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) = ((t-1)/(t+1)) i + sin(t)j + ln(64 - t^2)k is (-∞, -1) ∪ (-1, ∞) × (-∞, ∞) × (0, ∞).

Step-by-step explanation:

The domain of a vector function refers to the set of values for which the function is defined. In this case, we need to consider the restrictions on the variables in each component of the vector function.

For the first component, the expression (t-1)/(t+1) is defined for all real numbers except for t = -1, as it would result in division by zero. Therefore, the domain for the first component is (-∞, -1) ∪ (-1, ∞).

For the second component, the sine function is defined for all real numbers. Therefore, the domain for the second component is (-∞, ∞).

For the third component, the natural logarithm function is defined only for positive real numbers. Therefore, the domain for the third component is (0, ∞).

Combining these domains for each component, we obtain the overall domain of the vector function as (-∞, -1) ∪ (-1, ∞) × (-∞, ∞) × (0, ∞), which can be expressed in interval notation as (-∞, -1) ∪ (-1, ∞) × (-∞, ∞) × (0, ∞).a