- 1²) and ngle of =i+j. +j+ k. - use 48. If u and v are the vector functions in Exercise 47, use For- mula 5 of Theorem 3 to find -[u(t) × v(t)] dt 49. Find f'(2), where f(t) = u(t) v(t), u(2) = (1, 2, -1), u' (2) = (3, 0, 4), and v(t) = (t, t², 1³). 50. If r(t) = u(t) x v(t), where u and v are the vector functions in Exercise 49, find r' (2). 51. If r(t) = a cos wt + b sin wt, where a and b are constant vectors, show that r(t) x r' (t) = wa X b. " 52. If r is the vector function in Exercise 51, show that r"(t) + w²r(t) = 0. 53. Show that if r is a vector function such that r" exists, then -[r(t) x r'(t)] = r(t) × r"(t) dt 54. Find an expression for 55. If r(t) = 0, show that dt [u(t) (v(t) x w(t))]. = | r(t) | r(t) · r'(t). dt [Hint: r(t) 2 = r(t) · r(t)] 56. If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on a sphere with center the origin. 57. If u(t) = r(t) [r'(t) × r"(t)], show that u'(t) = r(t) [r'(t) × r"(t)] 58. Show that the tangent vector to a curve defined by a vector function r(t) points in the direction of increasing t. [Hint: Refer to Figure 1 and consider the cases h> 0 and h <0 separately.]
- 1²) and ngle of =i+j. +j+ k. - use 48. If u and v are the vector functions in Exercise 47, use For- mula 5 of Theorem 3 to find -[u(t) × v(t)] dt 49. Find f'(2), where f(t) = u(t) v(t), u(2) = (1, 2, -1), u' (2) = (3, 0, 4), and v(t) = (t, t², 1³). 50. If r(t) = u(t) x v(t), where u and v are the vector functions in Exercise 49, find r' (2). 51. If r(t) = a cos wt + b sin wt, where a and b are constant vectors, show that r(t) x r' (t) = wa X b. " 52. If r is the vector function in Exercise 51, show that r"(t) + w²r(t) = 0. 53. Show that if r is a vector function such that r" exists, then -[r(t) x r'(t)] = r(t) × r"(t) dt 54. Find an expression for 55. If r(t) = 0, show that dt [u(t) (v(t) x w(t))]. = | r(t) | r(t) · r'(t). dt [Hint: r(t) 2 = r(t) · r(t)] 56. If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on a sphere with center the origin. 57. If u(t) = r(t) [r'(t) × r"(t)], show that u'(t) = r(t) [r'(t) × r"(t)] 58. Show that the tangent vector to a curve defined by a vector function r(t) points in the direction of increasing t. [Hint: Refer to Figure 1 and consider the cases h> 0 and h <0 separately.]
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
49,55 pl
![t²) and
angle of
HOFU
DV
= i + j.
+j+ k.
use
48. If u and v are the vector functions in Exercise 47, use For-
mula 5 of Theorem 3 to find
d
dt
49. Find f'(2), where f(t) = u(t) v(t), u(2)= (1, 2, -1),
u'(2)=(3, 0, 4), and v(t) = (t, t2, 1³).
[u(t) × v(t)]
50. If r(t) = u(t) x v(t), where u and v are the vector functions
in Exercise 49, find r' (2).
51. If r(t)
= a cos wt + b sin wt, where a and b are constant
vectors, show that r(t) × r'(t) = wa X b.
52. If r is the vector function in Exercise 51, show that
r"(t) + w²r(t) = 0.
53. Show that if r is a vector function such that r" exists, then
T
[r(t) × r'(t)] = r(t) × r"(t)
d
dt
d
54. Find an expression for [u(t) · (v(t) × w(t))].
dt
BOMAT
55. If r(t) = 0, show that
d
-
| r(t) | =
1
| r(t) |
- r(t) · r'(t).
dt
[Hint: | r(t) |² = r(t) · r(t)]
56. If a curve has the property that the position vector r(t) is
always perpendicular to the tangent vector r'(t), show that
the curve lies on a sphere with center the origin.
57. If u(t) = r(t) [r'(t) × r"(t)], show that
u'(t) = r(t) [r'(t) × r"(t)]
58. Show that the tangent vector to a curve defined by a vector
function r(t) points in the direction of increasing t.
[Hint: Refer to Figure 1 and consider the cases h> 0 and
h <0 separately.]
BRUDE](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F962b9010-dc85-4404-92bb-863cbb6da4e7%2Fad892f09-160a-43f2-a634-a2e3b9b743f7%2F073t9nr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:t²) and
angle of
HOFU
DV
= i + j.
+j+ k.
use
48. If u and v are the vector functions in Exercise 47, use For-
mula 5 of Theorem 3 to find
d
dt
49. Find f'(2), where f(t) = u(t) v(t), u(2)= (1, 2, -1),
u'(2)=(3, 0, 4), and v(t) = (t, t2, 1³).
[u(t) × v(t)]
50. If r(t) = u(t) x v(t), where u and v are the vector functions
in Exercise 49, find r' (2).
51. If r(t)
= a cos wt + b sin wt, where a and b are constant
vectors, show that r(t) × r'(t) = wa X b.
52. If r is the vector function in Exercise 51, show that
r"(t) + w²r(t) = 0.
53. Show that if r is a vector function such that r" exists, then
T
[r(t) × r'(t)] = r(t) × r"(t)
d
dt
d
54. Find an expression for [u(t) · (v(t) × w(t))].
dt
BOMAT
55. If r(t) = 0, show that
d
-
| r(t) | =
1
| r(t) |
- r(t) · r'(t).
dt
[Hint: | r(t) |² = r(t) · r(t)]
56. If a curve has the property that the position vector r(t) is
always perpendicular to the tangent vector r'(t), show that
the curve lies on a sphere with center the origin.
57. If u(t) = r(t) [r'(t) × r"(t)], show that
u'(t) = r(t) [r'(t) × r"(t)]
58. Show that the tangent vector to a curve defined by a vector
function r(t) points in the direction of increasing t.
[Hint: Refer to Figure 1 and consider the cases h> 0 and
h <0 separately.]
BRUDE
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