 13.3.1: 16 Find the length of the curve.rt t, 3 cos t, 3 sin t 5 t 5rt
 13.3.2: 16 Find the length of the curve.rt 2t, t2, 13 t3 0 t 1rt
 13.3.3: 16 Find the length of the curve.rt s2 t i et j et k 0 t 1rt
 13.3.4: 16 Find the length of the curve.rt cos t i sin tj ln cos t k 0 t 4
 13.3.5: 16 Find the length of the curve.rt i t2 j t3 k 0 t 1rt
 13.3.6: 16 Find the length of the curve.rt 12t i 8t32 j 3t2 k 0 t 1rt
 13.3.7: 79 Find the length of the curve correct to four decimal places. (Us...
 13.3.8: 79 Find the length of the curve correct to four decimal places. (Us...
 13.3.9: 79 Find the length of the curve correct to four decimal places. (Us...
 13.3.10: Graph the curve with parametric equations , , . Find the total leng...
 13.3.11: . Let be the curve of intersection of the parabolic cylinder and th...
 13.3.12: Find, correct to four decimal places, the length of the curve of in...
 13.3.13: 1314 Reparametrize the curve with respect to arc length measured fr...
 13.3.14: 1314 Reparametrize the curve with respect to arc length measured fr...
 13.3.15: Suppose you start at the point and move 5 units along the curve , ,...
 13.3.16: Reparametrize the curve with respect to arc length measured from th...
 13.3.17: 1720 (a) Find the unit tangent and unit normal vectors and . (b) Us...
 13.3.18: 1720 (a) Find the unit tangent and unit normal vectors and . (b) Us...
 13.3.19: 1720 (a) Find the unit tangent and unit normal vectors and . (b) Us...
 13.3.20: 1720 (a) Find the unit tangent and unit normal vectors and . (b) Us...
 13.3.21: 2123 Use Theorem 10 to find the curvature.rt t3 j t2 krt
 13.3.22: 2123 Use Theorem 10 to find the curvature.rt t i t2 j et krt
 13.3.23: 2123 Use Theorem 10 to find the curvature.rt 3t i 4 sin t j 4 cos t...
 13.3.24: Find the curvature of at the point .
 13.3.25: . Find the curvature of at the point (1, 1, 1).
 13.3.26: Graph the curve with parametric equations , , and find the curvatur...
 13.3.27: 2729 Use Formula 11 to find the curvature.y x 4
 13.3.28: 2729 Use Formula 11 to find the curvature.y tan x y
 13.3.29: 2729 Use Formula 11 to find the curvature.y xex
 13.3.30: 3031 At what point does the curve have maximum curvature? What happ...
 13.3.31: 3031 At what point does the curve have maximum curvature? What happ...
 13.3.32: Find an equation of a parabola that has curvature 4 at the origin.
 13.3.33: (a) Is the curvature of the curve shown in the figure greater at or...
 13.3.34: ; 3435 Use a graphing calculator or computer to graph both the curv...
 13.3.35: ; 3435 Use a graphing calculator or computer to graph both the curv...
 13.3.36: 3637 Plot the space curve and its curvature function . Comment on h...
 13.3.37: 3637 Plot the space curve and its curvature function . Comment on h...
 13.3.38: 3839 Two graphs, and , are shown. One is a curve and the other is t...
 13.3.39: 3839 Two graphs, and , are shown. One is a curve and the other is t...
 13.3.40: . (a) Graph the curve . At how many points on the curve does it app...
 13.3.41: . The graph of is shown in Figure 12(b) in Section 13.1. Where do y...
 13.3.42: Use Theorem 10 to show that the curvature of a plane parametric cur...
 13.3.43: 43 45 Use the formula in Exercise 42 to find the curvature.3 x t2
 13.3.44: 43 45 Use the formula in Exercise 42 to find the curvature.x a cos ...
 13.3.45: 43 45 Use the formula in Exercise 42 to find the curvature.y et x e...
 13.3.46: Consider the curvature at for each member of the family of function...
 13.3.47: 47 48 Find the vectors , , and at the given point.rt t2,23 t3, t (1...
 13.3.48: 47 48 Find the vectors , , and at the given point.rt cos t, sin t, ...
 13.3.49: 4950 Find equations of the normal plane and osculating plane of the...
 13.3.50: 4950 Find equations of the normal plane and osculating plane of the...
 13.3.51: . Find equations of the osculating circles of the ellipse at the po...
 13.3.52: Find equations of the osculating circles of the parabola at the poi...
 13.3.53: At what point on the curve , , is the normal plane parallel to the ...
 13.3.54: Is there a point on the curve in Exercise 53 where the oscu lating ...
 13.3.55: Find equations of the normal and osculating planes of the curve of ...
 13.3.56: Show that the osculating plane at every point on the curve is the s...
 13.3.57: Show that the curvature is related to the tangent and normal vector...
 13.3.58: Show that the curvature of a plane curve is , where is the angle be...
 13.3.59: (a) Show that is perpendicular to . (b) Show that is perpendicular ...
 13.3.60: The following formulas, called the FrenetSerret formulas, are of f...
 13.3.61: Use the FrenetSerret formulas to prove each of the following. (Pri...
 13.3.62: Show that the circular helix , where and are positive constants, ha...
 13.3.63: Use the formula in Exercise 61(d) to find the torsion of the curve .
 13.3.64: Find the curvature and torsion of the curve , , at the point .
 13.3.65: The DNA molecule has the shape of a double helix (see Figure 3 on p...
 13.3.66: Lets consider the problem of designing a railroad track to make a s...
Solutions for Chapter 13.3: Arc Length and Curvature
Full solutions for Calculus: Early Transcendentals  7th Edition
ISBN: 9780538497909
Solutions for Chapter 13.3: Arc Length and Curvature
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780538497909. Since 66 problems in chapter 13.3: Arc Length and Curvature have been answered, more than 33425 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals , edition: 7. Chapter 13.3: Arc Length and Curvature includes 66 full stepbystep solutions.

Annual percentage yield (APY)
The rate that would give the same return if interest were computed just once a year

Annuity
A sequence of equal periodic payments.

Average rate of change of ƒ over [a, b]
The number ƒ(b)  ƒ(a) b  a, provided a ? b.

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

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

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

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

Difference identity
An identity involving a trigonometric function of u  v

Factored form
The left side of u(v + w) = uv + uw.

Identity matrix
A square matrix with 1’s in the main diagonal and 0’s elsewhere, p. 534.

Initial value of a function
ƒ 0.

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

Pascal’s triangle
A number pattern in which row n (beginning with n = 02) consists of the coefficients of the expanded form of (a+b)n.

Product of complex numbers
(a + bi)(c + di) = (ac  bd) + (ad + bc)i

Projection of u onto v
The vector projv u = au # vƒvƒb2v

Quotient of complex numbers
a + bi c + di = ac + bd c2 + d2 + bc  ad c2 + d2 i

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

Statute mile
5280 feet.

Unit circle
A circle with radius 1 centered at the origin.

Zero of a function
A value in the domain of a function that makes the function value zero.