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

Answer 1

Elijah Abram

#y-8e^2=20e^2(x-2)#

To find the equation of the tangent line, we need a point and a slope, then we can put it into point-slope form.

Point: #(2,f(2))# #f(2)=(2)^3e^(2)=8e^2# #(2,8e^2)#

Slope: #f'(2)# #f(x)=(x^3)(e^x)# #f'(x)=(x^3)(e^x)+(3x^2)(e^x)# #f'(x)=(x^2)(e^x)(x+3)#

#f'(2)=((2)^2)(e^2)(2+3)# #=(4)(e^2)(5)# #=20e^2#

Point-slope form:#y-y_1=m(x-x_1)# #y-8e^2=20e^2(x-2)#

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Answer 2

Emily Ammerman

The equation of the line tangent to f(x) = x^3e^x at x = 2 is y = 14e^2x - 4e^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.