 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 46650 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.

Absolute maximum
A value ƒ(c) is an absolute maximum value of ƒ if ƒ(c) ? ƒ(x) for all x in the domain of ƒ.

Anchor
See Mathematical induction.

Binomial coefficients
The numbers in Pascal’s triangle: nCr = anrb = n!r!1n  r2!

Dot product
The number found when the corresponding components of two vectors are multiplied and then summed

End behavior asymptote of a rational function
A polynomial that the function approaches as.

Heron’s formula
The area of ¢ABC with semiperimeter s is given by 2s1s  a21s  b21s  c2.

Inferential statistics
Using the science of statistics to make inferences about the parameters in a population from a sample.

Linear equation in x
An equation that can be written in the form ax + b = 0, where a and b are real numbers and a Z 0

Linear inequality in x
An inequality that can be written in the form ax + b < 0 ,ax + b … 0 , ax + b > 0, or ax + b Ú 0, where a and b are real numbers and a Z 0

Multiplicity
The multiplicity of a zero c of a polynomial ƒ(x) of degree n > 0 is the number of times the factor (x  c) (x  z 2) Á (x  z n)

n factorial
For any positive integer n, n factorial is n! = n.(n  1) . (n  2) .... .3.2.1; zero factorial is 0! = 1

nset
A set of n objects.

Quadratic regression
A procedure for fitting a quadratic function to a set of data.

Quotient identities
tan ?= sin ?cos ?and cot ?= cos ? sin ?

Rational zeros theorem
A procedure for finding the possible rational zeros of a polynomial.

Remainder theorem
If a polynomial f(x) is divided by x  c , the remainder is ƒ(c)

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

Scatter plot
A plot of all the ordered pairs of a twovariable data set on a coordinate plane.

Solution of a system in two variables
An ordered pair of real numbers that satisfies all of the equations or inequalities in the system

Unit vector
Vector of length 1.