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

Answer 1

Equation of normal at #x=pi/3# is #(3sqrt3-8)y-sqrt3x-2sqrt3pi+8pi/3+3sqrt3=0#

If #f(x)=tan^2x-3x#,
#f(pi/3)=tan^2(pi/3)-3xxpi/3=(sqrt3)^2-pi=3-pi#.
Hence normal should pass through #(3-pi,pi/3)#
Slope of the curve at #x# is given by differential of #f(x)# i.e.
#f'(x)=2tanx xxsec^2x-3#.
Hence slope of tangent at #x=pi/3# is
#f'(pi/3)=2tan(pi/3)xxsec^2(pi/3)-3=2sqrt3xx(2/sqrt3)^2-3=8/sqrt3-3=(8-3sqrt3)/sqrt3# and as product of slopes of tangent and normal should be #-1#
Hence, slope of normal is #sqrt3/(3sqrt3-8)#

Hence, equation of normal using point slope form is

#(y-pi/3)=sqrt3/(3sqrt3-8)(x-3+pi)# or
#(3sqrt3-8)y-sqrt3x-sqrt3pi+8pi/3+3sqrt3-sqrt3pi=0#
or #(3sqrt3-8)y-sqrt3x-2sqrt3pi+8pi/3+3sqrt3=0#
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Answer 2

The equation of the line normal to f(x) = tan^2x - 3x at x = pi/3 is y = -3√3(x - pi/3) + 3√3.

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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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