 13.13.1: Describe the curve defined by the vector function
 13.13.2: Use a computer to draw the curve with vector equation This curve is...
 13.13.3: Find the domain of the vector function. rts4 t 2 , e3t , lnt 1 13.1 E
 13.13.4: Find the domain of the vector function. rtt 2 t 2 i sin t j ln9 t 2...
 13.13.5: Find the limit lim tl0cos t, sin t, t ln t
 13.13.6: Find the limit lim tl0 e t 1 t , s1 t 1 t , 3 1 t lim
 13.13.7: Find the limit lim tl0 e3t i t 2 sin2 t j cos 2t k l
 13.13.8: Find the limit lim tl arctan t, e2t , ln t t
 13.13.9: Sketch the curve with the given vector equation. Indicate with an a...
 13.13.10: Sketch the curve with the given vector equation. Indicate with an a...
 13.13.11: Sketch the curve with the given vector equation. Indicate with an a...
 13.13.12: Sketch the curve with the given vector equation. Indicate with an a...
 13.13.13: Sketch the curve with the given vector equation. Indicate with an a...
 13.13.14: Sketch the curve with the given vector equation. Indicate with an a...
 13.13.15: Sketch the curve with the given vector equation. Indicate with an a...
 13.13.16: Sketch the curve with the given vector equation. Indicate with an a...
 13.13.17: Find a vector equation and parametric equations for the line segmen...
 13.13.18: Find a vector equation and parametric equations for the line segmen...
 13.13.19: Find a vector equation and parametric equations for the line segmen...
 13.13.20: Find a vector equation and parametric equations for the line segmen...
 13.13.21: Match the parametric equations with the graphs (labeled IVI). Give ...
 13.13.22: Match the parametric equations with the graphs (labeled IVI). Give ...
 13.13.23: Match the parametric equations with the graphs (labeled IVI). Give ...
 13.13.24: Match the parametric equations with the graphs (labeled IVI). Give ...
 13.13.25: Match the parametric equations with the graphs (labeled IVI). Give ...
 13.13.26: Match the parametric equations with the graphs (labeled IVI). Give ...
 13.13.27: Show that the curve with parametric equations , , lies on the cone ...
 13.13.28: Show that the curve with parametric equations , , is the curve of i...
 13.13.29: At what points does the curve intersect the paraboloid z x 2 y 2
 13.13.30: At what points does the helix intersect the sphere x 2 y 2 z2 5 r
 13.13.31: Use a computer to graph the curve with the given vector equation. M...
 13.13.32: Use a computer to graph the curve with the given vector equation. M...
 13.13.33: Use a computer to graph the curve with the given vector equation. M...
 13.13.34: Use a computer to graph the curve with the given vector equation. M...
 13.13.35: Graph the curve with parametric equations , , . Explain the appeara...
 13.13.36: Graph the curve with parametric equations Explain the appearance of...
 13.13.37: Show that the curve with parametric equations , , passes through th...
 13.13.38: Find a vector function that represents the curve of intersection of...
 13.13.39: Find a vector function that represents the curve of intersection of...
 13.13.40: Find a vector function that represents the curve of intersection of...
 13.13.41: Try to sketch by hand the curve of intersection of the circular cyl...
 13.13.42: Try to sketch by hand the curve of intersection of the parabolic cy...
 13.13.43: If two objects travel through space along two different curves, its...
 13.13.44: Two particles travel along the space curves Do the particles collid...
 13.13.45: Suppose and are vector functions that possess limits as and let be ...
 13.13.46: The view of the trefoil knot shown in Figure 8 is accurate, but it ...
 13.13.47: When you have finished your sketch, use a computer to draw the curv...
 13.13.48: Show that if and only if for every there is a number such that if t...
 13.13.49: (a) Find the derivative of . (b) Find the unit tangent vector at th...
 13.13.50: For the curve , find and sketch the position vector and the tangent...
 13.13.51: Find parametric equations for the tangent line to the helix with pa...
 13.13.52: Show that if (a constant), then is orthogonal to for all t
 13.13.53: If , then where is a vector constant of integration, and y M 2 0 rt...
 13.13.54: The figure shows a curve given by a vector function . r1 (a) Draw t...
 13.13.55: (a) Make a large sketch of the curve described by the vector functi...
 13.13.56: (a) Sketch the plane curve with the given vector equation. (b) Find...
 13.13.57: (a) Sketch the plane curve with the given vector equation. (b) Find...
 13.13.58: (a) Sketch the plane curve with the given vector equation. (b) Find...
 13.13.59: (a) Sketch the plane curve with the given vector equation. (b) Find...
 13.13.60: (a) Sketch the plane curve with the given vector equation. (b) Find...
 13.13.61: (a) Sketch the plane curve with the given vector equation. (b) Find...
 13.13.62: Find the derivative of the vector function rtt sin t, t 2 , t cos 2t
 13.13.63: Find the derivative of the vector function
 13.13.64: Find the derivative of the vector function
 13.13.65: Find the derivative of the vector function
 13.13.66: Find the derivative of the vector function
 13.13.67: Find the derivative of the vector function
 13.13.68: Find the derivative of the vector function
 13.13.69: Find the derivative of the vector function
 13.13.70: Find the derivative of the vector function
 13.13.71: Find the derivative of the vector function
 13.13.72: Find the derivative of the vector function rtt a b t c r
 13.13.73: Find the unit tangent vector at the point with the given value of t...
 13.13.74: Find the unit tangent vector at the point with the given value of t...
 13.13.75: Find the unit tangent vector at the point with the given value of t...
 13.13.76: Find the unit tangent vector at the point with the given value of t...
 13.13.77: If , find and rtt, t rt, T1, rt, rtrt. 2 , t 3 rt2 sin
 13.13.78: If , find , , and rte T0r0rtrt. 2t , e2t , te 2t rt
 13.13.79: Find parametric equations for the tangent line to the curvewith the...
 13.13.80: Find parametric equations for the tangent line to the curve with th...
 13.13.81: Find parametric equations for the tangent line to the curve with th...
 13.13.82: Find parametric equations for the tangent line to the curve with th...
 13.13.83: Find parametric equations for the tangent line to the curve with th...
 13.13.84: Find parametric equations for the tangent line to the curve with th...
 13.13.85: Find parametric equations for the tangent line to the curve with th...
 13.13.86: (a) Find the point of intersection of the tangent lines to the curv...
 13.13.87: The curves and intersect at the origin. Find their angle of interse...
 13.13.88: At what point do the curves and intersect? Find their angle of inte...
 13.13.89: Evaluate the integral y 1 0 16t 3 i 9t 2 j 25t 4 kdt
 13.13.90: Evaluate the integral y 1 0 4 1 t 2 j 2t 1 t 2 k dt
 13.13.91: Evaluate the integral y 2 0 3 sin2 t cos t i 3 sin t cos 2 t j 2 si...
 13.13.92: Evaluate the integral y 2 1 (t 2 i tst 1 j t sin t k) d
 13.13.93: Evaluate the integral y et i 2t j ln t kdt
 13.13.94: Evaluate the integral y cos t i sin t j t kdt
 13.13.95: Find if and r1i j 2
 13.13.96: Find if and r0i j k t
 13.13.97: Prove Formula 1 of Theorem 3
 13.13.98: Prove Formula 3 of Theorem 3
 13.13.99: Prove Formula 5 of Theorem 3
 13.13.100: Prove Formula 6 of Theorem 3
 13.13.101: If and , use Formula 4 of Theorem 3 to find d dt utvt ut
 13.13.102: If and are the vector functions in Exercise 45, use Formula 5 of Th...
 13.13.103: Show that if is a vector function such that exists, then d dt rtrtr...
 13.13.104: Find an expression for d dt utvtwt d dt
 13.13.105: If , show that d dt rt1 rt49. rt0 rtrt d dt u
 13.13.106: If a curve has the property that the position vector is always perp...
 13.13.107: If , show that utrtrtr t utrtrt
 13.13.108: Find the length of the arc of the circular helix with vector equati...
 13.13.109: Reparametrize the helix with respect to arc length measured from in...
 13.13.110: Show that the curvature of a circle of radius a is 1/a
 13.13.111: Find the curvature of the twisted cubic at a general point and at (...
 13.13.112: Find the curvature of the parabola at the points , , and 2,4
 13.13.113: Find the unit normal and binormal vectors for the circular helix rt...
 13.13.114: Find the equations of the normal plane and osculating plane of the ...
 13.13.115: Find and graph the osculating circle of the parabola at the origin.
 13.13.116: Find the length of the curve.
 13.13.117: Find the length of the curve.rt2t, t 0 t 1
 13.13.118: Find the length of the curve.rts2t i e 0 t 1 t
 13.13.119: Find the length of the curve.rtcos t i sin tj ln cos t k 0 t 4
 13.13.120: Find the length of the curve.rti t 0 t 1 2 j t
 13.13.121: Find the length of the curve.rt12t i 8t 0 t 1
 13.13.122: Find the length of the curve correct to four decimal places. (Use y...
 13.13.123: Find the length of the curve correct to four decimal places. (Use y...
 13.13.124: Find the length of the curve correct to four decimal places. (Use y...
 13.13.125: Graph the curve with parametric equations , , . Find the total leng...
 13.13.126: Let be the curve of intersection of the parabolic cylinder and the ...
 13.13.127: Find, correct to four decimal places, the length of the curve of in...
 13.13.128: Reparametrize the curve with respect to arc length measured from th...
 13.13.129: Reparametrize the curve with respect to arc length measured from th...
 13.13.130: Suppose you start at the point and move 5 units along the curve , ,...
 13.13.131: Reparametrize the curve with respect to arc length measured from th...
 13.13.132: a) Find the unit tangent and unit normal vectors and . (b) Use Form...
 13.13.133: a) Find the unit tangent and unit normal vectors and . (b) Use Form...
 13.13.134: a) Find the unit tangent and unit normal vectors and . (b) Use Form...
 13.13.135: a) Find the unit tangent and unit normal vectors and . (b) Use Form...
 13.13.136: Use Theorem 10 to find the curvature rtt 2 i t k
 13.13.137: Use Theorem 10 to find the curvature rtt i t j 1 t 2 k
 13.13.138: Use Theorem 10 to find the curvature rt3t i 4 sin t j 4 cos t k
 13.13.139: Find the curvature of at the point (1, 0, 0).
 13.13.140: Find the curvature of at the point (1, 1, 1).
 13.13.141: Graph the curve with parametric equations and find the curvature at...
 13.13.142: Use Formula 11 to find the curvature y 2x x
 13.13.143: Use Formula 11 to find the curvature y cos x 2
 13.13.144: Use Formula 11 to find the curvature y 4x 52
