 12.1: In Exercises 14, (a) find the domain of and (b) determine the value...
 12.2: In Exercises 14, (a) find the domain of and (b) determine the value...
 12.3: In Exercises 14, (a) find the domain of and (b) determine the value...
 12.4: In Exercises 14, (a) find the domain of and (b) determine the value...
 12.5: In Exercises 5 and 6, evaluate (if possible) the vectorvalued func...
 12.6: In Exercises 5 and 6, evaluate (if possible) the vectorvalued func...
 12.7: In Exercises 7 and 8, sketch the plane curve represented by the vec...
 12.8: In Exercises 7 and 8, sketch the plane curve represented by the vec...
 12.9: In Exercises 914, use a computer algebra system to graph the space ...
 12.10: In Exercises 914, use a computer algebra system to graph the space ...
 12.11: In Exercises 914, use a computer algebra system to graph the space ...
 12.12: In Exercises 914, use a computer algebra system to graph the space ...
 12.13: In Exercises 914, use a computer algebra system to graph the space ...
 12.14: In Exercises 914, use a computer algebra system to graph the space ...
 12.15: In Exercises 15 and 16, find vectorvalued functions forming the bo...
 12.16: In Exercises 15 and 16, find vectorvalued functions forming the bo...
 12.17: A particle moves on a straightline path that passes through the po...
 12.18: The outer edge of a spiral staircase is in the shape of a helix of ...
 12.19: In Exercises 19 and 20, sketch the space curve represented by the i...
 12.20: In Exercises 19 and 20, sketch the space curve represented by the i...
 12.21: In Exercises 21 and 22, evaluate the limit.limt4 t i 4 t j k
 12.22: In Exercises 21 and 22, evaluate the limit.LImt0 sin 2tti et j et k
 12.23: In Exercises 23 and 24, find the following.
 12.24: In Exercises 23 and 24, find the following.
 12.25: Writing The and components of the derivative of the vectorvalued f...
 12.26: Writing The component of the derivative of the vectorvalued functio...
 12.27: In Exercises 2730, find the indefinite integralcos t i t cos tj dt
 12.28: In Exercises 2730, find the indefinite integralln t i t ln tj k dt
 12.29: In Exercises 2730, find the indefinite integralcos t i sin tj t k dt
 12.30: In Exercises 2730, find the indefinite integraltj t 2 k i tj t k dt
 12.31: In Exercises 3134, evaluate the definite integral.223t i 2t 2 j t 3...
 12.32: In Exercises 3134, evaluate the definite integral.t j t sin tk dt
 12.33: In Exercises 3134, evaluate the definite integral.et2 i 3t2j k dt
 12.34: In Exercises 3134, evaluate the definite integral.11t3i arcsin tj t...
 12.35: In Exercises 35 and 36, find for the given conditions.rt 2t i e r0 ...
 12.36: In Exercises 35 and 36, find for the given conditions.rt sec t i ta...
 12.37: In Exercises 37 40, the position vector describes the path of an ob...
 12.38: In Exercises 37 40, the position vector describes the path of an ob...
 12.39: In Exercises 37 40, the position vector describes the path of an ob...
 12.40: In Exercises 37 40, the position vector describes the path of an ob...
 12.41: Linear Approximation In Exercises 41 and 42, find a set of parametr...
 12.42: Linear Approximation In Exercises 41 and 42, find a set of parametr...
 12.43: Projectile Motion In Exercises 43 46, use the model for projectile ...
 12.44: Projectile Motion In Exercises 43 46, use the model for projectile ...
 12.45: Projectile Motion In Exercises 43 46, use the model for projectile ...
 12.46: Projectile Motion In Exercises 43 46, use the model for projectile ...
 12.47: In Exercises 4754, find the velocity, speed, and acceleration at ti...
 12.48: In Exercises 4754, find the velocity, speed, and acceleration at ti...
 12.49: In Exercises 4754, find the velocity, speed, and acceleration at ti...
 12.50: In Exercises 4754, find the velocity, speed, and acceleration at ti...
 12.51: In Exercises 4754, find the velocity, speed, and acceleration at ti...
 12.52: In Exercises 4754, find the velocity, speed, and acceleration at ti...
 12.53: In Exercises 4754, find the velocity, speed, and acceleration at ti...
 12.54: In Exercises 4754, find the velocity, speed, and acceleration at ti...
 12.55: In Exercises 55 and 56, find a set of parametric equations for the ...
 12.56: In Exercises 55 and 56, find a set of parametric equations for the ...
 12.57: Satellite Orbit Find the speed necessary for a satellite to maintai...
 12.58: Centripetal Force An automobile in a circular traffic exchange is t...
 12.59: In Exercises 5962, sketch the plane curve and find its length over ...
 12.60: In Exercises 5962, sketch the plane curve and find its length over ...
 12.61: In Exercises 5962, sketch the plane curve and find its length over ...
 12.62: In Exercises 5962, sketch the plane curve and find its length over ...
 12.63: In Exercises 6366, sketch the space curve and find its length over ...
 12.64: In Exercises 6366, sketch the space curve and find its length over ...
 12.65: In Exercises 6366, sketch the space curve and find its length over ...
 12.66: In Exercises 6366, sketch the space curve and find its length over ...
 12.67: In Exercises 6770, find the curvature of the curve.rt 3ti 2tj
 12.68: In Exercises 6770, find the curvature of the curve.rt 2t i 3t
 12.69: In Exercises 6770, find the curvature of the curve.rt 2ti 12t2j t2k
 12.70: In Exercises 6770, find the curvature of the curve.rt 2ti 5 cos tj ...
 12.71: In Exercises 71 and 72, find the curvature of the curve at point Pr...
 12.72: In Exercises 71 and 72, find the curvature of the curve at point Pt...
 12.73: In Exercises 7376, find the curvature and radius of curvature of th...
 12.74: In Exercises 7376, find the curvature and radius of curvature of th...
 12.75: In Exercises 7376, find the curvature and radius of curvature of th...
 12.76: In Exercises 7376, find the curvature and radius of curvature of th...
 12.77: Writing A civil engineer designs a highway as shown in the figure. ...
 12.78: A line segment extends horizontally to the left from the point and ...
Solutions for Chapter 12: VectorValued Functions
Full solutions for Calculus  9th Edition
ISBN: 9780547167022
Solutions for Chapter 12: VectorValued Functions
Get Full SolutionsCalculus was written by and is associated to the ISBN: 9780547167022. This expansive textbook survival guide covers the following chapters and their solutions. Since 78 problems in chapter 12: VectorValued Functions have been answered, more than 36227 students have viewed full stepbystep solutions from this chapter. Chapter 12: VectorValued Functions includes 78 full stepbystep solutions. This textbook survival guide was created for the textbook: Calculus , edition: 9.

Acute angle
An angle whose measure is between 0° and 90°

Arrow
The notation PQ denoting the directed line segment with initial point P and terminal point Q.

Bounded above
A function is bounded above if there is a number B such that ƒ(x) ? B for all x in the domain of ƒ.

Composition of functions
(f ? g) (x) = f (g(x))

Conic section (or conic)
A curve obtained by intersecting a doublenapped right circular cone with a plane

Direction of an arrow
The angle the arrow makes with the positive xaxis

Ellipse
The set of all points in the plane such that the sum of the distances from a pair of fixed points (the foci) is a constant

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.

Higherdegree polynomial function
A polynomial function whose degree is ? 3

Horizontal asymptote
The line is a horizontal asymptote of the graph of a function ƒ if lim x: q ƒ(x) = or lim x: q ƒ(x) = b

Initial point
See Arrow.

Initial side of an angle
See Angle.

Line graph
A graph of data in which consecutive data points are connected by line segments

Mean (of a set of data)
The sum of all the data divided by the total number of items

Root of an equation
A solution.

Row operations
See Elementary row operations.

Sum of a finite arithmetic series
Sn = na a1 + a2 2 b = n 2 32a1 + 1n  12d4,

Terms of a sequence
The range elements of a sequence.

Translation
See Horizontal translation, Vertical translation.

Unit vector
Vector of length 1.