If r(t) = 4e3t, 2e−3t, 4te3t , find t(0), r''(0), and r'(t) · r''(t).

we are given

Calculation of r(0):

we can plug t=0

Calculation of r'(0):

Firstly, we will find derivative

we can simplify it

now, we can plug t=0

Calculation of r''(0):

we can find second derivative

now, we can plug t=0

we get

Calculation of r'(0) · r''(0):

............Answer