Solved: "Determine the interval(s) on which the vector-valued function..."

NompsypeFeplk

Answered question

2021-12-05

Determine the interval(s) on which the vector-valued function continuous.

r (t) = t i + \sqrt{t + 1} j + (t^{2} + 1) k

Answer & Explanation

Howell

Beginner2021-12-06Added 11 answers

Step1
Given the vector-valued function:
$r (t) = t i + \sqrt{(t + 1)} j + (^{{2}} + 1) k$
Now,
A vector valued function in the form of $r (t) = f (t) i + g (t) j + h (t) k$
is continuous at the intersection of points of continuity of all the components
$f (t), g (t) and h (t)$ .
Step 2
So by comparing the given function with the standard form:
$f (t) = t$
$g (t) = \sqrt{t + 1}$
$h (t) = t^{2} + 1$
Now,
$f (t) = t$
The function is continuous everywhere.
That is definet for $(- \infty, \infty) .$
$g (t) = \sqrt{t + 1}$
The function is defined only when $t \geq - 1$ .
$h (t) = t^{2} + 1$
The function is continuous everywhere.
That is defined for $(- \infty, \infty) .$
Therefore,
The required interval of continuity is the intersection of $(- \infty, \infty)$ and $[- 1, \infty)$ .
That is.
The required interval is $[- 1, \infty) .$
Thus,
The vector r(t) is continuous in interval $[- 1, \infty) .$

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