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10th Edition

Ron Larson + 1 other

ISBN: 9781285057095

Textbook Problem

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Arc Length and CurvatureConsider the vector-valued function

(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

(d) Use a graphing utility to graph the function K.

(e) Find (if possible)

(f) Using the result of part (e), make a conjecture about the graph of

**(a)**

To determine

**To graph:** The function

Explanation

**Given:**

The provided function is:

**Graph:**

The provided function is:

The above equation is written in parametric form as:

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,

**(b)**

To determine

**To calculate:** The arc length of the function,

**(c)**

To determine

**To calculate:** The curvature *K* as a function of *t* for the function

**(d)**

To determine

**To graph:** The curvature of the function,

**(e)**

To determine

**To calculate:** The limit of

**(f)**

To determine

A conjecture of graph of r ( t ) = 〈 t cos π t , t sin π t 〉 as t → ∞ by the use of the result of part (e).

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