Suppose that a curve C in the xy-plane is smoothly parametrizedbyr(t) = x(t)i + y(t)j (a

Chapter 15, Problem 57

(choose chapter or problem)

Suppose that a curve \(C\) in the \(xy-plane\) is smoothly parametrized by

\(r(t)=x(t) i+y(t) j\)            \((a \leq t \leq b)\)

In each 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 \(\left\|r^{\prime}(t)\right\|\) on \([a, b]\). Prove that \(0 \leq m\left(\max t_{k}\right) \leq \max s_{k} \leq M\left(\max t_{k}\right)\)

(b) Use part (a) to prove that \(\max s_{k} \rightarrow 0\) if and only if \(\max t_{k} \rightarrow 0\).

Equation Transcription:

Text Transcription:

C

Xy-plane

r(t) = x(t)i + y(t)j

(a leq t leq b)

m and M

|| r^prime (t) ||

[a,b]

0 leq m (max t_k) leq max s_k leq M (max t_k)

max s_k right arrow 0

max t_k right arrow 0