   Chapter 12.5, Problem 16E

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# 57095-12.5-16E-Question-Digital.docxInvestigation 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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