# 19. "Describe the motion of a particle with position (#x,y#) as #t# varies in the given interval" ?

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#x=5+2\cos\color(maroon)(cancel(t))\pit# , #y=3+2\sin\pit# , #1\let\le2#

*thanks to Ultrilliam for pointing it out,

#x=5+2\color(red)(\cos\pi\t)#

*thanks to Ultrilliam for pointing it out,

See explanation

for

for

Based on Wolfram Alpha, the particle moves counterclockwise in a circular manner from (

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In cartesian: circle with centre

Period

Set it up for the Pytharorean identity like this:

Pytharorean identity

From the graph, you can see that it's making the CCW journey along the bottom half of the circle in that time interval

graph{ (x-5)^2 + (y - 3)^2 = 2^2 [-10, 10, -5, 5]}

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To describe the motion of a particle with position (x, y) as t varies in the given interval, we need to analyze the functions that represent the particle's position with respect to time. Typically, this involves studying the parametric equations x(t) and y(t) that define the particle's position as functions of time t.

Once the parametric equations are known, we can examine how the x-coordinate and y-coordinate of the particle change as time varies within the given interval. This analysis helps us understand the particle's trajectory, speed, direction, and any other relevant characteristics of its motion during that interval.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you differentiate the following parametric equation: # x(t)=t-(t+1)e^t, y(t)= t^2-e^(t-1) #?
- How do you differentiate the following parametric equation: # x(t)=-te^t-2t, y(t)= 3t^3-4t #?
- How do you differentiate the following parametric equation: # x(t)=1/t, y(t)=lnt #?
- How do you find the vector equation and the parametric equations of the line that passes through the points A (3, 4) and B (5, 5)?
- For #f(t)= (sint,cost)# what is the distance between #f(pi/4)# and #f(pi)#?

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