 11.9.1: What is the curvature of a straight line?
 11.9.2: Explain the meaning of the curvature of a curve. Is it a scalar fun...
 11.9.3: Give a practical formula for computing the curvature.
 11.9.4: Interpret the principal unit normal vector of a curve. Is it a scal...
 11.9.5: Give a practical formula for computing the principal unit normal ve...
 11.9.6: Explain how to decompose the acceleration vector of a moving object...
 11.9.7: Explain how the vectors T, N, and B are related geometrically.
 11.9.8: How do you compute B?
 11.9.9: Give a geometrical interpretation of the torsion.
 11.9.10: How do you compute the torsion?
 11.9.11: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.12: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.13: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.14: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.15: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.16: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.17: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.18: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.19: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.20: 1120. Curvature Find the unit tangent vector T and the curvature k ...
 11.9.21: 2126. Alternative curvature formula Use the alternative curvature f...
 11.9.22: 2126. Alternative curvature formula Use the alternative curvature f...
 11.9.23: 2126. Alternative curvature formula Use the alternative curvature f...
 11.9.24: 2126. Alternative curvature formula Use the alternative curvature f...
 11.9.25: 2126. Alternative curvature formula Use the alternative curvature f...
 11.9.26: 2126. Alternative curvature formula Use the alternative curvature f...
 11.9.27: 2734. Principal unit normal vector Find the unit tangent vector T a...
 11.9.28: 2734. Principal unit normal vector Find the unit tangent vector T a...
 11.9.29: 2734. Principal unit normal vector Find the unit tangent vector T a...
 11.9.30: 2734. Principal unit normal vector Find the unit tangent vector T a...
 11.9.31: 2734. Principal unit normal vector Find the unit tangent vector T a...
 11.9.32: 2734. Principal unit normal vector Find the unit tangent vector T a...
 11.9.33: 2734. Principal unit normal vector Find the unit tangent vector T a...
 11.9.34: 2734. Principal unit normal vector Find the unit tangent vector T a...
 11.9.35: 3540. Components of the acceleration Consider the following traject...
 11.9.36: 3540. Components of the acceleration Consider the following traject...
 11.9.37: 3540. Components of the acceleration Consider the following traject...
 11.9.38: 3540. Components of the acceleration Consider the following traject...
 11.9.39: 3540. Components of the acceleration Consider the following traject...
 11.9.40: 3540. Components of the acceleration Consider the following traject...
 11.9.41: 4144. Computing the binormal vector and torsion In Exercises 2730, ...
 11.9.42: 4144. Computing the binormal vector and torsion In Exercises 2730, ...
 11.9.43: 4144. Computing the binormal vector and torsion In Exercises 2730, ...
 11.9.44: 4144. Computing the binormal vector and torsion In Exercises 2730, ...
 11.9.45: 4548. Computing the binormal vector and torsion Use the definitions...
 11.9.46: 4548. Computing the binormal vector and torsion Use the definitions...
 11.9.47: 4548. Computing the binormal vector and torsion Use the definitions...
 11.9.48: 4548. Computing the binormal vector and torsion Use the definitions...
 11.9.49: Explain why or why not Determine whether the following statements a...
 11.9.50: Special formula: Curvature for y _ f 1x2 Assume that f is twice dif...
 11.9.51: 5154. Curvature for y f 1x2 Use the result of Exercise 50 to find t...
 11.9.52: 5154. Curvature for y f 1x2 Use the result of Exercise 50 to find t...
 11.9.53: 5154. Curvature for y f 1x2 Use the result of Exercise 50 to find t...
 11.9.54: 5154. Curvature for y f 1x2 Use the result of Exercise 50 to find t...
 11.9.55: Special formula: Curvature for plane curves Show that the parametri...
 11.9.56: 5659. Curvature for plane curves Use the result of Exercise 55 to f...
 11.9.57: 5659. Curvature for plane curves Use the result of Exercise 55 to f...
 11.9.58: 5659. Curvature for plane curves Use the result of Exercise 55 to f...
 11.9.59: 5659. Curvature for plane curves Use the result of Exercise 55 to f...
 11.9.60: 6063. Same paths, different velocity The position functions of obje...
 11.9.61: 6063. Same paths, different velocity The position functions of obje...
 11.9.62: 6063. Same paths, different velocity The position functions of obje...
 11.9.63: 6063. Same paths, different velocity The position functions of obje...
 11.9.64: 6467. Graphs of the curvature Consider the following curves. a. Gra...
 11.9.65: 6467. Graphs of the curvature Consider the following curves. a. Gra...
 11.9.66: 6467. Graphs of the curvature Consider the following curves. a. Gra...
 11.9.67: 6467. Graphs of the curvature Consider the following curves. a. Gra...
 11.9.68: Curvature of ln x Find the curvature of f 1x2 = ln x, for x 7 0, an...
 11.9.69: Curvature of ex Find the curvature of f 1x2 = ex and find the point...
 11.9.70: Circle and radius of curvature Choose a point P on a smooth curve C...
 11.9.71: 7174. Finding radii of curvature Find the radius of curvature (see ...
 11.9.72: 7174. Finding radii of curvature Find the radius of curvature (see ...
 11.9.73: 7174. Finding radii of curvature Find the radius of curvature (see ...
 11.9.74: 7174. Finding radii of curvature Find the radius of curvature (see ...
 11.9.75: Curvature of the sine curve The function f 1x2 = sin nx, where n is...
 11.9.76: Parabolic trajectory In Example 7 it was shown that for the parabol...
 11.9.77: Parabolic trajectory Consider the parabolic trajectory x = 1V0 cos ...
 11.9.78: Relationship between T, N, and a Show that if an object accelerates...
 11.9.79: Zero curvature Prove that the curve r1t2 = 8a + btp, c + dtp, e + f...
 11.9.80: Practical formula for N Show that the definition of the principal u...
 11.9.81: Maximum curvature Consider the superparabolas fn1x2 = x2n, where n ...
 11.9.82: Alternative derivation of the curvature Derive the computational fo...
 11.9.83: Computational formula for B Use the result of part (a) of Exercise ...
 11.9.84: Torsion formula Show that the formula defining the torsion, t =  d...
 11.9.85: Descartes fourcircle solution Consider the four mutually tangent c...
Solutions for Chapter 11.9: Calculus: Early Transcendentals 2nd Edition
Full solutions for Calculus: Early Transcendentals  2nd Edition
ISBN: 9780321947345
Solutions for Chapter 11.9
Get Full SolutionsThis textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 2. Since 85 problems in chapter 11.9 have been answered, more than 61424 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321947345. Chapter 11.9 includes 85 full stepbystep solutions.

