Question

Find the domain of the following vector-valued functions. $$\mathbf{r}(t)=\cos 2 t \mathbf{i}+e^{\sqrt{t}} \mathbf{j}+\frac{12}{t} \mathbf{k}$$

Short answer

Answer: The domain of the given vector-valued function is \(t > 0\).

Step-by-step solution

Analyze the first component

The first component is given by \(\cos 2t\). The cosine function is defined for all real numbers, so there are no domain restrictions for this component.

Analyze the second component

The second component is given by \(e^{\sqrt{t}}\). The exponential function is defined for all real numbers, but since we have \(\sqrt{t}\), we need to ensure \(t \geq 0\). So the domain restriction for this component is \(t \geq 0\).

Analyze the third component

The third component is given by \(\frac{12}{t}\). This function is undefined when \(t = 0\), so the domain restriction for this component is \(t \neq 0\).

Combine domain restrictions

To find the overall domain of the vector-valued function, we need to combine the domain restrictions found from each component. From Step 2, we have \(t \geq 0\). From Step 3, we have \(t \neq 0\). Combining these restrictions, we get the overall domain as \(t > 0\).

Final Answer

The domain of the given vector-valued function is \(t > 0\).