24. Find the curvature of r(t) = (t², In t, t In t) at the point (1, 0, 0). 25. Find the curvature of r(t) = (t, t², t³) at the point (1,1,1). x = cost, 26. Graph the curve with parametric equations x = v = sin t. z = sin 5t and find the curvature at the
24. Find the curvature of r(t) = (t², In t, t In t) at the point (1, 0, 0). 25. Find the curvature of r(t) = (t, t², t³) at the point (1,1,1). x = cost, 26. Graph the curve with parametric equations x = v = sin t. z = sin 5t and find the curvature at the
C++ for Engineers and Scientists
4th Edition
ISBN:9781133187844
Author:Bronson, Gary J.
Publisher:Bronson, Gary J.
Chapter13: Structures
Section: Chapter Questions
Problem 4PP
Related questions
Question
25
Transcribed Image Text:868
13.3 EXERCISES
1-6 Find the length of the curve.
1. r(t) = (1, 3 cos 1, 3 sin 1), -5≤1≤5
2. r(t) = (21, 1², 1³), 0≤1≤1
CHAPTER 13 Vector Functions
3. r(t) = √√√2ti+e'j+e¹¹k, 0≤i≤l
4. r(t) = cos ti+ sin tj + In cos tk, 0≤t</4
5. r(t) = i + t²j+ 1³k, 0≤1≤1
6. r(t) = t² i + 9tj + 41³/2 k, 1≤1≤4
7-9 Find the length of the curve correct to four decimal places.
(Use a calculator to approximate the integral.)
7. r(t) = (1², 1³, 14),
0≤ t ≤2
8. r(t) = (t, e¹, te ¹), 1< t <3
9. r(t) = (cos πt, 2t, sin 2πt), from (1, 0, 0) to (1, 4, 0)
at you
turning at
10. Graph the curve with parametric equations x = sin t,
y = sin 2t, z = sin 3t. Find the total length of this curve
correct to four decimal places.
11. Let C be the curve of intersection of the parabolic cylinder
x² = 2y and the surface 3z xy. Find the exact length of C
from the origin to the point (6, 18, 36).
12. Find, correct to four decimal places, the length of the curve
of intersection of the cylinder 4x² + y² = 4 and the plane
x+y+z = 2.
fior
13-14 (a) Find the arc length function for the curve measured
from the point P in the direction of increasing t and then
reparametrize the curve with respect to arc length starting from
P, and (b) find the point 4 units along the curve (in the direction
of increasing t) from P.
er nigno Sil IR ST
-
13. r(t) = (5 t)i + (4t − 3)j + 3t k,
14. r(t) = e' sin ti + e' cos tj + √√2e'k,
r(t)
=
=
15. Suppose you start at the point (0, 0, 3) and move 5 units
along the curve x = 3 sin t, y = 4t, z = 3 cos t in the posi-
tive direction. Where are you now?
2020 in
16. Reparametrize the curve
2
1² + 1
-
P(4, 1, 3)
P(0, 1, √2)
1i+
snitluse
2t
t² + 1
with respect to arc length measured from the point (1, 0)
in the direction of increasing t. Express the reparametriza-
curve?
tion in its simplest form. What can you conclude about the
17-20
(b) Use Formula 9 to find the curvature.
(a) Find the unit tangent and unit normal vectors T(t) and N(t).
17. r(t) = (1, 3 cos t, 3 sin t)
18. r(t) = (t2, sin t t cos t, cost + t sin t), t> 0
19. r(t) =(√2t, e', e^¹)
20. r(t) = (1, 11², 1²)
21-23 Use Theorem 10 to find the curvature.
21. r(t) = t³j+t² k
22. r(t) = ti + t²j+e'k
23. r(t) = √6t² i + 2t j + 2t³ k
24. Find the curvature of r(t) = (t², In t, t In t) at the
point (1, 0, 0).
C
25. Find the curvature of r(t) = (t, t², t³) at the point (1, 1, 1).
26. Graph the curve with parametric equations x = cost,
y = sin t, z = sin 5t and find the curvature at the
point (1, 0, 0).
pa signi
27-29 Use Formula 11 to find the curvature.
27. y = x+
28. y tan x
30-31 At what point does the curve have maximum curvature?
What happens to the curvature as x → ∞?
30. y = ln x
31. y = e* simb
32. Find an equation of a parabola that has curvature 4 at the
origin.
YA
1
33. (a) Is the curvature of the curve C shown in the figure
greater at P or at Q? Explain.
(b) Estimate the curvature at P and at Q by sketching the
osculating circles at those points.
1
29. y = xe*
1
P
Q
C
Expert Solution
Step 1
The curvature of a curve r(t) is given by the formula:
k(t) = |r'(t) × r''(t)| / |r'(t)|^3
where × denotes the cross product and | | denotes the magnitude of a vector.
To find the curvature of the curve r(t) = (t, t^2, t^3), we first need to find its first and second derivatives:
r'(t) = (1, 2t, 3t^2)
r''(t) = (0, 2, 6t)
Then, at the point (1, 1, 1), we have:
r'(1) = (1, 2, 3)
r''(1) = (0, 2, 6)
The magnitude of r'(1) is:
|r'(1)| = sqrt(1^2 + 2^2 + 3^2) = sqrt(14)
The cross product of r'(1) and r''(1) is:
r'(1) × r''(1) = (-18, 0, 2)
The magnitude of r'(1) × r''(1) is:
|r'(1) × r''(1)| = sqrt((-18)^2 + 0^2 + 2^2) = sqrt(328)
Therefore, the curvature of r(t) at the point (1,1,1) is:
k(1) = |r'(1) × r''(1)| / |r'(1)|^3
= sqrt(328) / (sqrt(14))^3
= 4sqrt(14) / 49
Hence, the curvature of r(t) at the point (1,1,1) is 4sqrt(14) / 49.
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Similar questions
Recommended textbooks for you
C++ for Engineers and Scientists
Computer Science
ISBN:
9781133187844
Author:
Bronson, Gary J.
Publisher:
Course Technology Ptr
C++ for Engineers and Scientists
Computer Science
ISBN:
9781133187844
Author:
Bronson, Gary J.
Publisher:
Course Technology Ptr