Explanation:

The acceleration of a particle is its change in velocity; therefore, to find its acceleration at a particular point in time, it is necessary to take the derivative of the velocity function and evaluate it for that value. In short, we want。

Differentiate:

Apply the sum rule:

Apply the chain rule by setting:

This is the acceleration function; substitute 6 for:

Explanation:

The acceleration function of a particle with respect to time is the derivativeof the velocity function of the particle.

Find this derivative by first using the product rule:

Apply the product rule:

Evaluate:

feet per second squared.

Explanation:

To determine the velocity function - the变动率of position - we must take the derivative of the position function:

The derivative was found using the following rules:

,

To find the velocity at a certain value of x - the instantaneous rate of change of position - we simply plug in the given value of x into the velocity function:

Explanation:

To find the rate of change of crystal growth at a given time, we must take the derivative of the growth rate function and evaluate it at a specific time.

The derivative of the function is

and was found using the following rules:

,

Now, to find the growth rate at t=5, simply plug in this value for t into the derivative function:

Explanation:

If p(t) gives the position of a planet as a function of time, find the function which models the planet's acceleration.

Velocity is the first derivative of position. Acceleration is the first derivative of velocity.

因此,我们不能ed to do to solve this problem is to find the second derivative of p(t).

We can do this via the power rule and the rule for differentiating sine and cosine.

1)

2)

So, we these rules in mind, we get:

So our velocity function is:

Next, differentiate v(t) to get a(t).

So, our acceleration function is

Explanation:

If p(t) gives the position of a planet as a function of time, find the planet's acceleration when t=0.

Velocity is the first derivative of position. Acceleration is the first derivative of velocity.

因此,我们不能ed to do to solve this problem is to find the second derivative of p(t) and then plug in 0 for t and solve.

We can do this via the power rule and the rule for differentiating sine and cosine.

1)

2)

So, we these rules in mind, we get:

So our velocity function is:

Next, differentiate v(t) to get a(t).

So, our acceleration function is

Next, plug in 0 and simplify.

Our answer is 12. Since we are not given any units, we can leave it as 12

Explanation:

The car's acceleration is the instantaneous rate of change in its velocity with respect to time (), so we can find the value of the cars acceleration at any time by taking the derivative of the velocity equation

Evaluating at, we get

Explanation:

We are given the function describing the position of the body given a time "t":

We are also given the interval that the function can be applied over:

First, we are tasked to find the speed of the body at each of the endpoints (0 and 2 seconds, respectively).

To figure this, we must understand thatspeed is the absolute value of velocity。

To find the velocity function, we must take thederivativeof the position function with respect to time:

Now that we have the function of velocity given a time "t", we can find the speed of the body given a time "t" by simply taking the absolute value of the velocity function.

Now, to find the speed of the endpoints:

We can repeat the same process to find the speed at 2 seconds.

For part a, the speeds at 0 and 2 seconds are3 and 1 [m/s], respectively.

Part b asks us to find the acceleration at the endpoint times (0 and 2 seconds). To do this, we must first understand thatto find acceleration at a time "t", we must take the derivative, with respect to time, of the velocity function。

From part a, we found the velocity function:

Thus, to find acceleration, we derive this function with respect to time.

The result of our derivation tells us that no matter what time is plugged into the function, the acceleration shall always return 2[m/s^2].

So, for part b, the acceleration at 0 and 2 seconds are2 and 2 [m/s^2], respectively.

Explanation:

We are given the position equation:

To find the acceleration when the velocity is equal to zero, we must first find the function to describe the velocity given a time "t".

We know thatthe velocity function is found by deriving the position function with respect to time。

Thus, the velocity function is given by the equation:

To find the acceleration when the velocity is equal to zero, we must set the velocity function equal to zero and solve for the times.

(multiply c factor by the a coefficient)

(divide out the original a coefficient)

Thus, the velocity is zero at both 3 and 1 seconds. The question asks for the acceleration at these times.

To do the next part, we must understand thatthe acceleration function is the derivative of the velocity function, with respect to time。我们已经知道速度函数:

So, if we derive this function with respect to time and plug in our times, we will know the acceleration when the velocity is equal to zero.

Our derivation tells us that given any time "t", the acceleration of the body can be found by the function:

So, to find the acceleration at times 3 and 1, we simply plug in the values to our acceleration function above.

所以,加速度在时间1是6 (m / s ^ 2)和the acceleration at time 3 is 6[m/s^2].

Explanation:

This problem, although nottechnicallycomplex, requires some understanding of the derivative and how it relates to elementary physics.

For part a)

Given a function that returns an objects position at time t, we can take thederivative of the position functionto find thevelocity function

We are given the position function:

We can apply thegeneral power rule for derivativesto find the derivative of this function:

And from our understanding that thederivative of the position functionisthe velocity function,we can make the substitution

Thus, ourpart a answeris:

For part b)

Furthermore, we can take the derivative of thevelocity functionto find ouracceleration function。

We found the velocity function to be:

Thus, the derivative of the velocity function (the acceleration function) is simply:

The full answer should be: