Question

Find the domain of the vector function. (Enter your answer in interval notation.)

r(t) = ((t-1)/(t+1)) i + sin tj + ln(64 ? t²)k

Answer 1

The domain of the vector function r(t) is all real numbers t except where the denominators are zero and the argument of the logarithm is positive, resulting in the interval (-8, -1) ∪ (-1, 8).

Step-by-step explanation:

The student has asked to find the domain of the vector function r(t) = ((t-1)/(t+1)) i + sin(t) j + ln(64 - t²) k. The domain of a vector function is the set of all possible values of t that allows the function to produce real number outputs for each component of the vector.

To find the domain of r(t), we need to examine each component of the vector separately:

• The first component (t-1)/(t+1) is defined for all real numbers t except where the denominator is zero, which is t = -1.
• The second component sin(t) is defined for all real numbers t.
• The third component ln(64 - t²) requires the argument of the logarithm to be positive. Therefore, 64 - t² > 0, which translates to -8 < t < 8.

Combining these conditions, the domain of r(t) is the intersection of all the individual domains, which is (-8, -1) ∪ (-1, 8) in interval notation.