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CalculusCalculus10th Edition

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Chapter 12, Problem 13PS
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Problem 14PS
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
1 views

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 0≤t≤2.

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 0≤t≤2.

(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 limt→∞K.

(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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Sect-12.1 P-1ESect-12.1 P-2ESect-12.1 P-3ESect-12.1 P-4ESect-12.1 P-5ESect-12.1 P-6ESect-12.1 P-7ESect-12.1 P-8ESect-12.1 P-9ESect-12.1 P-10E
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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P-93ESect-12.5 P-94ESect-12.5 P-95ESect-12.5 P-96ECh-12 P-1RECh-12 P-2RECh-12 P-3RECh-12 P-4RECh-12 P-5RECh-12 P-6RECh-12 P-7RECh-12 P-8RECh-12 P-9RECh-12 P-10RECh-12 P-11RECh-12 P-12RECh-12 P-13RECh-12 P-14RECh-12 P-15RECh-12 P-16RECh-12 P-17RECh-12 P-18RECh-12 P-19RECh-12 P-20RECh-12 P-21RECh-12 P-22RECh-12 P-23RECh-12 P-24RECh-12 P-25RECh-12 P-26RECh-12 P-27RECh-12 P-28RECh-12 P-29RECh-12 P-30RECh-12 P-31RECh-12 P-32RECh-12 P-33RECh-12 P-34RECh-12 P-35RECh-12 P-36RECh-12 P-37RECh-12 P-38RECh-12 P-39RECh-12 P-40RECh-12 P-41RECh-12 P-42RECh-12 P-43RECh-12 P-44RECh-12 P-45RECh-12 P-46RECh-12 P-47RECh-12 P-48RECh-12 P-49RECh-12 P-50RECh-12 P-51RECh-12 P-52RECh-12 P-53RECh-12 P-54RECh-12 P-55RECh-12 P-56RECh-12 P-57RECh-12 P-58RECh-12 P-59RECh-12 P-60RECh-12 P-61RECh-12 P-62RECh-12 P-63RECh-12 P-64RECh-12 P-65RECh-12 P-66RECh-12 P-67RECh-12 P-68RECh-12 P-69RECh-12 P-70RECh-12 P-71RECh-12 P-1PSCh-12 P-2PSCh-12 P-3PSCh-12 P-4PSCh-12 P-5PSCh-12 P-6PSCh-12 P-7PSCh-12 P-8PSCh-12 P-9PSCh-12 P-10PSCh-12 P-11PSCh-12 P-12PSCh-12 P-13PSCh-12 P-14PS

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