 13.13.145: At what point does the curve have maximum curvature? What happens t...
 13.13.146: At what point does the curve have maximum curvature? What happens t...
 13.13.147: Find an equation of a parabola that has curvature 4 at the origin. ...
 13.13.148: Use a graphing calculator or computer to graph both the curve and i...
 13.13.149: Use a graphing calculator or computer to graph both the curve and i...
 13.13.150: Two graphs, and , are shown. One is a curve and the other is the gr...
 13.13.151: Two graphs, and , are shown. One is a curve and the other is the gr...
 13.13.152: (a) Graph the curve . At how many points on the curve does it appea...
 13.13.153: The graph of is shown in Figure 12(b) in Section 13.1. Where do you...
 13.13.154: Use Theorem 10 to show that the curvature of a plane parametric cur...
 13.13.155: Use the formula in Exercise 40 to find the curvature
 13.13.156: Use the formula in Exercise 40 to find the curvature
 13.13.157: Find the vectors , , and at the given point.
 13.13.158: Find the vectors , , and at the given point.
 13.13.159: Find equations of the normal plane and osculating plane of the curv...
 13.13.160: Find equations of the normal plane and osculating plane of the curv...
 13.13.161: Find equations of the osculating circles of the ellipse at the poin...
 13.13.162: Find equations of the osculating circles of the parabola at the poi...
 13.13.163: At what point on the curve , , is the normal plane parallel to the ...
 13.13.164: Is there a point on the curve in Exercise 49 where the osculating p...
 13.13.165: Show that the curvature is related to the tangent and normal vector...
 13.13.166: Show that the curvature of a plane curve is , where is the angle be...
 13.13.167: (a) Show that is perpendicular to . (b) Show that is perpendicular ...
 13.13.168: The following formulas, called the FrenetSerret formulas, are of f...
 13.13.169: Use the FrenetSerret formulas to prove each of the following. (Pri...
 13.13.170: Show that the circular helix , where and are positive constants, ha...
 13.13.171: Use the formula in Exercise 55(d) to find the torsion of the curve ...
 13.13.172: Find the curvature and torsion of the curve , , at the point .
 13.13.173: The DNA molecule has the shape of a double helix (see Figure 3 on p...
 13.13.174: Lets consider the problem of designing a railroad track to make a s...
 13.13.175: The position vector of an object moving in a plane is given by Find...
 13.13.176: Find the velocity, acceleration, and speed of a particle with posit...
 13.13.177: A moving particle starts at an initial position with initial veloci...
 13.13.178: An object with mass that moves in a circular path with constant ang...
 13.13.179: A projectile is fired with angle of elevation and initial velocity ...
 13.13.180: A projectile is fired with muzzle speed and angle of elevation from...
 13.13.181: A particle moves with position function . Find the tangential and n...
 13.13.182: The table gives coordinates of a particle moving through space alon...
 13.13.183: The figure shows the path of a particle that moves with position ve...
 13.13.184: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.185: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.186: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.187: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.188: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.189: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.190: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.191: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.192: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.193: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.194: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.195: Find the velocity, acceleration, and speed of a particle with the g...
 13.13.196: Find the velocity and position vectors of a particle that has the g...
 13.13.197: Find the velocity and position vectors of a particle that has the g...
 13.13.198: a) Find the position vector of a particle that has the given accele...
 13.13.199: a) Find the position vector of a particle that has the given accele...
 13.13.200: The position function of a particle is given by . When is the speed...
 13.13.201: What force is required so that a particle of mass has the position ...
 13.13.202: A force with magnitude 20 N acts directly upward from the plane on...
 13.13.203: Show that if a particle moves with constant speed, then the velocit...
 13.13.204: A projectile is fired with an initial speed of 500 ms and angle of ...
 13.13.205: Rework Exercise 23 if the projectile is fired from a position 200 m...
 13.13.206: A ball is thrown at an angle of to the ground. If the ball lands 90...
 13.13.207: A gun is fired with angle of elevation . What is the muzzle speed i...
 13.13.208: A gun has muzzle speed . Find two angles of elevation that can be u...
 13.13.209: A batter hits a baseball 3 ft above the ground toward the center fi...
 13.13.210: A medieval city has the shape of a square and is protected by walls...
 13.13.211: A ball with mass 0.8 kg is thrown southward into the air with a spe...
 13.13.212: Water traveling along a straight portion of a river normally flows ...
 13.13.213: Another reasonable model for the water speed of the river in Exerci...
