What is the slope of #f(t) = ((t-2)^3,2t)# at #t =-1#?
Slope at
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To find the slope of ( f(t) = ((t-2)^3, 2t) ) at ( t = -1 ), we first need to find the derivative of ( f(t) ) with respect to ( t ), then evaluate it at ( t = -1 ).
The derivative of ( f(t) ) with respect to ( t ) is given by:
( f'(t) = \left( \frac{d}{dt}((t-2)^3), \frac{d}{dt}(2t) \right) )
( = \left( 3(t-2)^2, 2 \right) )
Now, evaluate ( f'(t) ) at ( t = -1 ):
( f'(-1) = \left( 3((-1)-2)^2, 2 \right) )
( = \left( 3(-3)^2, 2 \right) )
( = \left( 3(9), 2 \right) )
( = \left( 27, 2 \right) )
Therefore, the slope of ( f(t) ) at ( t = -1 ) is ( 27 ).
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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.
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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