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Find the curvature at the point (4, 4, −1). x = 4t, y = 4t³/², z = −t²

User Edrina
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Final answer:

The curvature at the point (4, 4, -1) can be found by differentiating the given position vector and applying the formula for curvature.

Step-by-step explanation:

The curvature of a curve in three-dimensional space can be found using the formula:

k = ||T'(t)|| / ||r'(t)||

where k is the curvature, T'(t) is the derivative of the unit tangent vector, and r'(t) is the derivative of the position vector.

By differentiating the given position vector, we can find the derivatives:

x'(t) = 4, y'(t) = 6t^(1/2), and z'(t) = -2t

At t = 1, we have x'(1) = 4, y'(1) = 6, and z'(1) = -2.

Thus, the curvature at the point (4, 4, -1) is:

k = ||T'(1)|| / ||r'(1)||.

User Simon Urbanek
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