 13.13.214: Find the tangential and normal components of the acceleration vecto...
 13.13.215: Find the tangential and normal components of the acceleration vecto...
 13.13.216: Find the tangential and normal components of the acceleration vecto...
 13.13.217: Find the tangential and normal components of the acceleration vecto...
 13.13.218: Find the tangential and normal components of the acceleration vecto...
 13.13.219: Find the tangential and normal components of the acceleration vecto...
 13.13.220: The magnitude of the acceleration vector is . Use the figure to est...
 13.13.221: If a particle with mass moves with position vector , then its angul...
 13.13.222: The position function of a spaceship is and the coordinates of a sp...
 13.13.223: A rocket burning its onboard fuel while moving through space has ve...
 13.13.224: What is a vector function? How do you find its derivative and its i...
 13.13.225: What is the connection between vector functions and space curves?
 13.13.226: How do you find the tangent vector to a smooth curve at a point? Ho...
 13.13.227: If and are differentiable vector functions, is a scalar, and is a r...
 13.13.228: How do you find the length of a space curve given by a vector funct...
 13.13.229: a) What is the definition of curvature? (b) Write a formula for cur...
 13.13.230: (a) Write formulas for the unit normal and binormal vectors of a sm...
 13.13.231: a) How do you find the velocity, speed, and acceleration of a parti...
 13.13.232: State Keplers Laws.
 13.13.233: The curve with vector equation is a line.
 13.13.234: The derivative of a vector function is obtained by differentiating ...
 13.13.235: If and are differentiable vector functions, then d dt utvt u tv t ut
 13.13.236: If is a differentiable vector function, then d dt rtr t r
 13.13.237: If is the unit tangent vector of a smooth curve, then the curvature is
 13.13.238: The binormal vector is BtNtTt
 13.13.239: Suppose is twice continuously differentiable. At an inflection poin...
 13.13.240: If for all , the curve is a straight line
 13.13.241: If for all , then is a constant
 13.13.242: If for all , then is orthogonal to for all .
 13.13.243: The osculating circle of a curve C at a point has the same tangent ...
 13.13.244: Different parametrizations of the same curve result in identical ta...
 13.13.245: (a) Sketch the curve with vector function (b) Find and
 13.13.246: Let . (a) Find the domain of . (b) Find . (c) Find
 13.13.247: Find a vector function that represents the curve of intersection of...
 13.13.248: Find parametric equations for the tangent line to the curve , , at ...
 13.13.249: If , evaluate x 1 0 rtt rtdt 2
 13.13.250: Let be the curve with equations , , . Find (a) the point where inte...
 13.13.251: Use Simpsons Rule with to estimate the length of the arc of the cur...
 13.13.252: Find the length of the curve rt2t 32 , cos 2t, sin 2t 0 0 t 1
 13.13.253: The helix intersects the curve at the point . Find the angle of int...
 13.13.254: Reparametrize the curve with respect to arc length measured from th...
 13.13.255: For the curve given by , find (a) the unit tangent vector (b) the u...
 13.13.256: Find the curvature of the ellipse , at the points and .
 13.13.257: Find the curvature of the curve at the point .
 13.13.258: Find an equation of the osculating circle of the curve at the origi...
 13.13.259: Find an equation of the osculating plane of the curve , , at the po...
 13.13.260: The figure shows the curve traced by a particle with position vecto...
 13.13.261: A particle moves with position function . Find the velocity, speed,...
 13.13.262: A particle starts at the origin with initial velocity . Its acceler...
 13.13.263: An athlete throws a shot at an angle of to the horizontal at an ini...
 13.13.264: Find the tangential and normal components of the acceleration vecto...
 13.13.265: A disk of radius is rotating in the counterclockwise direction at a...
 13.13.266: In designing transfer curves to connect sections of straight railro...
 13.13.267: A particle moves with constant angular speed around a circle whose ...
 13.13.268: A circular curve of radius on a highway is banked at an angle so th...
 13.13.269: A projectile is fired from the origin with angle of elevation and i...
 13.13.270: (a) A projectile is fired from the origin down an inclined plane th...
 13.13.271: A ball rolls off a table with a speed of 2 fts. The table is 3.5 ft...
 13.13.272: Find the curvature of the curve with parametric equations x y ) dt ...
 13.13.273: If a projectile is fired with angle of elevation and initial speed ...
 13.13.274: A cable has radius and length and is wound around a spool with radi...
Solutions for Chapter 13: VECTOR FUNCTIONS
Full solutions for Calculus: Early Transcendentals  6th Edition
ISBN: 9780495011668
Solutions for Chapter 13: VECTOR FUNCTIONS
Get Full SolutionsChapter 13: VECTOR FUNCTIONS includes 274 full stepbystep solutions. Since 274 problems in chapter 13: VECTOR FUNCTIONS have been answered, more than 37331 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 6. This expansive textbook survival guide covers the following chapters and their solutions. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780495011668.

