# For #f(t)= (t^2,t^3)# what is the distance between #f(1)# and #f(3)#?

The distance between them is given by Pythagoras Theorem.

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To find the distance between ( f(1) ) and ( f(3) ), we first need to calculate the values of ( f(t) ) at ( t = 1 ) and ( t = 3 ).

( f(1) = (1^2, 1^3) = (1, 1) )

( f(3) = (3^2, 3^3) = (9, 27) )

Then, we can use the distance formula in two dimensions, which states that the distance between two points ( (x_1, y_1) ) and ( (x_2, y_2) ) is given by:

[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Applying this formula to our problem:

[ \text{Distance} = \sqrt{(9 - 1)^2 + (27 - 1)^2} = \sqrt{8^2 + 26^2} = \sqrt{64 + 676} = \sqrt{740} ]

So, the distance between ( f(1) ) and ( f(3) ) is ( \sqrt{740} ) or approximately ( 27.20 ).

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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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