given curve and... Problem 30ES: Find an arc length parametrization of the curve that has the same orientation as the given curve and... Problem 31ES: Show that the arc length of the circular helix x=acost,y=asint,z=ctfor0tt0ist0a2+c2. Problem 32ES: Use the result in Exercise 31 to show the circular helix r=acosti+asintj+ctk can be expressed as... Problem 33ES: Find an arc length parametrization of the cycloid x=atasinty=aacost0t2 with 0,0 as the reference... Problem 34ES: Show that in cylindrical coordinates a curve given by the parametric equation r=rt,=t,z=ztforatb has... Problem 35ES: In each part, use the formula in Exercise 34 to find the arc length of the curve.... Problem 36ES: Show That in spherical coordinates a curve given by the parametric equation =t,=t,=tforatb has arc... Problem 37ES: In each part, use the formula in Exercise 36 to find the arc length of the curve.... Problem 38ES: h (a) Sketch the graph of rt=ti+t2j. Show that rt is a smooth vector-valued function but the change... Problem 39ES: Find a change of parameter t=g for the semicircle rt=costi+sintj0t Such that (a) the semicircle is... Problem 40ES: What change of parameter t=g would you make if you wanted to trace the graph of rt0t1 in the... Problem 41ES: As illustrated in the accompanying figure, copper cable with a diameter of 12 inch is to be wrapped... Problem 42ES: Let rt=cost,sint,t3/2.Findartbdsdtc02rtdt. Problem 43ES: Let rt=lnti+2tj+t2k.Findartbdsdtc13rtdt. Problem 44ES: Let rt=t2i+t3j (seeFigure12.3.1) . Let t be the angle between rt and i. Show that tast0andt0ast0+ Problem 45ES: Prove: If rt is a smoothly parametrized function, then the angles between rt and the vector i, j,... Problem 46ES: Prove the vector form of the chain rule for 2-space (Theorem 12.3.2) by expressing rt in terms of... Problem 47ES format_list_bulleted