To find the acceleration of the particle, we must take the first derivative of the velocity function:

The derivative was found using the following rule:

Now, we evaluate the acceleration function at the given point:

Acceleration is the rate of change of velocity, so we must integrate the acceleration function to find the velocity function:

The integration was performed using the following rules:

, 

To find the integration constant C, we must use the initial condition given:

Our final answer is

The velocity of a particle is given by v(t). Find the function which models the particle's acceleration.

To find the acceleration from a velocity function, simply take the derivative.

In this case, we are given v(t), and we need to find v'(t) because v'(t)=a(t).

To find v'(t), we need to use the power rule. 

For each term, simply multiply by the exponent, and then subtract one from the exponent. Constant terms will drop out, linear terms will become constants, and so on.

So, our answer is:

The velocity of a particle is given by v(t). Find the particle's acceleration when t=3.

To find the acceleration from a velocity function, simply take the derivative.

In this case, we are given v(t), and we need to find v'(t) because v'(t)=a(t).

To find v'(t), we need to use the power rule. 

For each term, simply multiply by the exponent, and then subtract one from the exponent. Constant terms will drop out, linear terms will become constants, and so on.

So, our acceleration function is:

Now, plug in 3 for t and solve.

So, our answer is 52.

Let's examine the limit

first.

and 

,

so by L'Hospital's Rule, 

Since ,

Now, for each , ; therefore, 

By the Squeeze Theorem, 

and 

Therefore, by L'Hospital's Rule, we can find  by taking the derivatives of the expressions in both the numerator and the denominator:

Similarly, 

So 

But  for any , so 

and 

Therefore, by L'Hospital's Rule, we can find  by taking the derivatives of the expressions in both the numerator and the denominator:

Similarly, 

so

and 

Therefore, by L'Hospital's Rule, we can find  by taking the derivatives of the expressions in both the numerator and the denominator:

We start at x = 0 and move to x=2, with a step size of 1. Essentially, we approximate the next step by using the formula:

.

So applying Euler's method, we evaluate using derivative: 

And two step sizes, at x = 1 and x = 2.

And thus the evaluation of p at x = 2, using Euler's method, gives us p(2) = 2.

Using Euler's method with  means that we use two iterations to get the approximation. The general iterative formula is 

where each  is

  is an approximation of , and , for this differential equation. So we have

So our approximation of  is