To find when the particle's position has maxima and minima, take the derivative of the position function with respect to time to find velocity, and find the times when this velocity is equal to zero:

For the position function

The velocity function is

This function is equal to zero at time . Since it is positive prior and negative afterwards, this indicates a maximum position.

Now, to find the time when the particle is first halfway to this position, use the equation;

The solutions to the equation are 

So the time when the particle reaches half of its maximum position is therefore:

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

Exponential and sin functions integrate as follows:

Note that after integration of a function, we always include a constant of integration, usually denoted as . Furthermore, if multiple functions are added/multiplied together, a single integration only requires one constant of integration, even though there are multiple terms.

Using this knowledge, the position function can be found 

The constant of integration can be found via knowledge of the initial position:

This gives the definite integral

At time 

Position can be found by integrating the acceleration function twice, though it's usually useful to find velocity by integrating acceleration once, and then integrating the velocity to find position, so that we may deal with constants of integration as they appear.

For the acceleration function

The integral is found to be

Upon integrating a function, we must always add a constant of integration. Note that the integral of a constant  will be found to be 

We're told the rock is dropped, and may assume a zero initial velocity:

Now, the position function can be found by integrating once more, noting the property that :

Recall that we were given an initial height:

At time 

The kinematic equations in physics are actually derived in this manner, presuming a constant acceleration.

Position can be found by integrating velocity with respect to time. For velocity functions:

The position functions are

These can be found via the properties 

And  is a constant of integration.

These constants of integration can be found by using the initial conditions given, namely

; 

; 

; 

So the definite integrals are tus:

At at time 

; 

; 

; 

Position can be found by integrating the velocity function with respect to time:

We are given the velocity function

To integrate this, it is necessary to know how to integrate exponential functions:

The velocity function has a sum of exponential functions, and for sums, each element can be integrated independently:

(Note that for a single integration, only one constant of integration is needed regardless of the number of terms)

To find the constant of integration, utilize the initial condition:

Now to find the farthest potential position, take the limit of this function as t approaches infinity:

Since the class of possible position functions is given by the antiderivative of the velocity function:

=

Now, impose the initial condition:

so:

and

Note that  is equivalent to  in radians because they differ by an integer multiple of . So we have 

Given two points:

The midpoint of  is given as:

Using our given coordinates, the midpoint is found by simple application of this formula:

This is one of the answer choices.

To find the position function from the velocity function, take the integral of the velocity function, using the power rule for integration:

Applying this formula to the given equation, remembering to multiply by the constants in each term:

Find the value of  by plugging  into .

Therefore

Evaluate this at  to find the position.

In order to find the position function from the velocity function we must take the integral of the velocity function since

.

When taking the integral, we use the trigonometric property which states

.

As such,

Therefore the position function is

.

If the function b(s) models the acceleration of a ping pong ball being thrown across a table, find the function which models the ball's position if the velocity when  is 108 and the position when  is 45.

We are asked to relate acceleration and position. This may seem confusing, but if we remember that acceleration is the first derivative of velocity, we know that we should start by integrating b(s).

Now we have the following:

Which is almost our velocity function, but we need to use our given conditions to find c.

We are told that when s=3, the velocity is equal to 108. Let's plug those numbers in and solve for c!

Which makes our velocity function:

Now, from our velocity function we can integrate to get out position function. We know this because velocity is the first derivative of position. So...

We'll call our position function P(s), for position. 

We are almost there, but we know that , so we need to find "c"

Simplify:

Making our position function: