 12.1: In Exercises 14, (a) find the domain of r and (b) determine the val...
 12.2: In Exercises 14, (a) find the domain of r and (b) determine the val...
 12.3: In Exercises 14, (a) find the domain of r and (b) determine the val...
 12.4: In Exercises 14, (a) find the domain of r and (b) determine the val...
 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.
 12.22: In Exercises 21 and 22, evaluate the limit.
 12.23: In Exercises 23 and 24, find the following. (a) (b) (c) (d) (e) (f)
 12.24: In Exercises 23 and 24, find the following. (a) (b) (c) (d) (e) (f)
 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 integral.
 12.28: In Exercises 2730, find the indefinite integral.
 12.29: In Exercises 2730, find the indefinite integral.
 12.30: In Exercises 2730, find the indefinite integral.
 12.31: In Exercises 31 and 32, find for the given conditions.
 12.32: In Exercises 31 and 32, find for the given conditions.
 12.33: In Exercises 3336, evaluate the definite integral.
 12.34: In Exercises 3336, evaluate the definite integral.
 12.35: In Exercises 3336, evaluate the definite integral.
 12.36: In Exercises 3336, evaluate the definite integral.
 12.37: In Exercises 37 and 38, the position vector r describes the path of...
 12.38: In Exercises 37 and 38, the position vector r describes the path of...
 12.39: Linear Approximation In Exercises 39 and 40, find a set of parametr...
 12.40: Linear Approximation In Exercises 39 and 40, find a set of parametr...
 12.41: Projectile Motion In Exercises 41 44, use the model for projectile ...
 12.42: Projectile Motion In Exercises 41 44, use the model for projectile ...
 12.43: Projectile Motion In Exercises 41 44, use the model for projectile ...
 12.44: Projectile Motion In Exercises 41 44, use the model for projectile ...
 12.45: In Exercises 4552, find the velocity, speed, and acceleration at ti...
 12.46: In Exercises 4552, find the velocity, speed, and acceleration at ti...
 12.47: In Exercises 4552, find the velocity, speed, and acceleration at ti...
 12.48: In Exercises 4552, find the velocity, speed, and acceleration at ti...
 12.49: In Exercises 4552, find the velocity, speed, and acceleration at ti...
 12.50: In Exercises 4552, find the velocity, speed, and acceleration at ti...
 12.51: In Exercises 4552, find the velocity, speed, and acceleration at ti...
 12.52: In Exercises 4552, find the velocity, speed, and acceleration at ti...
 12.53: In Exercises 53 and 54, find a set of parametric equations for the ...
 12.54: In Exercises 53 and 54, find a set of parametric equations for the ...
 12.55: Satellite Orbit Find the speed necessary for a satellite to maintai...
 12.56: Centripetal Force An automobile in a circular traffic exchange is t...
 12.57: In Exercises 5760, sketch the plane curve and find its length over ...
 12.58: In Exercises 5760, sketch the plane curve and find its length over ...
 12.59: In Exercises 5760, sketch the plane curve and find its length over ...
 12.60: In Exercises 5760, sketch the plane curve and find its length over ...
 12.61: In Exercises 6164, sketch the space curve and find its length over ...
 12.62: In Exercises 6164, sketch the space curve and find its length over ...
 12.63: In Exercises 6164, sketch the space curve and find its length over ...
 12.64: In Exercises 6164, sketch the space curve and find its length over ...
 12.65: In Exercises 65 and 66, use a computer algebra system to find the l...
 12.66: In Exercises 65 and 66, use a computer algebra system to find the l...
 12.67: In Exercises 6770, find the curvature of the curve.
 12.68: In Exercises 6770, find the curvature of the curve.
 12.69: In Exercises 6770, find the curvature of the curve.
 12.70: In Exercises 6770, find the curvature of the curve.
 12.71: In Exercises 7174, find the curvature and radius of curvature of th...
 12.72: In Exercises 7174, find the curvature and radius of curvature of th...
 12.73: In Exercises 7174, find the curvature and radius of curvature of th...
 12.74: In Exercises 7174, find the curvature and radius of curvature of th...
 12.75: Writing A civil engineer designs a highway as shown in the figure. ...
 12.76: A line segment extends horizontally to the left from the point Anot...
 12.1: The cornu spiral is given by and The spiral shown in the figure was...
 12.2: Let be the tangent line at the point to the graph of the curve as s...
 12.3: A bomber is flying horizontally at an altitude of 3200 feet with a ...
 12.4: Repeat Exercise 3 if the bomber is facing away from the launch site...
 12.5: Consider one arch of the cycloid as shown in the figure. Let be the...
 12.6: Consider the cardioid as shown in the figure. Let be the arc length...
 12.7: If is a nonzero differentiable function of prove that
 12.8: A communications satellite moves in a circular orbit around Earth a...
 12.9: In Exercises 911, use the binormal vector defined by the equation F...
 12.10: In Exercises 911, use the binormal vector defined by the equation F...
 12.11: In Exercises 911, use the binormal vector defined by the equation (...
 12.12: A highway has an exit ramp that begins at the origin of a coordinat...
 12.13: Consider the vectorvalued function (a) Use a graphing utility to g...
 12.14: You want to toss an object to a friend who is riding a Ferris wheel...
Solutions for Chapter 12: VectorValued Functions
Full solutions for Calculus  8th Edition
ISBN: 9780618502981
Solutions for Chapter 12: VectorValued Functions
Get Full SolutionsThis textbook survival guide was created for the textbook: Calculus, edition: 8. This expansive textbook survival guide covers the following chapters and their solutions. Since 90 problems in chapter 12: VectorValued Functions have been answered, more than 78786 students have viewed full stepbystep solutions from this chapter. Chapter 12: VectorValued Functions includes 90 full stepbystep solutions. Calculus was written by and is associated to the ISBN: 9780618502981.

Dependent event
An event whose probability depends on another event already occurring

Discriminant
For the equation ax 2 + bx + c, the expression b2  4ac; for the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the expression B2  4AC

Divergence
A sequence or series diverges if it does not converge

Double inequality
A statement that describes a bounded interval, such as 3 ? x < 5

Elimination method
A method of solving a system of linear equations

Equivalent systems of equations
Systems of equations that have the same solution.

Explicitly defined sequence
A sequence in which the kth term is given as a function of k.

Exponential decay function
Decay modeled by ƒ(x) = a ? bx, a > 0 with 0 < b < 1.

Factor
In algebra, a quantity being multiplied in a product. In statistics, a potential explanatory variable under study in an experiment, .

Fitting a line or curve to data
Finding a line or curve that comes close to passing through all the points in a scatter plot.

Linear regression line
The line for which the sum of the squares of the residuals is the smallest possible

Magnitude of a real number
See Absolute value of a real number

Radian
The measure of a central angle whose intercepted arc has a length equal to the circle’s radius.

Reciprocal of a real number
See Multiplicative inverse of a real number.

Slant asymptote
An end behavior asymptote that is a slant line

Slope
Ratio change in y/change in x

Standard form of a polar equation of a conic
r = ke 1 e cos ? or r = ke 1 e sin ? ,

Triangular form
A special form for a system of linear equations that facilitates finding the solution.

Unit vector
Vector of length 1.

Vertex of a parabola
The point of intersection of a parabola and its line of symmetry.