# For #f(t)= (lnt-e^t, t^2/e^t)# what is the distance between #f(2)# and #f(4)#?

with

so the distance is given by

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To find the distance between the points ( f(2) ) and ( f(4) ), we first need to evaluate the function ( f(t) ) at ( t = 2 ) and ( t = 4 ). Then, we can use the distance formula in two dimensions to calculate the distance between these points.

Let's find ( f(2) ): [ f(2) = (\ln(2) - e^2, \frac{2^2}{e^2}) ]

And ( f(4) ): [ f(4) = (\ln(4) - e^4, \frac{4^2}{e^4}) ]

Now, we'll use the distance formula to find the distance between these points: [ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Substitute the coordinates of ( f(2) ) and ( f(4) ) into the formula: [ \text{Distance} = \sqrt{(\ln(4) - e^4 - \ln(2) + e^2)^2 + (\frac{4^2}{e^4} - \frac{2^2}{e^2})^2} ]

Simplify the expression: [ \text{Distance} = \sqrt{(\ln(4) - \ln(2) - e^4 + e^2)^2 + (\frac{16}{e^4} - \frac{4}{e^2})^2} ]

Evaluate the expressions inside the square root: [ \text{Distance} = \sqrt{(\ln(4) - \ln(2) - e^4 + e^2)^2 + (\frac{16}{e^4} - \frac{4}{e^2})^2} ] [ = \sqrt{(\ln(\frac{4}{2}) - (e^4 - e^2))^2 + (\frac{16e^2 - 4e^4}{e^4})^2} ] [ = \sqrt{(\ln(2) - (e^4 - e^2))^2 + (\frac{4(e^2 - e^4)}{e^4})^2} ] [ = \sqrt{(\ln(2) - e^4 + e^2)^2 + (\frac{4(e^2 - e^4)}{e^4})^2} ]

Calculate the expression: [ \text{Distance} \approx \sqrt{(0.6931 - 54.6 + 7.389)^2 + (\frac{4(7.389 - 54.6)}{54.6})^2} ] [ \approx \sqrt{(-46.5179)^2 + (-0.2523)^2} ] [ \approx \sqrt{2161.831 + 0.06372} ] [ \approx \sqrt{2161.89472} ] [ \approx 46.49 ]

So, the distance between ( f(2) ) and ( f(4) ) is approximately ( 46.49 ).

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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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