Determine the interval on which the vector-valued function $r ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }$ is continuous.
A) $( 9 , \infty )$
B) $( 5 , \infty )$
C) $[ 5 , \infty )$
D) $[ 9 , \infty )$
E) $( - 5 , \infty )$

Question

Quizplus · Accepted Answer

The function is continuous where all its components are continuous. The exponential functions $9e^{-t}$ and $e^{-t}$ are continuous for all real numbers, but $\ln(t-5)$ is only defined and continuous for $t > 5$. Therefore, the entire vector-valued function is continuous for $t > 5$, which corresponds to the interval $(5, \infty)$.

Determine the Interval on Which the Vector-Valued Function r(t)=9e−ti+e−tj+ln⁡(t−5)kr ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }r(t)=9e−ti+e−tj+ln(t−5)k

Determine the Interval on Which the Vector-Valued Function $r ( t ) = 9 e ^ { - t } \mathbf { i } + e ^ { - t } \mathbf { j } + \ln ( t - 5 ) \mathbf { k }$