Question

What is the instantaneous velocity of an object moving in accordance to `f(t)= (sin^2t,t-cos3t)` at `t=pi/4`?

Answer 1

`(1,1+3/sqrt(2))`

Explanation:

If you have the law of motion, the instant velocity is given by its derivative.
You derive a vectorial function by deriving every output; so

# f'(t)=(d/dt sin^2(t), d/dt t-cos(3t)) #

and those derivatives are easily computed:

# f'(t)=(2sin(t)cos(t), 1+3sin(3t)) #

Now you only need to evaluate `f'(\pi/4)`, which gives you

# f'(\pi/4)=(2sin(\pi/4)cos(\pi/4), 1+3sin((3\pi)/4)) #

# =(2 1/sqrt(2) 1/sqrt(2), 1+3/sqrt(2)) #

# =(1,1+3/sqrt(2)) #