A particle moves on the x axis so that it's position at any time t is greater than or equal to 0 is given by x(t)=2te^-t.
A) find velocity for any t is greater or equal to 0
velocity = dx/dt
just use the product rule
To find the velocity of the particle at any time t, we need to take the derivative of the position function x(t) with respect to time t.
The position function is x(t) = 2te^(-t).
To find the derivative, we can use the product and chain rule.
Let's start by using the product rule. The product rule states that for two functions u(t) and v(t), the derivative of their product is given by:
(d/dt)(u(t)v(t)) = u'(t)v(t) + u(t)v'(t).
In this case, u(t) = 2t and v(t) = e^(-t).
Taking the derivatives of these functions, we have u'(t) = 2 and v'(t) = -e^(-t).
Now let's apply the product rule:
(d/dt)(2te^(-t)) = (2)(e^(-t)) + (2t)(-e^(-t)).
Simplifying this expression, we have:
= 2e^(-t) - 2te^(-t).
Therefore, the velocity of the particle at any time t is given by:
v(t) = 2e^(-t) - 2te^(-t).
This is the equation for the velocity function.