Suppose that a curve C in the x y - plane is smoothly parametrized by r ( t ) = x ( t ) i + y ( t ) j ( a ≤ t ≤ b ) In each part, part refer to the notation used in the derivation of Formula (9). (a) Let m and M denote the respective minimum and maximum values of r' ( t ) on [a,b] . Prove that 0 ≤ m ( max Δ t k ) ≤ max Δ s k ≤ M ( max Δ t k ) (b) Use part (a) to prove that max Δ s k → 0. if and only if max Δ t k → 0.
Suppose that a curve C in the x y - plane is smoothly parametrized by r ( t ) = x ( t ) i + y ( t ) j ( a ≤ t ≤ b ) In each part, part refer to the notation used in the derivation of Formula (9). (a) Let m and M denote the respective minimum and maximum values of r' ( t ) on [a,b] . Prove that 0 ≤ m ( max Δ t k ) ≤ max Δ s k ≤ M ( max Δ t k ) (b) Use part (a) to prove that max Δ s k → 0. if and only if max Δ t k → 0.
Suppose that a curve C in the
x
y
-
plane
is smoothly parametrized by
r
(
t
)
=
x
(
t
)
i
+
y
(
t
)
j
(
a
≤
t
≤
b
)
In each part, part refer to the notation used in the derivation of Formula (9).
(a) Let m and Mdenote the respective minimum and maximum values of
r'
(
t
)
on [a,b]
. Prove that 0
≤
m
(
max
Δ
t
k
)
≤
max
Δ
s
k
≤
M
(
max
Δ
t
k
)
(b) Use part (a) to prove that max
Δ
s
k
→
0.
if and only if max
Δ
t
k
→
0.
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