A particle moves along a linear path (left/right) and its position, relative to its starting location is given by the function s(t)=13t^3−2t^2+3t , t≥0.

a) Determine the particle's velocity and acceleration at any time t.
b) When does the particle change direction? You must prove there is a change of direction.

it can't change direction without stopping for an instant.

velocity is ds/dt, so when does s'=0?

To determine the particle's velocity, we need to find the derivative of the position function, s(t). The velocity, v(t), is the derivative of s(t).

a)
The derivative of s(t) is found by applying the power rule and sum rule of derivatives:

s'(t) = d/dt (13t^3 - 2t^2 + 3t)
= 39t^2 - 4t + 3

Therefore, the particle's velocity, v(t), is given by v(t) = 39t^2 - 4t + 3.

To determine the particle's acceleration, we need to find the derivative of the velocity function, v(t). The acceleration, a(t), is the derivative of v(t).

The derivative of v(t) is found by applying the power rule and sum rule of derivatives:

v'(t) = d/dt (39t^2 - 4t + 3)
= 78t - 4

Therefore, the particle's acceleration, a(t), is given by a(t) = 78t - 4.

b)
To find when the particle changes direction, we need to determine when the velocity, v(t), changes sign. A change in sign of the velocity indicates a change in direction.

Setting v(t) = 0 and solving for t:

39t^2 - 4t + 3 = 0

This is a quadratic equation, which can be solved using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 39, b = -4, and c = 3. Plugging in these values into the quadratic formula, we get:

t = (-(-4) ± √((-4)^2 - 4(39)(3))) / (2(39))
t = (4 ± √(16 - 468)) / 78
t = (4 ± √(-452)) / 78

We have a negative value under the square root, indicating that there are no real solutions to this equation. This means the particle never changes direction.

Therefore, the particle does not change direction.

Note: The proof for the change in direction is based on the fact that the particle is moving along a linear path (left/right) and its position function is a polynomial function. If the particle were to follow a different path, such as a curve, the determination of a change in direction would require considering the higher-order derivatives and curve characteristics.