# If the velocity of an object is given by # v(t) = 3t^2 - 22t + 24 # and #s(0)=0# then how do you find the displacement at time #t#?

# s = t^3 - 11t^2 + 24t #

We have:

We are aware of:

Hence, [A] can be expressed as a differential equation:

We may "separate the variables" because this is separable "as is" and obtain:

which we can combine directly to obtain:

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To find the displacement at time ( t ), you need to integrate the velocity function ( v(t) ) with respect to time ( t ). The displacement function ( s(t) ) can be obtained by integrating the velocity function:

[ s(t) = \int v(t) , dt ]

Given that ( v(t) = 3t^2 - 22t + 24 ), integrate this expression with respect to ( t ) to find ( s(t) ).

[ s(t) = \int (3t^2 - 22t + 24) , dt ]

[ s(t) = \frac{1}{3}t^3 - \frac{22}{2}t^2 + 24t + C ]

Where ( C ) is the constant of integration. Since ( s(0) = 0 ), we can find ( C ) by substituting ( t = 0 ) into the displacement function:

[ 0 = \frac{1}{3}(0)^3 - \frac{22}{2}(0)^2 + 24(0) + C ]

[ C = 0 ]

Therefore, the displacement function is:

[ s(t) = \frac{1}{3}t^3 - 11t^2 + 24t ]

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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.

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