Angular speed
Speed of rotation, typically measured in radians or revolutions per unit time

Boxplot (or boxandwhisker plot)
A graph that displays a fivenumber summary

Common ratio
See Geometric sequence.

Continuous at x = a
lim x:a x a ƒ(x) = ƒ(a)

Coterminal angles
Two angles having the same initial side and the same terminal side

Cycloid
The graph of the parametric equations

Integrable over [a, b] Lba
ƒ1x2 dx exists.

Local extremum
A local maximum or a local minimum

Magnitude of a vector
The magnitude of <a, b> is 2a2 + b2. The magnitude of <a, b, c> is 2a2 + b2 + c2

Onetoone function
A function in which each element of the range corresponds to exactly one element in the domain

Outliers
Data items more than 1.5 times the IQR below the first quartile or above the third quartile.

Parametric equations
Equations of the form x = ƒ(t) and y = g(t) for all t in an interval I. The variable t is the parameter and I is the parameter interval.

Polar form of a complex number
See Trigonometric form of a complex number.

Polynomial in x
An expression that can be written in the form an x n + an1x n1 + Á + a1x + a0, where n is a nonnegative integer, the coefficients are real numbers, and an ? 0. The degree of the polynomial is n, the leading coefficient is an, the leading term is anxn, and the constant term is a0. (The number 0 is the zero polynomial)

Rational zeros
Zeros of a function that are rational numbers.

System
A set of equations or inequalities.

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

Whole numbers
The numbers 0, 1, 2, 3, ... .

xyplane
The points x, y, 0 in Cartesian space.

Zero matrix
A matrix consisting entirely of zeros.