Question

What is the instantaneous velocity of an object moving in accordance to `f(t)= (sin(t+pi),cos(3t-pi/4))` at `t=(pi)/3`?

Answer 1

The instantaneous velocity of `f(t) => for t=pi/3` is:
`f(t) = (-sqrt(3)/2, -sqrt(2)/2)`

Explanation:

Given: `f(t)=(sin(t+pi),cos(3t-pi/4))`

Required: The instantaneous velocity at `t=(pi)/3`

Solution Strategy:
The instantaneous velocity at any time is `f(t)` itself so you need
Evaluate the `f(t)=f(t)|_(t=pi/3)=f(pi/3)`

`color(green)("Evaluate at at " t=(pi)/3)`
`f(t)=(sin(t+pi),cos(3t-pi/4))=(-sin(t),cos(3t-pi/4))`
`f(t)=(-sin(pi/3),cos(3pi/3-pi/4))`
`f'(t) = (-sqrt(3)/2, -sqrt(2)/2)`

`color(red)(Remark)`: This is deceptively easy problem. Most students are easily tricked to think that `f(t)` is not the instantaneous velocity at any time. But `f(t)` is indeed the instantaneous velocity at anytime. It is asking what it is at a given point.