Acceration is the derivative of velocity. This means that to go from acceleration to velocity we must take an integral. The integral of  is . Knowing this we can solve the problem.

The constant at the end is the inital velocity so the final answer is 

Velocity is the derivative of position. To find the velocity function all we have to do is take the fisrst derivative.

The derivative of  is .

So the velocity function is

To find the velocity at a certain time, we must plug that time into the function.

When the question asks when the particle begins to move in an opposite direction, it's referring to a change in sign of the particle's velocity.

Velocity can be found as the time derivative of position.

For the position function

The velocity is found to be

The initial velocity is positive, since

So we'll want to find where it first becomes negative; should this happen, it'll be right after the function equals zero. It may help to factor the equation:

The velocity reaches zero at two times; , and is negative between them. It can be said then that the particle first changes directions at time

Begin by finding a function for velocity. Velocity is given as the integral of acceleration with respect to time.

For the acceleration function

The integral can be performed knowing that 

The velocity function is then

The constant of integration is unknown as of this moment, so disregard it for the time being.

The distance traveled over an interval of time is given by the integral of the velocity function over said interval of time. For the interval of time , the distance traveled can be calculated as follows:

Now we're told that the distance traveled over this interval is , so we can solve for the constant of integration:

Putting this back into our velocity equation, we can find the initial velocity:

The velocity function can be found by integrating the acceleration function with respect to time.

For the acceleration function

Note that 

The velocity function is

To find this constant of integration, use what we're told in the problem: namely that the particle comes to rest (a zero velocity) at time :

So knowing the full velocity function

All that remains is to find the time when the velocity is :

There are two roots, . However, since a negative time is not possible, we elect the positive value:

Velocity can be found by taking the derivative of the position function. For the position function

Note that  and that the derivative a constant is zero.

The velocity function can thus be found to be

Currently the values of  and  are unknown, but we're given some information to find them. First, the initial velocity:

Next, we're told the velocity at time :

Now we can find the velocity at time :

If the cars distance traveled is given by the equation , we can see that the derivative  is another way of writing the equation for .  Because of this relation, we can see that at the start of the car's travel (), the car was traveling at   .  If we plug in a three for time (three seconds after the start) we can see that we come to our answer, .

Acceleration is the derivative of velocity. This makes velocity the integral of acceleration. The integral of  is . This makes the velocty function

We must include the  because the integral is indefinite and there are no given values.

Velocity is the first derivative of position. The derivative of  is . This means that the velocity function is given by 

We must plug in  to get the velocity at that time.

Velocity of a particle can be found by taking the derivative of the position function with respect to time. Recall that a derivative gives the rate of change of some parameter, relative to the change of some other variable. When we take the derivative of position with respect to time, we are evaluating how position changes over time; i.e velocity!

To take the derivative of the position function

We'll need to make use of the following derivative rule:

Derivative of an exponential: 

Note that u may represent large functions, and not just individual variables!

Using the above properties, the velocity function is