What is the equation of the line normal to #f(x)= x^3/e^x # at #x=2#?

Answer 1

#y = 8/e^2 -e^2/4(x-2)#

The equation of the line normal to the graph of the function:

#y=f(x)#
at ath point #(bar x, f(barx)) # is given by:
#y=f(bar x)-1/(f'(barx)) (x-barx)#

In our case:

#f(x) = x^3/e^x=x^3e^(-x)#
#f(2) = 8/e^2#
#f'(x) = 3x^2e^-x -x^3e^-x#
#f'(2) = 12/e^2-8/e^2=4/e^2#

So the normal line is:

#y = 8/e^2 -e^2/4(x-2)#
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Answer 2

To find the equation of the line normal to ( f(x) = \frac{x^3}{e^x} ) at ( x = 2 ), follow these steps:

  1. Find the derivative of ( f(x) ) using the quotient rule.
  2. Evaluate the derivative at ( x = 2 ) to find the slope of the tangent line.
  3. Find the negative reciprocal of the slope obtained in step 2 to get the slope of the normal line.
  4. Use the point-slope form of a line with the point ( (2, f(2)) ) and the slope obtained in step 3 to find the equation of the normal line.
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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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