Let r(t) = (t² + 2)i + e ^t j + ( + e ^t )k. Find B(t) for t = 0. Approximate your calculations to three decimal places. A) 2i + 0.707j + 0.707k B) 0.707j - 0.707k C) 2i + 0.577j + 0.577k D) 0.577j - 0.577k E) 1.334i - 1.334j - 1.334k

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To find B(t), we need to take the derivative of r(t) and then normalize it. r'(t) = (²)i + (e)j + (e)k To normalize r'(t), we need to find its magnitude: |r'(t)| = sqrt((²)^2 + (e)^2 + (e)^2) |r'(t)| = sqrt((²)^2 + 2*(e)^2) |r'(t)| = sqrt((2)^2 + (2e)^2) |r'(t)| = 2sqrt(()^2 + (e)^2) So, B(t) = r'(t) / |r'(t)| B(t) = [(²)i + (e)j + (e)k] / [2sqrt(()^2 + (e)^2)] After simplification, we get: B(t) = (² / 2)i + (0.577j - 0.577k) Therefore, the answer is D) 0.577j - 0.577k

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

Let R(t) = (T² + 2)i + E ^t J