Answer:
a. True
Step-by-step explanation:
To find the arc length of a curve, we use the formula for arc length:
L = ∫[a,b] ||r'(t)|| dt
Given that r(t) defines a smooth curve C on [0,1] such that r(0) = ⟨2,1,4⟩ and r(1) = ⟨0,3,5⟩, we can find the arc length over the interval [0,1] by integrating ||r'(t)|| over the interval [0,1].
Let's calculate the values:
r'(t) = dr/dt = <-2, 2, 1>
||r'(t)|| = sqrt((-2)^2 + 2^2 + 1^2) = sqrt(4 + 4 + 1) = sqrt(9) = 3
Therefore, the arc length of C over the interval [0,1] is:
L = ∫[0,1] 3 dt = 3(t)|[0,1] = 3(1) - 3(0) = 3
The arc length of C over the interval [0,1] is exactly 3. Therefore, the statement "the arc length of C over the interval [0,1] cannot be less than 3" is true.
Therefore, the answer is:
a. True
