   Chapter 12, Problem 13PS

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Textbook Problem
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# Arc Length and CurvatureConsider the vector-valued function r ( t ) = 〈 t cos π t , t sin π t 〉 ,       0 ≤ t ≤ 2. (a) Use a graphing utility to graph the function.(b) Find the length of the arc in part (a).(c) Find the curvature K as a function of t . Find the curvature at t = 0 , t = 1 , and t = 2 .(d) Use a graphing utility to graph the function K.(e) Find (if possible) lim t → ∞ K . (f) Using the result of part (e), make a conjecture about the graph of r as t → ∞ .

(a)

To determine

To graph: The function r(t)=tcosπt,tsinπt where 0t2.

Explanation

Given:

The provided function is:

r(t)=tcosπt,tsinπt

Graph:

The provided function is:

r(t)=tcosπti+tsinπtj

The above equation is written in parametric form as:

x(t)=tcosπty(t)=tsinπt

Now, to plot the graph of the function by the use Ti83 as below:

Step1: Press the button “MODE”.

Step2: Select “PAR” to use parametric form.

Step3: Press the button “Y=”.

Step4: Enter the functions,

X1=Tcos(πT)Y1=

(b)

To determine

To calculate: The arc length of the function, r(t)=tcosπt,tsinπt where 0t2.

(c)

To determine

To calculate: The curvature K as a function of t for the function r(t)=tcosπt,tsinπt. Also, find the curvature at t=0,t=1 and t=2.

(d)

To determine

To graph: The curvature of the function, K=π(π2t2+2)(π2t2+1)32.

(e)

To determine

To calculate: The limit of limtK.

(f)

To determine
A conjecture of graph of r(t)=tcosπt,tsinπt as t by the use of the result of part (e).

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