How do you convert the parametric equation to rectangular form #x = e^t#, #y = e^(-t)#?

Answer 1

#y = 1/x#

Short way... #x = e^t# #y = e^(-t) = 1/e^t = 1/x#
Long winded but maybe more interesting way... #x = e^t implies t = ln x# #y = e^(-t) implies -t = ln y#
#implies ln x = - ln y#
#ln x + ln y = 0#
#ln xy = 0#
#xy = e^0 = 1#
#y = 1/x#
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Answer 2

To convert the parametric equations ( x = e^t ) and ( y = e^{-t} ) to rectangular form, solve for ( t ) in terms of ( x ) and ( y ), then substitute these expressions back into the equations to eliminate ( t ) and express ( y ) in terms of ( x ).

Solving for ( t ) in terms of ( x ) and ( y ):

  1. From ( x = e^t ), take the natural logarithm of both sides to get ( t = \ln(x) ).
  2. From ( y = e^{-t} ), take the natural logarithm of both sides to get ( -t = \ln(y) ). Multiply both sides by -1 to get ( t = -\ln(y) ).

Now substitute these expressions back into the original parametric equations:

  1. Substitute ( t = \ln(x) ) into ( y = e^{-t} ) to get ( y = e^{-\ln(x)} ).
  2. Simplify ( e^{-\ln(x)} ) to ( \frac{1}{x} ).

Therefore, the rectangular form of the parametric equations is ( y = \frac{1}{x} ).

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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.

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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