Question

Find the domain of the vector function. (Enter your answer in interval notation.) r(t) = t − 2 t + 2 i + sin(t)j + ln(36 − t2)k

Answer 1

Answer:

The domain of the vector function is :  ( -6, -2) ∪ ( -2, 6)

Step-by-step explanation:

The given vector function  can be correctly expressed as:

$r(t) = \dfrac{t-2}{t+2}i + sin \ tj + In(36-t^2) k$

The domain of the given function can be determined by finding the domain of each respective component.

To start with $\dfrac{t-2}{t+2}$

Not defined, t = -2

Thus, the domain of the first component of the vector = $( - \infty , -2) \cup ( 2, \infty )$

The 2nd component = sin t

No restriction on t

The 3rd component of the function is ㏑(36 - t²)

Let recall that natural is defined for +ve numbers,

i.e.

36 - t² > 0

(6 + t) (6 - t) > 0

Thus, it satisfies the inequality  -6 < t < 6

The third domain = (-6,6)

Overall;

The domain of the vector function is :  ( -6, -2) ∪ ( -2, 6)

Answer 2

Domain of a function is the set all possible inputs for that function.The domain of the given vector function is,

$(-6, -2) \cap (-2,6)$

What is domain?

Domain of a function is the set all possible inputs for that function.

Given information-

The vector function given in the problem is,

$r(t)=\dfrac{t-1}{t+2} \hat i++sin(t)\hat j+ln(49-t^2)\hat k$

To find the domain of the above vector function, we need to find the domain of each function of vector quantity.

Lets find the domain of $\hat i$ first which is given by,

$\dfrac{t-2}{t+2}$

This function is defined at all values except, t equals to -2 as the function is not defined at,

$t=-2$

Hence the domain of component $\hat i$ is,

$(-\infty, -2) \cap (\infty,2)$

Lets find the domain of $\hat j$ first which is given by,

$\sin(t)$

Two trigonometry function sine and cosine defined for all real number.

This function is defined at all real numbers. Hence the domain of component $\hat j$ is

Lets find the domain of $\hat k$ first which is given by,

$\ln(36-t^2)$

Set the above argument greater than zero to find where the expression is defined.Thus,

$(36-t^2)>0\\(6+t)(6-t)>0$

Therefore,

The domain of it should be $(-6,6)$

Hence the domain of the given vector function is,

$(-6, -2) \cap (-2,6)$

Learn more about the domain here;

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