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
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, ∞).