What is the instantaneous velocity of an object with position at time t equal to # f(t)= (1/(t4),sqrt(5t3)) # at # t=2 #?
Since velocity is defined as the rate of change in position, we may find the velocity by differentiating the position function with respect to time as follows :
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To find the instantaneous velocity of an object at ( t = 2 ) given the position function ( f(t) = \left(\frac{1}{t  4}, \sqrt{5t  3}\right) ), we need to find the derivative of the position function with respect to time ( t ) and then evaluate it at ( t = 2 ).
The position function ( f(t) ) can be expressed as ( f(t) = \left(\frac{1}{t  4}, \sqrt{5t  3}\right) ).
The derivative of ( f(t) ) with respect to ( t ) is ( f'(t) = \left(\frac{d}{dt}\left(\frac{1}{t  4}\right), \frac{d}{dt}\left(\sqrt{5t  3}\right)\right) ).
Now, let's find the derivatives of each component function:

For the first component, ( \frac{d}{dt}\left(\frac{1}{t  4}\right) ): [ \frac{d}{dt}\left(\frac{1}{t  4}\right) = \frac{1}{{(t  4)}^2} ]

For the second component, ( \frac{d}{dt}\left(\sqrt{5t  3}\right) ): [ \frac{d}{dt}\left(\sqrt{5t  3}\right) = \frac{1}{2\sqrt{5t  3}} \cdot \frac{d}{dt}(5t  3) = \frac{5}{2\sqrt{5t  3}} ]
So, the derivative of ( f(t) ) with respect to ( t ) is ( f'(t) = \left(\frac{1}{{(t  4)}^2}, \frac{5}{2\sqrt{5t  3}}\right) ).
Now, we evaluate ( f'(t) ) at ( t = 2 ): [ f'(2) = \left(\frac{1}{{(2  4)}^2}, \frac{5}{2\sqrt{5(2)  3}}\right) = \left(\frac{1}{(2)^2}, \frac{5}{2\sqrt{10  3}}\right) = \left(\frac{1}{4}, \frac{5}{2\sqrt{7}}\right) ]
So, the instantaneous velocity of the object at ( t = 2 ) is ( \left(\frac{1}{4}, \frac{5}{2\sqrt{7}}\right) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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