Common difference
See Arithmetic sequence.

Complements or complementary angles
Two angles of positive measure whose sum is 90°

Compound fraction
A fractional expression in which the numerator or denominator may contain fractions

Divisor of a polynomial
See Division algorithm for polynomials.

End behavior
The behavior of a graph of a function as.

equation of an ellipse
(x  h2) a2 + (y  k)2 b2 = 1 or (y  k)2 a2 + (x  h)2 b2 = 1

Geometric sequence
A sequence {an}in which an = an1.r for every positive integer n ? 2. The nonzero number r is called the common ratio.

Graph of an inequality in x and y
The set of all points in the coordinate plane corresponding to the solutions x, y of the inequality.

Limaçon
A graph of a polar equation r = a b sin u or r = a b cos u with a > 0 b > 0

Linear system
A system of linear equations

Measure of an angle
The number of degrees or radians in an angle

Median (of a data set)
The middle number (or the mean of the two middle numbers) if the data are listed in order.

nth power of a
The number with n factors of a , where n is the exponent and a is the base.

Paraboloid of revolution
A surface generated by rotating a parabola about its line of symmetry.

Permutation
An arrangement of elements of a set, in which order is important.

Richter scale
A logarithmic scale used in measuring the intensity of an earthquake.

Supply curve
p = ƒ(x), where x represents production and p represents price

Transformation
A function that maps real numbers to real numbers.

Unit vector
Vector of length 1.

zaxis
Usually the third dimension in Cartesian space.