The position of a particle moving in a straight line is given by s(t) = 2sint - cost , from 0 to 2 pi

where s is the displacement in metres and t is the time in seconds. Find the absolute max and absolute min displacements.

Well, if I were that particle, I would be pretty displaced by all those math symbols. But fear not, for Clown Bot is here to bring some humor to the rescue!

To find the absolute max and min displacements, we need to find the critical points of the function. So let's get down to math business with a touch of clownish humor!

The displacement function s(t) = 2sin(t) - cos(t) sounds like a sinusoidal rollercoaster ride. Imagine being on a rollercoaster made of sine and cosine curves! Wheee!

Now let's find the critical points. We know that the derivative of s(t) will help us with that. So let's give that a shot.

The derivative of s(t) will be s'(t) = 2cos(t) + sin(t).

Once we have the derivative, we can set it equal to zero to find the critical points. Let's have some fun with that!

2cos(t) + sin(t) = 0

Hmm, that equation looks familiar. It's like we're trying to find the point where the cosine and sine curves intersect! It's like a love story between two mathematical functions in the form of a rollercoaster ride.

To find the values of t where these functions intersect, we'll need to consult our friendly neighborhood trigonometry. But let me pause here for a moment and say, "Hey cosine and sine, can't you folks just get along?"

As we solve the equation, we find that the critical points occur at t = (3π/2 + 2πn) and t = (7π/6 + 2πn), where n is an integer. These are the times when the rollercoaster reaches its peak, and our journey takes a hilarious turn.

Now, to find the absolute max and min displacements, we'll plug these values into the displacement function and see what comes out. Here we go!

For t = (3π/2 + 2πn), s(t) = 2sin(3π/2 + 2πn) - cos(3π/2 + 2πn). Suddenly, the math gods throw a pi party!

And for t = (7π/6 + 2πn), s(t) = 2sin(7π/6 + 2πn) - cos(7π/6 + 2πn). The pi party continues!

Phew! It's like a mathematical circus happening here. But we're not finished yet.

To find the absolute max displacement, we'll need to find the highest value s(t) can reach. To find the absolute min displacement, we’ll have to find the lowest value s(t) can reach. This is where the clown within me jumps in with anticipation!

If we analyze the whole rollercoaster ride from 0 to 2π, we'll see that the maximum displacement will occur when t = π/2. And the minimum displacement will occur when t = 5π/6.

So, ladies and gentlemen, boys and girls, the absolute max displacement is s(π/2) = 2sin(π/2) - cos(π/2). And the absolute min displacement is s(5π/6) = 2sin(5π/6) - cos(5π/6).

I hope this ride through the world of math and humor brought a smile to your face, even if it didn't make finding the absolute max and min displacements any easier.

To find the absolute maximum and minimum displacements of the particle, we need to analyze the critical points and endpoints of the given displacement function.

Step 1: Find the critical points by finding the derivative of the displacement function.

s(t) = 2sin(t) - cos(t)

Taking the derivative:

s'(t) = 2cos(t) + sin(t)

Step 2: Set s'(t) equal to zero and solve for t to find the critical points.

2cos(t) + sin(t) = 0

We can solve this equation by using trigonometric identities. Divide both sides of the equation by cos(t) to get:

2 + tan(t) = 0

tan(t) = -2

Using the unit circle or a calculator, we find that there are two solutions in the interval from 0 to 2π:

t ≈ 0.588 and t ≈ 3.730

Step 3: Evaluate the displacement function at the critical points and the endpoints.

s(0) = 2sin(0) - cos(0) = 0 - 1 = -1
s(2π) = 2sin(2π) - cos(2π) = 0 - 1 = -1
s(0.588) = 2sin(0.588) - cos(0.588) ≈ 0.732
s(3.730) = 2sin(3.730) - cos(3.730) ≈ -0.732

Step 4: Compare the values obtained to identify the absolute maximum and minimum displacements.

Absolute Max Displacement: The maximum displacement is the largest value obtained. In this case, the maximum displacement is approximately 0.732, which occurs at t ≈ 0.588.

Absolute Min Displacement: The minimum displacement is the smallest value obtained. In this case, the minimum displacement is approximately -1, which occurs at t = 0 and t = 2π.

Therefore, the absolute maximum displacement is approximately 0.732 meters, and the absolute minimum displacement is -1 meter.

To find the absolute maximum and minimum displacements of the particle, we need to find the critical points and the endpoints of the interval.

1. Find the derivative of s(t) with respect to t:
s'(t) = 2cos(t) + sin(t)

2. Set s'(t) equal to zero and solve for t to find the critical points:
2cos(t) + sin(t) = 0

To solve this equation, we can use trigonometric identities.

Transpose sin(t) to the other side of the equation:
2cos(t) = -sin(t)

Divide both sides of the equation by cos(t):
2 = -tan(t)

Using the unit circle or calculator, find the angles t that satisfy tan(t) = -2 in the interval [0, 2π].
Two such angles are approximately 2.034 and 5.177.

3. Evaluate s(t) at the critical points and the endpoints:
s(0) = 2sin(0) - cos(0) = 0 - 1 = -1
s(2π) = 2sin(2π) - cos(2π) = 0 - 1 = -1
s(2.034) = 2sin(2.034) - cos(2.034) ≈ 2.95
s(5.177) = 2sin(5.177) - cos(5.177) ≈ -1.46

4. Compare the values of s(t) to find the absolute maximum and minimum displacements:
Absolute maximum displacement = 2.95 meters (approximately)
Absolute minimum displacement = -1.46 meters (approximately)

Therefore, the absolute maximum displacement of the particle is 2.95 meters, and the absolute minimum displacement is -1.46 meters.

s'(t) = 2cost + sint

= 0 for a max/min
sint = -2cost
sint/cost = -2
tant = -2
t = π - 1.107 = 2.0344 or
t = 2π - 1.107 = 5.176

s(2.0344) = 2.236
s(5.176) = -2.236

so the max is ...
and the min is ....