Answer to Question #228939 in Calculus for fouzi

Question

 Find the domain of r(t) and the value of r(t0).4. r(t) = 2e−t ,sin−1 t, ln(1 − t); t0 = 0

Answer 1

Let us find the domain of $r(t)$, where

$r(t) = (2e^{−t}, \sin^{−1} t, \ln(1 − t)).$

Taking into account that the domain of the function $2e^{−t}$ is the set $\R$ of all real numbers, the domain of the function $\sin^{−1} t$ is the set $[-1,1]$ and the domain of the function $\ln(1 − t)$ consists of all $t\in\R$ such that $1-t>0,$ that is $t<1,$ we conclude that the domain of the vector-function $r(t)$ is the set $[-1,1).$

Let us find the value of $r(t_0)$ for $t_0=0:$

$r(t_0)=r(0)=(2e^{0}, \sin^{−1} 0, \ln(1 − 0))=(2,0,0).$