a. Show that n(t) = -g(t)i + (t)j and -n(t) = g(t)i - (t)j are both normal

a. Show that n(t) = -g′(t)i + ƒ′(t)j and -n(t) = g′(t)i - ƒ′(t)j are both normal to the curve r(t) = ƒ(t)i + g(t)j at the point (ƒ(t), g(t)).

To obtain N for a particular plane curve, we can choose the one of n or -n from part (a) that points toward the concave side of the curve, and make it into a unit vector. (See Figure 13.19.) Apply this method to find N for the following curves.

In Figure 13.19

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b. r(t) = ti + e²tj c. r(t) r(t) = √4 − t²i + tj, −2 ≤ t ≤ 2