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

The Normal Line is

Determine now the Normal Line using

after simplification the final answer:

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The equation of the normal line of f(x)=(x-2)^(3/2)-x^3 at x=2 is y = -4x + 8.

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The equation of the normal line to the function ( f(x) = (x - 2)^{\frac{3}{2}} - x^3 ) at ( x = 2 ) can be found by first determining the slope of the tangent line at ( x = 2 ) and then using the negative reciprocal of this slope to find the slope of the normal line. Finally, the equation of the normal line can be written in point-slope form using the point of tangency ( (2, f(2)) ).

First, find ( f'(x) ), the derivative of ( f(x) ), and evaluate it at ( x = 2 ) to find the slope of the tangent line. Then, take the negative reciprocal of this slope to find the slope of the normal line. Finally, use the point-slope form of a line to write the equation of the normal line.

( f'(x) = \frac{3}{2}(x - 2)^{\frac{1}{2}} - 3x^2 )

Evaluate ( f'(2) ) to find the slope of the tangent line:

( f'(2) = \frac{3}{2}(2 - 2)^{\frac{1}{2}} - 3(2)^2 ) ( f'(2) = -12 )

The slope of the normal line is the negative reciprocal of the slope of the tangent line:

( m_{\text{normal}} = \frac{-1}{-12} = \frac{1}{12} )

Using the point-slope form of a line with the point ( (2, f(2)) ) and the slope ( \frac{1}{12} ), the equation of the normal line is:

( y - f(2) = \frac{1}{12}(x - 2) )

Substitute ( x = 2 ) into ( f(x) ) to find ( f(2) ):

( f(2) = (2 - 2)^{\frac{3}{2}} - 2^3 ) ( f(2) = 0 - 8 = -8 )

So the equation of the normal line is:

( y + 8 = \frac{1}{12}(x - 2) )

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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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