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Lengths of related curves Suppose a curve is given by r(t)
Chapter 11, Problem 49AE(choose chapter or problem)
Lengths of related curves Suppose a curve is given by \(\mathbf{r}(t)=\langle f(t), g(t)\rangle\), where f' and g' are continuous for \(a \leq t \leq b\). Assume the curve is traversed once for \(a \leq t \leq b\) and the length of the curve between (f(a), g(a)) and (f(b), g(b)) is L. Prove that for any nonzero constant c the length of the curve defined by \(\mathbf{r}(t)=\langle c f(t), c g(t)\rangle \text { for } a \leq t \leq b \text { is }|c| L\).
Questions & Answers
QUESTION:
Lengths of related curves Suppose a curve is given by \(\mathbf{r}(t)=\langle f(t), g(t)\rangle\), where f' and g' are continuous for \(a \leq t \leq b\). Assume the curve is traversed once for \(a \leq t \leq b\) and the length of the curve between (f(a), g(a)) and (f(b), g(b)) is L. Prove that for any nonzero constant c the length of the curve defined by \(\mathbf{r}(t)=\langle c f(t), c g(t)\rangle \text { for } a \leq t \leq b \text { is }|c| L\).
ANSWER:Solution 49AE
Step 1 of 3:
In this problem we need to prove that for any non zero constant ‘C’ the length of the curve defined by r(t) = , for a , is |C|L.
Given ; A curve r(t) = ,where andare continuous, for a .Assume the curve is traversed once , for a , and the length of the curve between (f(a), g(a)) and (f(b) , g(b)) is L.
But , we know that If f has a continuous derivatives on the interval [a , b]. Then the arc length of a curve r(t) = is ;
L =dt = dt ………….(1)