Find the domain of the vector function r(t)= sqrt(4 - t ^ 2) i + sqrt(t) j - 2 / (sqrt(1 + t)) k.

Question

answer in interval notation

Answer 1

The domain of r(t) is dependent on the range of each of the three component functions:

1. The domain of f(t) = sqrt(4 - t^2) is [-2, 2] because the radicand must be non-negative.
2. The domain of g(t) = sqrt(t) is [0, ∞) because the radicand must be non-negative.
3. The domain of h(t) = -2 / sqrt(1 + t) is (-1, ∞) because the denominator must be positive and non-zero.

Thus, the domain of r(t) is the intersection of the domains of f(t), g(t), and h(t), which is [0, 2].

In interval notation, the domain of r(t) is [0, 2].