For the following parameterized curve, find the unit tangent vector T(t) at the given value of t. r(t) = < 8 t,10,3 sine 2 t >, for 0

The tangent vector for is:

Step-by-step explanation:

The function to be used is

The unit tangent vector is the gradient of divided by its norm, that is:

Where is the gradient operator, whose definition is:

The components of the gradient function of are, respectively:

, and

For :

, and

The norm of the gradient function of is:

For :

The tangent vector for is:

it's undefined because anything multiplied by 0 is 0. that said, 7 days is 168 hours

ending amount = beginning amount / (2^n)

where "n" = number of half-lives (or elapsed time / half-life)

ending amount = 50 mg / (2 ^10 / 5.3)

ending amount = 50 mg / (2 ^ 1.8867924528)

ending amount = 50 m / 3.6981210597

ending amount = 13.5203794557 mg

or about 13.5 mg remain

source: