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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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57095-12.5-16E-Question-Digital.docx

Investigation Repeat Exercise 15 for the vector-valued function r ( t ) = 6 cos ( π t / 4 ) i + 2 sin ( π t / 4 ) j + t k

(a)

To determine

To calculate: The length of the curve, r(t)=6cos(πt4)i+2sin(πt4)j+tk on the interval [0,2] by finding the length of the line segment connecting to its endpoints.

Explanation

Given:

The curve, r(t)=6cos(πt4)i+2sin(πt4)j+tk

Formula used:

The distance between the vectors, r1=a1i+b1j+c1k and r2=a2i+b2j+c2k is given by,

r2r1=(a2a1)2+(b2b1)2+(c2c1)2

Calculation:

Consider the curve, r(t)=6cos(πt4)i+2sin(πt4)j+tk on the interval [0,2].

Now, calculate the endpoints.

First put t=0 in the function, r(t)=6cos(πt4)i+2sin(πt4)j+tk

(b)

To determine

The length of the curve, r(t)=6cos(πt4)i+2sin(πt4)j+tk on the interval [0,2] by summing the lengths of the line segments connecting the terminal points of the vectors, r(0),r(0.5),r(1),r(1.5),r(2).

(c)

To determine

The description of how an individual might obtain a more accurate approximation of the curve, r(t)=6cos(πt4)i+2sin(πt4)j+tk, by continuing the process in part (a) and part (b).

(d)

To determine

The arc length of the curve, r(t)=6cos(πt4)i+2sin(πt4)j+tk on the interval [0,2] using a graphing utility and also compare the result with the answers of part (a) and part (b).

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Sect-12.1 P-11ESect-12.1 P-12ESect-12.1 P-13ESect-12.1 P-14ESect-12.1 P-15ESect-12.1 P-16ESect-12.1 P-17ESect-12.1 P-18ESect-12.1 P-19ESect-12.1 P-20ESect-12.1 P-21ESect-12.1 P-22ESect-12.1 P-23ESect-12.1 P-24ESect-12.1 P-25ESect-12.1 P-26ESect-12.1 P-27ESect-12.1 P-28ESect-12.1 P-29ESect-12.1 P-30ESect-12.1 P-31ESect-12.1 P-32ESect-12.1 P-33ESect-12.1 P-34ESect-12.1 P-35ESect-12.1 P-36ESect-12.1 P-37ESect-12.1 P-38ESect-12.1 P-39ESect-12.1 P-42ESect-12.1 P-41ESect-12.1 P-40ESect-12.1 P-43ESect-12.1 P-44ESect-12.1 P-45ESect-12.1 P-46ESect-12.1 P-47ESect-12.1 P-48ESect-12.1 P-49ESect-12.1 P-50ESect-12.1 P-51ESect-12.1 P-52ESect-12.1 P-53ESect-12.1 P-54ESect-12.1 P-55ESect-12.1 P-56ESect-12.1 P-57ESect-12.1 P-58ESect-12.1 P-59ESect-12.1 P-60ESect-12.1 P-61ESect-12.1 P-62ESect-12.1 P-63ESect-12.1 P-64ESect-12.1 P-65ESect-12.1 P-66ESect-12.1 P-67ESect-12.1 P-68ESect-12.1 P-69ESect-12.1 P-70ESect-12.1 P-71ESect-12.1 P-72ESect-12.1 P-73ESect-12.1 P-74ESect-12.1 P-75ESect-12.1 P-76ESect-12.1 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