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What is the slope of #f(t) = (t-2,2t)# at #t =-1#?

Answer 1

Samantha Adkison

Slope equals #dy/dx = (dy"/"dt)/(dx"/"dt)#

#dy/dt = d/dt(2t) = 2#

#dx/dt = d/dt(t-2) = 1#

#therefore dy/dx = 2/1 = 2#

This means that no matter what #t# is, the slope of the curve at time #t# will always be 2.

So #dy/dx]_(t=1) = 2#

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Answer 2

Evelyn Agard

To find the slope of the function ( f(t) = (t - 2, 2t) ) at ( t = -1 ), you need to calculate the derivative of ( f(t) ) with respect to ( t ) and then evaluate it at ( t = -1 ).

The derivative of ( f(t) ) can be found by differentiating each component separately:

[ f(t) = (t - 2, 2t) ] [ \frac{df}{dt} = \left(\frac{d}{dt}(t - 2), \frac{d}{dt}(2t)\right) ] [ = (1, 2) ]

Now, evaluate the derivative at ( t = -1 ):

[ \frac{df}{dt} \bigg|{t=-1} = (1, 2) \bigg|{t=-1} = (1, 2) ]

So, the slope of the function ( f(t) = (t - 2, 2t) ) at ( t = -1 ) is ( (1, 2) ).

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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.