Final answer:
The domain of the vector function is −5<t≤6
Step-by-step explanation:
To find the domain of the vector function
r(t)=⟨ 36−t 2,e −4t ,ln(t+5)⟩, we need to consider the restrictions imposed by the square root and the natural logarithm functions.
Square Root Function:
For the square root function to be defined, the expression inside the square root must be non-negative:
36−t 2 ≥0
Solving for
36t 2 ≤36−6≤t≤6
Exponential Function:
The exponential function −4t
is defined for all real values of
Natural Logarithm Function:
The natural logarithm function
ln(t+5) is defined only for positive values inside the logarithm. So, t+5>0: t>−5
Considering all these conditions together, the domain of the vector function r(t) is:−5<t≤6
Therefore, the domain is t belonging to the open interval (−5,6].