Let r(t) = (t² + 2)i + (e ^t + 4)j + e ^t k. Find N(t) for t = 0. Approximate your calculations to three decimal places. A) 0 B) i C) 0.577j + 0.577k D) i + 0.577j + 0.577k E) i - 1.732j - 1.732k

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Quizplus · Accepted Answer

To find the unit normal vector N(t), we first need to find the derivative of r(t), r'(t), and then the unit tangent vector T(t) = r'(t)/|r'(t)|. The normal vector N(t) is the derivative of T(t) with respect to t, normalized. At t = 0, r'(t) = 2ti + e^t j + e^t k, so r'(0) = 0i + 1j + 1k. The magnitude |r'(0)| = sqrt(0^2 + 1^2 + 1^2) = sqrt(2). Thus, T(0) = (1/sqrt(2))j + (1/sqrt(2))k. Differentiating T(t) and normalizing gives N(0) = 0.577j + 0.577k.

Let R(t) = (T2 + 2)i + (E t

Let R(t) = (T² + 2)i + (E ^t