Chapter 1. calc_tutorial_13_4_019

1.1 Problem Statement

{2,4,6,8}
{3,5,7}
9*$b
$a*$b
9*$a
3*$a
$a/2
$d/2
$e/2
pow($g,2)+pow($b,2)
pow($c,2)+3*pow($h,2)+pow($i,2)
9+pow($b,2)
pow($d,2)+pow($f,2)
round(pow($k,0.5)/pow($j,1.5),5)
round(pow($m,0.5)/pow($l,1.5),5)

Find the curvature of r(t) = <$a sin t, cos 3t, $bt> at t = (π / 3) and t = (π / 2).

1.2 Step 1

Question Sequence

Question 1.1

Recall the formula for the curvature of a regular parametrization, r(t).

Thus we will need to find r'(t) and r''(t).

Find r'(t).

r'(t) = <nc1ItEz0kR4=cos t, -3 sin 3t, iSba6t70dtA=>

Find r''(t).

r''(t) = <IHLqJALES1Y=sin t, qnmvdj7HGaQ= cos 3t, 0>

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Find the cross product r'(t) × r''(t). Let the tables below represent the 3 x 3 matrix A and the 2 x 2 matrices B, C, and D, in that order.

i j k
$a cos t -3 sin 3t $b
-$a sin t -9 cos 3t 0
Table : Matrix A
-3 sin 3t $b
-9 cos 3t 0
Table : Matrix B
$a cos t $b
-$a sin t 0
Table : Matrix C
$a cos t -3 sin 3t
-$a sin t -9 cos 3t
Table : Matrix D

TABLE

r'(t) × r''(t) = det(A) = det(B)i - det(C)j + det(D)k

= (SFgqQUkJGdg= cos 3t)i + (U2GIbglD1oM= sin t)j + (8SMCMW9N2Ew= cos t cos 3t - or6dmYWrEbA= sin t sin 3t)k

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Evaluate r'(t) and r'(t) × r''(t) at t = (π / 3) and t = (π / 2). Round your answers to five decimal places.

r'(π / 3) = <$g, 0, $b>

r'(π / 3) × r''(π / 3) = <YWSdEbNFjeo=, zBNzILz/DvM=√3, zDsgB75cNfg=>

r'(π / 2) = <0, 607M7xmPORU=, iSba6t70dtA=>

r'(π / 2) × r''(π / 2) = <0, iXKaZpK/a+Q=, or6dmYWrEbA=>

Correct.
Incorrect.

1.5 Step 4

Question Sequence

Question 1.4

Evaluate the magnitudes ||r'(t)|| and ||r'(t) × r''(t)|| at t = (π / 3) and t = (π / 2).

||r'(π / 3|| = √(bEz/iYnXpnk=)

||r'(π / 3) × r''(π / 3)|| = √(5VKs1keauzU=)

||r'(π / 2)|| = √(Q2IgEMvK+Uo=)

||r'(π / 2) × r''(π / 2)|| = √(xf1an8qan6o=)

Correct.
Incorrect.

1.6 Step 5

Question Sequence

Question 1.5

Compute the curvature, κ(t), at t = (π / 3) and t = (π / 2). Round your answers to five decimal places.

κ(π / 3) = (||r'(π / 3) × r''(π / 3)|| / ||r'(π / 3)||3)

= (√$k)/(√$j)3

≈ S66BSo5Vlp0=

κ(π / 2) = (||r'(π / 2) × r''(π / 2)|| / ||r'(π / 2)||3)

= (√$m)/(√$l)3

≈ fhf0phNsVCo=

Correct.
Incorrect.