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What is the equation of the normal line of #f(x)=(1-x)e^(3x)-e^x# at #x=1#?

Answer 1

Charles Anctil

graph{((1-x)e^(3x)-e^x-y)(y+e-1/(e(e^2+1))(x-1))=0 [-3.844, 4.926, -3.28, 1.103]}

#y+e=1/(e(e^2+1))(x-1)#

The general equation for normal is

#y-f(x_0)=-1/f^'(x_0)(x-x_0)#

#x_0 = 1#

#f(x_0)=f(1)=-e#

#f^'(x)=-e^x(3xe^(2x)+1-2e^(2x))#

#f^'(x_0)=f^'(1)=-e(3e^2+1-2e^2)=-e(e^2+1)#

#y+e=1/(e(e^2+1))(x-1)#

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Answer 2

Christopher Agresta

The equation of the normal line of f(x)=(1-x)e^(3x)-e^x at x=1 is y = -2x + 2.

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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.