What is the position of a particle at time #t=2# if a particle moves along the x axis so that at any time #t>0#, the velocity is given by #v(t)=4-6t^2#, and at position #x=7# at #t=1#?

Answer 1
The answer is: #x=-3#.
The law that gives the position of the particle at the time #t# is given by the integral:
#x(t)=int(4-6t^2)dt=4t-6t^3/3+c=4t-2t^3+c#.
To find #c#, we can use the intial conditions:
#x(1)=7#,

so:

#7=4-2+crArrc=5#.

So the law becomes:

#x(t)=4t-2t^3+5#.
Now we can use it to find the positition at the time #t=2#:
#x(2)=8-16+5=-3#.
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Answer 2

To find the position of the particle at ( t = 2 ), we first integrate the velocity function to get the displacement function.

[ v(t) = 4 - 6t^2 ]

Integrate ( v(t) ) with respect to ( t ) to get the displacement function ( x(t) ):

[ x(t) = \int (4 - 6t^2) , dt ]

[ x(t) = 4t - 2t^3 + C ]

Given that the particle is at position ( x = 7 ) at ( t = 1 ), we can use this information to find the constant ( C ):

[ x(1) = 4(1) - 2(1)^3 + C = 4 - 2 + C = 7 ] [ C = 5 ]

Now, we have the displacement function:

[ x(t) = 4t - 2t^3 + 5 ]

To find the position of the particle at ( t = 2 ), substitute ( t = 2 ) into the displacement function:

[ x(2) = 4(2) - 2(2)^3 + 5 ]

[ x(2) = 8 - 16 + 5 ]

[ x(2) = -3 ]

Therefore, the position of the particle at ( t = 2 ) is ( x = -3 ).

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Answer from HIX Tutor

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.

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.

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