calc_tutorial_13_4_019

 
Problem Statement

{2,4,6,8}
{3,5,7}
9*7
6*7
9*6
3*6
6/2
42/2
54/2
pow(3,2)+pow(7,2)
pow(63,2)+3*pow(21,2)+pow(27,2)
9+pow(7,2)
pow(42,2)+pow(18,2)
round(pow(6021,0.5)/pow(58,1.5),5)
round(pow(2088,0.5)/pow(58,1.5),5)

Find the curvature of r(t) = <6 sin t, cos 3t, 7t> at t = (π / 3) and t = (π / 2).

 
Step 1

Question Sequence

Question 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) = <cos t, -3 sin 3t, >

Find r''(t).

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

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 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
6 cos t -3 sin 3t 7
-6 sin t -9 cos 3t 0
Table : Matrix A
-3 sin 3t 7
-9 cos 3t 0
Table : Matrix B
6 cos t 7
-6 sin t 0
Table : Matrix C
6 cos t -3 sin 3t
-6 sin t -9 cos 3t
Table : Matrix D

TABLE

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

= ( cos 3t)i + ( sin t)j + ( cos t cos 3t - sin t sin 3t)k

Correct.
Incorrect.

 
Step 3

Question Sequence

Question 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) = <3, 0, 7>

r'(π / 3) × r''(π / 3) = <, √3, >

r'(π / 2) = <0, , >

r'(π / 2) × r''(π / 2) = <0, , >

Correct.
Incorrect.

 
Step 4

Question Sequence

Question 4

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

||r'(π / 3|| = √()

||r'(π / 3) × r''(π / 3)|| = √()

||r'(π / 2)|| = √()

||r'(π / 2) × r''(π / 2)|| = √()

Correct.
Incorrect.

 
Step 5

Question Sequence

Question 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)

= (√6021)/(√58)3

≈

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

= (√2088)/(√58)3

≈

Correct.
Incorrect.