"Find the domain of the vector function. r(t)=t−2t+2i+sin⁡tj+ln⁡(9−t2)k" was answered by our community

College
Algebra
Linear algebra
Vectors and spaces

Bergen

Answered question

2021-09-28

Find the domain of the vector function.

r (t) = t - \frac{2}{t} + 2 i + \sin t j + \ln (9 - t^{2}) k

Answer & Explanation

brawnyN

Skilled2021-09-29Added 91 answers

$r (t) = \frac{t - 2}{t + 2} i + \sin t j + \ln (9 - t^{2}) k$
The domain has to exclude anything that will make the function undefined.
For $\frac{t - 2}{t + 2}$ , the denominator can't be 0 so we exclude $t = - 2$
For $\sin t$ it is defined for all real numbers so there is nothing to exclude for this part.
For $\ln (9 - t^{2})$ , we can only take the logarithm of positive numbers so $9 - t^{2}$ must be $> 0$
$0 = 9 - t^{2}$
$= (3 - t) (3 + t)$
$T = - 3, 3$
Since $y = 9 - t^{2}$ is a downward opening parabola we know that it is only $> 0$ in between but not including the zeros, $- 3 < t < 3$ or in interval notation $(- 3, 3)$
Now combine everything together. We just have to exclude -2 from the interval we found in the last part.
$- 3 < t < - 2$ or $- 2 < t < 3$
of in interval notation
$(- 3, 2) \cup (- 2, 3)$
Result: $(- 3, 2) \cup (- 2, 3)$

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