Find the domain of vector function r (t)= sqrt(4 - t ^ 2) i + sqrt(t) j - 2/(sqrt(1+t)) k. State your answer in interval notation

Answer 1

To find the domain of a vector function, we need to look for values of t that would make any of the components undefined.

For the first component, we have a square root of (4 - t^2), which is undefined whenever the expression inside the root is negative. So, we need to solve the inequality:

4 - t^2 ≥ 0

This can be factorized as:

(2 + t)(2 - t) ≥ 0

The solutions are: t ≤ -2 or t ≥ 2.

For the second component, we have a square root of t, which is defined for any non-negative value of t.

For the third component, we have a fraction with a square root in the denominator. This is undefined whenever the denominator becomes zero. So, we need to solve the equation:

1 + t = 0

The solution is t = -1.

Putting everything together, the domain of r(t) is the intersection of the domains of its components. Therefore, the domain is given by:

t ≤ -2 or -1 < t < 1 or t > 1

In interval notation, this can be written as:

(-∞, -2]∪(-1,1)∪[1, ∞).