In order to find the acceleration of a given point, you must first find the derivative of the position function which gives you the velocity function:

Then, you differentiate the velocity function to get the acceleration function:

Then, find  when .

The answer is: 

Acceleration is the change in velocity (which itself is the change in position), so we are looking to solve for the h''(6).

To find the second derivative we must use the power rule for differentiation twice.

The power rule states,

.

Applying the power rule once we find the velocity of the function.

Applying the power rule a second time we find the acceleration of the function.

From here, substitute six in for t to solve for the specific acceleration.

Since each position function is simply a polynomial, their respctive accelerations can easily be found by applying the power rule twice.

You will notice that each answer reduces to a constant when derived twice. There is only one answer for which the acceleration equals 16 at time t=5 (in fact, the acceleration equals 16 at all values of t).

In fact, because all monomials which include a power of x that is less than 2 will reduce to 0 (either from the first or the second derivation), we are only interested in what happens to the leading term throughout the derivations.

INCORRECT ANSWERS

(I) 

(II)

(III)

CORRECT ANSWER

Of each of the functions listed, the correct answer is the only function for which the acceleration equation reduces to a constant.

The second derivatives of the other functions clearly do not represent a constant acceleration of zero. In addition to this, each of the remaining functions haveexclusively positive accelerations (greater than zero) when t is positive. As small, and perhaps unfamiliar as  is, it is certainly positive, and for this reason the only possible answer to this question is  .

In order to find the acceleration of a particle at a certain point, we must find the second derivative.

To find the derivative we must use the trigonometric rules of differentiation for cosine and sine which states,

This is the first derivative, which gives us the velocity: 

We then differentiate the first derivative to get:

Then, find the value of  when :

Therefore, the answer is: 

In order to find the acceleration of a particle at a certain point, we must find the second derivative.

For this particular function we will need to apply the chain rule and power rule which state,

This is the first derivative, which gives us the velocity: 

We then differentiate the first derivative to get: 

Then, find the value of   when :

Therefore, the answer is: 

In order to find the acceleration of a particle at a certain point, we must find the second derivative.

To find the derivative we must use the power rule which states,

This is the first derivative, which gives us the velocity:  

We then differentiate the first derivative to get: 

Then, find the value of   when :

Therefore, the answer is: 

In order to find the acceleration of a particle at a certain point, we must find the second derivative.

To find the derivative we must use the power rule which states,

Also recall that the derivative of .

This is the first derivative, which gives us the velocity: 

We then differentiate the first derivative to get:

Then, find the value of  when :

Therefore, the answer is: 

To find acceleration, you must first find velocity, which is the derivative of position. To take the derivative, you must multiply the leading coefficient by the exponent and then subtract 1 from the exponent. Therefore, the velocity function is: . To find acceleration, take the derivative of velocity: . Then set that expression equal to 0 to get your answer: .

In order to find the acceleration of any point given a position function, we must first find the derivative of the position function and then the derivative of the velocity function giving us the acceleration function, .

This is the position function: 

The derivative gives us the velocity function:  

The derivative of the velocity function gives us the acceleration function: 

You can then find the acceleration of a given point by substituting it in for the variable: 

Therefore, the velocity at  is: