Find a piecewise smooth parametrization of the path C. r(t) = ti + tj 0 ≤ t ≤ 1 _____????______ 1 ≤ t ≤ 2?

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1 Answer
Aug 5, 2018

r(t) = { (that(i)+that(j) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ " for " 0 <= t <= 1), ((2-t)hat(i)+(2-t)^2hat(j) " for " 1 <= t <= 2) :}

Explanation:

Note that the curved portion of the curve C is a parabola with horizontal axis.

If it were parameterised from (0, 0) to (1, 1) for t in [0, 1] then we could write:

r(t) = that(i)+t^2hat(j)

As it is, we want to parameterise it in the opposite direction for t in the range [1, 2].

Hence we want:

r(t) = (2-t)hat(i)+(2-t)^2hat(j)

So the piecewise parameterisation can be written:

r(t) = { (that(i)+that(j) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ " for " 0 <= t <= 1), ((2-t)hat(i)+(2-t)^2hat(j) " for " 1 <= t <= 2) :}