A particle moves along a line so that at any time t its position is given by x(t)=2(pi)t+cos(2(pi)t). Determine the particle's maximum velocity.

max velocity occurs when the second derivative is zero

so ...
x(t) = 2πt + cos(2πt)
x '(t) = 2π - 2π(sin 2πt)
x ''(t) = 0 - 4π^2(cos 2πt)
= 0 for a max/min of velocity
cos 2πt = 0
2πt = π/2 or 2πt = 3π/2
t = 1/4 or t = 3/4

if t= 1/4
v(1/4) = x '(1/4) = 2π - 2π(sin π/2) = 2π - 2π = 0
if t=3/4
v(3/4) = 2π - 2π(sin 3π/2) = 2π + 2π = 4π

Well, well, well, let's find the maximum velocity of this particle, shall we?

First, we need to find the velocity function. The velocity function is simply the derivative of the position function. So, let's take the derivative of x(t) with respect to t.

x'(t) = 2(pi) - sin(2(pi)t) * 2(pi)

Now, to find the maximum velocity, we need to find where the derivative is equal to zero or undefined. But wait, we have a periodic function sin(2(pi)t) in there, and it never equals zero or is undefined. So, we can conclude that the velocity will never be zero or undefined.

Since there's no maximum or minimum velocity, we can't determine the particle's maximum velocity in this case. But hey, don't be too disappointed, at least the particle won't stop or disappear into thin air!

To determine the particle's maximum velocity, we need to find the derivative of its position function with respect to time, which gives us the velocity function.

Given the position function:
x(t) = 2πt + cos(2πt)

To find the velocity function v(t), we take the derivative of x(t) with respect to t:

v(t) = d/dt (2πt + cos(2πt))

Differentiating each term in the expression, we get:

v(t) = 2π - sin(2πt)(2π)

Since the velocity function v(t) is the derivative of the position function x(t), the velocity function tells us the rate at which the particle is moving at any given time.

To find the maximum velocity, we can set the first derivative of the velocity function equal to zero and solve for t:

2π - sin(2πt)(2π) = 0

Dividing both sides by 2π, we get:

1 - sin(2πt) = 0

Simplifying further, we have:

sin(2πt) = 1

The maximum velocity occurs when sin(2πt) is equal to 1. Since sin(2πt) repeats every 2π radians, we can solve for t by setting 2πt equal to the angle that evaluates to sin(1), which is π/2:

2πt = π/2

Solving for t, we divide both sides by 2π:

t = π/4π

Simplifying, we get:

t = 1/4

So, the particle's maximum velocity occurs at t = 1/4. To find the maximum velocity, we substitute this value back into the velocity function:

v(1/4) = 2π - sin(2π(1/4))(2π)

Let's calculate the maximum velocity using this equation.

To determine the particle's maximum velocity, we need to find the derivative of its position equation, x(t), with respect to time (t). The derivative of a function represents the rate of change of that function.

To find the derivative of x(t), we can take the derivative term by term. The derivative of 2(pi)t is simply 2(pi), as the derivative of t with respect to t is 1. The derivative of cos(2(pi)t) requires the chain rule: the derivative of the outer function (cosine) times the derivative of the inner function (2(pi)t). The derivative of cos(x) is -sin(x), so the derivative of cos(2(pi)t) is -sin(2(pi)t) times the derivative of 2(pi)t, which is 2(pi).

Therefore, the velocity function, v(t), which represents the particle's velocity at any given time t, is given by v(t) = 2(pi) - 2(pi)sin(2(pi)t).

To find the particle's maximum velocity, we can analyze the velocity function. The maximum or minimum points occur where the derivative of the velocity function is equal to zero or does not exist. Since the derivative of v(t) is continuous, we only need to check where it equals zero.

Setting the derivative of v(t) equal to zero:

0 = d(v(t))/dt = d/dt (2(pi) - 2(pi)sin(2(pi)t))
0 = 2(pi) - 2(pi)cos(2(pi)t)

Then, we solve for t:

2(pi)cos(2(pi)t) = 2(pi)
cos(2(pi)t) = 1

The cosine of any multiple of 2(pi) is equal to 1. Therefore, t can be any real number.

Since the cosine term does not affect the value of the particle's maximum velocity, we can ignore it. Therefore, the maximum velocity of the particle is given by |2(pi)|, which simplifies to 2(pi).

So, the particle's maximum velocity is 2(pi).