What is the equation of the line normal to # f(x)=2/(x-1)^2-2x+4# at # x=-2#?

Answer 1

Equation of normal is #y = 27/50 x+ 2093/225#

#f(x) =2/(x-1)^2-2 x +4 ; x = -2#
#f(-2) =2/(-2-1)^2-2 *(-2) +4 = 2/9+4+4=74/9 #
The point is #(-2, 74/9)# at which normal to be drawn.
#f^'(x) =-4/(x-1)^3-2 # . Slope of tangent at #x=-2# is
#f^'(-2) =-4/(-2-1)^3-2 = 4/27-2= -50/27 #
Slope of normal at #x=-2# is #m= 27/50#
Equation of normal at point #(-2, 74/9)# is
#y - 74/9 = 27/50 (x+2)# or
#y = 27/50 x+ 27/25 +74/9# or
#y = 27/50 x+ 2093/225# [Ans]
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Answer 2

To find the equation of the line normal to the function f(x) at x=-2, we need to determine the slope of the tangent line at that point.

First, we find the derivative of f(x) using the power rule and chain rule:

f'(x) = -4(x-1)^(-3) - 2

Next, we substitute x=-2 into the derivative to find the slope of the tangent line at x=-2:

f'(-2) = -4(-2-1)^(-3) - 2

Simplifying this expression, we get:

f'(-2) = -4(-3)^(-3) - 2

Now, we can find the slope of the line normal to f(x) at x=-2 by taking the negative reciprocal of the slope of the tangent line:

m_normal = -1 / f'(-2)

Substituting the value of f'(-2) into the equation, we have:

m_normal = -1 / (-4(-3)^(-3) - 2)

Simplifying further, we get:

m_normal = -1 / (-4/27 - 2)

To simplify the denominator, we convert -2 into a fraction with a common denominator of 27:

m_normal = -1 / (-4/27 - 54/27)

Combining the fractions, we have:

m_normal = -1 / (-58/27)

To divide by a fraction, we multiply by its reciprocal:

m_normal = -1 * (27/-58)

Finally, we simplify the expression:

m_normal = 27/58

Therefore, the equation of the line normal to f(x) at x=-2 is y = (27/58)(x+2) + b, where b is the y-intercept of the 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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