Calculus 2nd Edition Rogawski

750 C H A P T E R 14 CALCULUS OF VECTOR-VALUED FUNCTIONS
As a check, let’s verify that r 1 (s) has unit speed:
∥〈
∥∥∥
∥r 1 ′ (s)∥ = − 4 4s
sin
5 5 , 4 4s
cos
5 5 , 3 〉∥ √ ∥∥∥ 16 4s
= sin2
5 25 5 + 16 4s
cos2
25 5 + 9
25 = 1
14.3 SUMMARY
• The length s of a path r(t) = ⟨x(t), y(t), z(t)⟩ for a ≤ t ≤ b is
∫ b
∫ b √
s = ∥r ′ (t)∥ dt = x ′ (t) 2 + y ′ (t) 2 + z ′ (t) 2 dt
• Arc length function: s(t) =
a
∫ t
a
a
∥r ′ (u)∥ du
• Speed is the derivative of distance traveled with respect to time:
v(t) = ds
dt = ∥r′ (t)∥
• The velocity vector v(t) = r ′ (t) points in the direction of motion [provided that r ′ (t) ̸=
0] and its magnitude v(t) = ∥r ′ (t)∥ is the object’s speed.
• We say that r(s) is an arc length parametrization if ∥r ′ (s)∥ = 1 for all s. In this case,
the length of the path for a ≤ s ≤ b is b − a.
• If r(t) is any parametrization such that r ′ (t) ̸= 0 for all t, then
r 1 (s) = r(g(s))
is an arc length parametrization, where t = g(s) is the inverse of the arc length function.
14.3 EXERCISES
Preliminary Questions
1. At a given instant, a car on a roller coaster has velocity vector
r ′ = ⟨25, −35, 10⟩ (in miles per hour). What would the velocity vector
be if the speed were doubled? What would it be if the car’s direction
were reversed but its speed remained unchanged?
2. Two cars travel in the same direction along the same roller coaster
(at different times). Which of the following statements about their velocity
vectors at a given point P on the roller coaster is/are true?
(a) The velocity vectors are identical.
(b) The velocity vectors point in the same direction but may have different
lengths.
Exercises
In Exercises 1–6, compute the length ofthe curve over the given interval.
1. r(t) = ⟨3t,4t − 3, 6t + 1⟩, 0 ≤ t ≤ 3
2. r(t) = 2ti − 3tk, 11 ≤ t ≤ 15
3. r(t) = 〈 2t,ln t,t 2〉 , 1 ≤ t ≤ 4
(c) The velocity vectors may point in opposite directions.
3. A mosquito flies along a parabola with speed v(t) = t 2 . Let L(t)
be the total distance traveled at time t.
(a) How fast is L(t) changing at t = 2?
(b) Is L(t) equal to the mosquito’s distance from the origin?
4. What is the length of the path traced by r(t) for 4 ≤ t ≤ 10 if r(t)
is an arc length parametrization?
4. r(t) = 〈 2t 2 + 1, 2t 2 − 1,t 3〉 , 0 ≤ t ≤ 2
5. r(t) = ⟨t cos t,t sin t,3t⟩, 0 ≤ t ≤ 2π
6. r(t) = ti + 2tj + (t 2 − 3)k, 0 ≤ t ≤ 2. Use the formula:
∫ √
t 2 + a 2 dt = 1 √
2 t t 2 + a 2 + 1 2 a2 ln ( t +
√t 2 + a 2)