Final answer:
The domain of the vector function r(t) is all real numbers except t = 0 and t = 1. This results from the natural logarithm requiring positive arguments and the third component being undefined at t = 1. The correct answer is C) t ≠ 1, considering the natural logarithm constraint.
Step-by-step explanation:
The student is asked to find the domain of the vector function r(t) = cos(t), ln(t²), eᵗ/(t-1). The domain of a function is the set of all possible input values (usually t) for which the function is defined.
The first component, cos(t), is defined for all real numbers, so it does not impose any restrictions on the domain. The second component, ln(t²), requires that t² be greater than zero because the natural logarithm is undefined for zero and negative numbers. Since t² is always non-negative, t must be different from zero (t > 0 or t < 0). The third component, eᵗ/(t-1), is undefined when the denominator is zero, which occurs at t = 1. Therefore, t cannot be 1.
Combining these conditions, the domain of the function is all real numbers with the exception of t = 0 and t = 1. The correct answer is t > 0 and t ≠ 1, which corresponds to option C) t ≠ 1 if we consider that for the natural logarithm t also needs to be positive.